Skip to main content

Bartleby Sitemap - Textbook Solutions

All Textbook Solutions for University Physics Volume 1

Check Your Understanding Restate 4.79105kg using a metric prefix such that the resulting number is bigger than one but less than 1000.Check Your Understanding Light navels about 9 Pm in a year. Given that a year is about 3107s , what is the speed of light in meters per second?Check Your Understanding We know horn Figure 1.4 that the diameter of Earth is on the order of 107m , so the order of magnitude of its surface area is 1014m2 . What is that in square kilometers (that is, km2)? (Try doing this both by converting 107m to km and then squaring it and then by converting 1014m2 directly to square kilometers. You should get the same answer both ways.)Check Your Understanding Given that 1 lb (pound) is 4.45 N. were the numbers being output by SM_FORCES too big or too small?Check Your Understanding Suppose we want the formula for the volume of a sphere. The two expressions commonly mentioned in elementary discussions of spheres are 4r2 and 4r3/3 . One is the volume of a sphere of radius r and the other is its surface area. Which one is the volume?Check Your Understanding Is the equation v=atdimensionally consistent? One further point thin needs to be mentioned is the effect of the operations of calculus on dimensions. We have seen that dimensions obey the rules of algebra, just like units, but what happens when we take the derivative of one physical quantity with respect to another or integrate a physical quantity over another? The derivative of a function is just the slope of the line tangent to its graph and slopes are ratios, so for physical quantities vand t , we hive that the dimension of the derivative of vwith respect to us just the ratio of the dimension of v over that of t : [dvdt]=[vt] . Similarly, since integrals are just sums of products, the dimension of the integral of vwith respect to t is simply the dimension of vtimes the dimension of t : [vdt]=[v][t] . By the same reasoning, analogous rules hold for the units of physical quantities derived from other quantities by integration or differentiation.Check Your Understanding Figure 1.4 says the mass of the atmosphere is 1019kg . Assuming the density of the atmosphere is 1kg/m3 , estimate the height of Earth’s atmosphere. Do you think your answer is an underestimate or an overestimate? Explain why. How many piano tuners are there in New York City? How many leaves are on that tree? If you are studying photosynthesis or thinking of writing a smartphone app tot piano tuners, then the answers to these questions might be of great interest to you. Otherwise, you probably couldn’t care less what the answers are. However, these are exactly the soils of estimation problems that people in various tech industries have been asking potential employees to evaluate their quantitative reasoning skills. If building physical intuition and evaluating quantitative claims do not seem like sufficient reasons tot you to practice estimation problems, how about the fact that being good at them just might land you a high-paying job? For practice estimating relative lengths, areas, and volumes, check out this PhET (https://openstaxcollege.org/l/21lengthgame) simulation, titled “Estimation.”Check Your Understanding A high school track coach has just purchased a new stopwatch. The stopwatch manual states the stopwatch has an uncertainty of ±0.05 s. Runners on the track coach’s learn regularly clock 100-rn sprints of 11.49 s to 15.01 s. At the school’s last track meet, the first-place sprinter came in at 12.04 s and the second-place sprinter came in a 12.07 s. Will the coach’s new stopwatch be helpful in timing the sprint team? Why or why not?The Scope and Scale of Physics What is physics?The Scope and Scale of Physics Some have described physics as a “search for simplicity.” Explain why this might be an appropriate description.The Scope and Scale of Physics If two different theories describe experimental observations equally well, can one be said to be more valid than the other (assuming both use accepted rules of logic)?The Scope and Scale of Physics What determines the validity of a theory?The Scope and Scale of Physics Certain criteria must be satisfied if a measurement or observation is to be believed. Will the criteria necessarily be as strict for an expected result as for an unexpected result?The Scope and Scale of Physics Can the validity of a model be limited or must it be universally valid? How does this compare with the required validity of a theory or a law?Units and Standards Identify some advantages of metric units.Units and Standards What are the SI base units of length, mass, and time?Units and Standards What is the difference between a base unit and a derived unit? (b) What is the difference between a base quantity and a derived quantity? (c) What is the difference between a base quantity and a base unit?For each of the following scenarios, refer to Figure 1.4 and Table 1.2 to determine which metric prefix on the meter is most appropriate for each of the following scenarios. (a) You want to tabulate the mean distance from the Sun for each planet in the solar system. (b) You want to compare the sizes of some common viruses to design a mechanical filter capable of blocking the pathogenic ones. (c) You want to list the diameters of all the elements on the periodic table. (d) You want to list the distances to all the stars that have now received any radio broadcasts sent from Earth 10 years ago. 1.2 Units and Standards Significant Figures (a) What is the relationship between the precision and the uncertainty of a measurement? (b) What is the relationship between the accuracy and the discrepancy of a measurement?Solving Problems in Physics What information do you need to choose which equation or equations to use to solve a problem?Solving Problems in Physics What should you do after obtaining a numerical answer when solving a problem?The Scope and Scale of Physics Find the order of magnitude of the following physical quantities. (a) The mass of Earth’s atmosphere: 5.11018kg : (b) The mass of the Moon’s atmosphere: 25,000kg ; (c) The mass of Earth’s hydrosphere: 1.41021kg : (d) The mass of Earth: 5.971024kg : (e) The mass of the Moon: 7.341022kg : (f) The Earth-Moon distance (semimajor axis): 3.84108m : (g) The mean Earth-Sun distance: 1.51011m : (h) The equatorial radius of Earth: 6.38106m : (i) The mass of an electron: 9.111031kg : (j) The mass of a proton: 1.671027kg : (k) The mass of the Sun: 1.991030kg .Use the orders of magnitude you found in the previous problem to answer the following questions to within an order of magnitude. (a) How many electrons would it take to equal the mass of a proton? (b) How many Earths would it take to equal the mass of the Sun? (c) How many Earth-Moon distances would it take to cover the distance from Earth to the Sun? (d) How many Moon atmospheres would it take to equal the mass of Earth’s atmosphere? (e) How many moons would it take to equal the mass of Earth? (f) How many protons would it take to equal the mass of the Sun? For the remaining questions, you need to use Figure 1.4 to obtain the necessary orders of magnitude of lengths, masses, and times.Roughly how many heartbeats are there in a lifetime?A generation is about one-third of a lifetime. Approximately how many generations have passed since the year 0 AD?Roughly how many times longer than the mean life of an extremely unstable atomic nucleus is the lifetime of a human?Calculate the approximate number of atoms in a bacterium. Assume the average mass of an atom in the bacterium is 10 times the mass of a proton.(a) Calculate the number of cells in a hummingbird assuming the mass of an average cell is 10 times the mass of a bacterium. (b) Making the same assumption, how many cells are there in a human?Assuming one nerve impulse must end before another can begin, what is the maximum firing rate of a nerve in impulses pet second?About how many floating-point operations can a supercomputer perform each year?Roughly how many floating-point operations can a supercomputer perform in a human lifetime?The following times are given using metric prefixes on the base SI unit of time: the second. Rewrite them in scientific notation without the prefix. For example, 47Ts would be rewritten as 4.71013s . (a) 980Ps ; (b) 980fs ; (c) 17ns ; (d) 577s .The following times are given in seconds. Use metric prefixes to rewrite them so the numerical value is greater than one but less than 1000. For example, 7.9102s could be written as either 7.9cs or 7.9ms . (a) 9.57105s : (b 0.045s ; (c) 5.5107s ; (d) 3.16107s .The following lengths are given using metric prefixes on the base SI unit of length: the meter. Rewrite them in scientific notation without the prefix. For example, 4.2Pm would be rewritten as 4.21015m (a) 89Tm ; (b) 89pm ; (c) 711mm ; (d) 0.45m .The following lengths are given in meters. Use metric prefixes to rewrite them so the numerical value is bigger than one but less than 1000. For example, 7.9102m could be written either as 7.9cm or 79mm . (a) 7.9107m : (b) 0.0074m ; (c) 8.81011m : (d) 1.631013m .The following masses are written using metric prefixes on the gram. Rewrite them in scientific notation in terms of the SI base unit of mass: the kilogram. For example, 40Mg would be written as 4104kg . (a) 23mg ; (b) 320Tg ; (c) 42ng ; (d) 7g ; (e) 9Pg .The following masses are given in kilograms. Use metric prefixes on the gram to rewrite them so the numerical value is bigger than one but less than 1000. For example, 7104kg could be written as 70cg or 700mg . (a) 3.8105kg ; (b) 2.31017kg ; (c) 2.41011kg ; (d) 81015kg ; (e) 4.2103kg .The volume of Earth is on the order of 1021m3 . (a) What is this in cubic kilometers (km3) ? (b) What is it in cubic miles (mi3) ? (c) What is it in cubic centimeters (cm3) ?The speed limit on some interstate highways is roughly 100 km/h. (a) What is this in meters per second? (b) How many miles per hour is this?A car is traveling at a speed of 33 m/s. (a) What is its speed in kilometers per hour? (b) Is it exceeding the 90 km/h speed limit?In SI units, speeds are measured in meters per second (m/s). But, depending on where you live, you’re probably mole comfortable of thinking of speeds in terms of either kilometers per hour (km/h) or miles per hour (mi/h). In this problem, you will see that 1 m/s is roughly 4 km/h or 2 mi/h, which is handy to use when developing your physical Intuition. More precisely, show that (a) 1.0m/s=3.6km/h and 1.0m/s=2.2mi/h .American football is played on a 100-yd-long field, excluding the end zones. How long is the field in meters? (Assume that 1m=3.281ft .)Soccer fields vary in size. A large soccer field is 115 m long and 85.0 m wide. What is its area in square feet? (Assume that 1m=3.281ft .)What is the height in meters of a person who is 6ft 1.0 in. tall?Mount Everest, at 29,028ft , is the tallest mountain on Earth. What is its height in kilometers? (Assume that 1m=3.281ft .)The speed of sound is measured to be 342 m/s on a certain day. What is this measurement in kilometers per hour?Tectonic plates are large segments of Earth’s mist that move slowly. Suppose one such plate has an average speed of 4.0 cm/yr. (a) What distance does it move in 1.0 s at this speed? (b) What is its speed in kilometers pet million years?The average distance between Earth and the Sun is 1.51011m . (a) Calculate the average speed of Earth in its orbit (assumed to be circular) in meters per second. (b) What is this speed in miles per hour?The density of nuclear matter is about 1018kg/m3 . Given that 1 mL is equal in volume to cm3 , what is the density of nuclear matter in megagrams per microliter (that is, Mg/L )?The density of aluminum Is 2.7g/cm3 . What is the density in kilograms per cubic meter?A commonly used turn of mass in the English system Is the pound-mass, abbreviated Ibm, where 1 Ibm = 0.454 kg. What is the density of water in pound-mass per cubic foot?A furlong is 220 yd. A fortnight is 2 weeks. Convert a speed of one furlong per fortnight to millimeters per second.It takes 2 radians (rad) to get around a circle, which Is the same as 360 . How many radians are in 1 ?Light travels a distance of about 3108m/s . A light-minute is the distance light travels in 1 mm. If the Sun is 1.51011m from Earth, how far away Is it in light- minutes?A light-nanosecond is the distance light travels in 1 ns. Convert 1 ft to light-nanoseconds.An electron has a mass of 9.1110-31kg . A proton has a mass of 1.6710-27kg . What is the mass of a proton in electron-masses?A fluid ounce is about 30mL. What is the voIume of a 12 fl-oz can of soda pop in cubic meters?A student is trying to remember some formulas from geometry. In what follows, assume Ais area, Vis volume, and all other variables are lengths. Determine which formulas are dimensionally consistent. (a) V=r2h; (b) A=2r2+2rh; (c) V=0.5bh(d) V=d2; (e) V=d3/6.Consider the physical quantities s,v,a, and t with dimensions [s]=L,[v]=LT1,[a]=LT2 and [t]=L . Determine whether each of the following equations is dimensionally consistent. (a)v2=2as;(b)s=vt2+0.5at2;(c)v=s/t;(d)a=v/t Consider the physical quantities m,s,v,a, and t with dimensions [m]=M,[s]=L,[v]=LT1 and [a]=LT2 . Assuming each of the following equations is dimensionally consistent, find the dimension of the quantity on the left-hand side of the equation: (a)F=ma;(b)K=0.5mv2;(c)p=mv;(d)W=mas;(e)L=mvr Suppose quantity s is a length and quantity t is a time. Suppose the quantities vand a are defined by v=ds/dt and a=dv/dt . (a) What is the dimension of v ? (b) What is the dimension of the quantity a ? What are the dimensions of (c) vdt , (d) adt, and (e) da/dt?Suppose [V]=L3,[]=ML3, and [t]=T . (a) What is the dimension of dV? (b) What is the dimension of dV/dt? (c) What is the dimension of (dV/dt)?The arc length formula says the length sof arc subtended by angle in a circle of radius r is given by the equation s=r . What are the dimensions of (a) s, (b) r , and (c) ?Estimates and Fermi Calculations Assuming the human body is made primarily of water, estimate the volume of a person.Assuming the human body is primarily made of water, estimate the number of molecules in it. (Note that water has a molecular mass of 18 g/mol and there are roughly 1024 atoms in a mole)Estimate the mass of air in a classroom.Estimate the number of molecules that make up Earth, assuming an average molecular mass of 30 g/mol. (Note there are on the order of 1024 objects per mole.)Estimate the surface area of a person.Roughly how many solar systems would it take to tile the disk of the Milky Way?(a) Estimate the density of the Moon. (b) Estimate the diameter of the Moon. (c) Given that the Moon subtends at an angle of about half a degree in the sky, estimate its distance from Earth.The average density of the Sun is on the order 103kg/m3 . (a) Estimate the diameter of the Sun. (b) Given that the Sun subtends at an angle of about half a degree in the sky, estimate its distance from Earth.Estimate the mass of a virus.A floating-point operation is a single arithmetic operation such as addition, subtraction, multiplication, division. (a) Estimate the maximum number of floating- point operations a human being could possibly perform in a lifetime. (b) How long would it take a supercomputer to perform that many floating-point operations?Consider the equation 4000/400=10.0 . Assuming the number of significant figures in the answer is correct, what can you say about the number of significant figures In 4000 and 400?Suppose your bathroom scale reads your mass as 65 kg with a 3% uncertainty. What is the uncertainty in your mass (In kilograms)?A good-quality measuring tape can be off by 0.50cm over a distance of 20m . What is its percent uncertainty?An infant’s pulse rate is measured to be 1305 beats/mm. What is the percent uncertainty in this measurement?(a) Suppose that a person has an average heart rate of 72.0 beats/mm. How many beats does he or she have in 2.0 years? (b) In 2.00 years? (c) In 2.000 years?A can contains 375 mL of soda. How much is left after 308 mL is removed?State how many significant figures are proper In the results of the following calculations: (a) (106.7)(98.2)/(46.210)(1.01) ; (b) (18.7)2 ; (c) (1.601019)(3712)(a) How many significant figures are in the numbers 99 and 100.? (b) If the uncertainty in each number is 1, what is the percent uncertainty in each? (c) Which is a more meaningful way to express the accuracy of these two numbers: significant figures or percent uncertainties?(a) If your speedometer has an uncertainty of 2.0 km/h at a speed of 90 km/h, what is the percent uncertainty? (b) If it has the same percent uncertainty when it reads 60 km/h, what Is the range of speeds you could be going?(a) A person’s blood pressure is measured to be 1202mmHg . What is its percent uncertainty? (b) Assuming the same percent uncertainty, what is the uncertainty in a blood pressure measurement of 80 mm He?A person measures his or her heart rate by counting the number of beats in 30s. If 401 beats are counted in 30.00.5s , what is the heart rate and its uncertainty in beats per minute?What is the area of a circle 3.102 cm in diameter?Determine the number of significant figures in the following measurements: (a) 0.0009 , (b) 15,450.0 , (c) 6103 , (d) 87.990 , and (e) 30.42 .Perform the following calculations and express your answer using the correct number of significant digits. (a) A man has two bags weighing 13.5 lb and one bag with a weight of 10.2 lb. What is the total weight of the bags? (b) The force Fon an object is equal to its mass mmultiplied by its acceleration a . If a wagon with mass 55 kg accelerates at a rate of 0.0255m/s2 , what is the force on the wagon? (The unit of force is called the newton and it is expressed with the symbol N.)Consider the equation y=mt+b, where the dimension of y is length and the dimension of t is time, and mand bare constants. What are the dimensions and SI units of (a) mand (b) b ?Consider the equation s=s0+v0t+a0t2/2+j0t3/6+s0t4/24+ct5/120 , were s is a length and t is a time. What are the dimensions and SI units of (a) s0 , (b) v0 , (c) a0 , (d) j0 , (e) s0, and (f) c ?(a) A car speedometer has a 5% uncertainty. What is the range of possible speeds when it reads 90 km/h? (b) Convert this range to miles per how. Note 1 km = 0.6214 mi.A marathon runner completes a 42.188-km course in 2 h, 30 min, and 12s. There is an uncertainty of 25 m in the distance traveled and an uncertainty of 1 s in the elapsed time. (a) Calculate the percent uncertainty in the distance. (b) Calculate the percent uncertainty in the elapsed time. (C) What is the average speed in meters per second? (d) What is the uncertainty in the average speed?The sides of a small rectangular box are measured to be 1.800.1cm,2.050.02cm, and 3.10.1cm long. Calculate its volume and uncertainty in cubic centimeters.When nonmetric units we used in the United Kingdom, a unit of mass called the pound-mass (lbm) was used, where 1 lbm = 0.4539 kg. (a) If there is an uncertainty of 0.0001 kg in the pound-mass unit, what is its percent uncertainty? (b) Based on that percent uncertainty, what mass in pound-mass has an uncertainty of 1 kg when converted to kilograms?The length and width of a rectangular room are measured to be 3.9550.005m and 10500.005m . Calculate the area of the room and its uncertainty in square mete A car engine moves a piston with a circular cross-section of 73000.002cm in diameter a distance of 3.2500.001cm to compress the gas in the cylinder. (a) By what amount is the gas decreased in volume in cubic centimeters? (b) Find the uncertainty in this volume.The first atomic bomb was detonated on July 16, 1945, at the Trinity test site about 200 mi south of Los Alamos. In 1947, the U.S. government declassified a film reel of the explosion. From this film reel, British physicist G.I. Taylor was able to determine the rate at which the radius of the fireball from the blast grew. Using dimensional analysis, he was then able to deduce the amount of energy released in the explosion, which was a closely guarded secret at the time. Because of this, Taylor did not publish his results until 1950. This problem challenges you to recreate this famous calculation. (a) Using keen physical insight developed from years of experience, Taylor decided the radius rof the fireball should depend only on time since the explosion, t, the density of the air, , and the energy of the initial explosion, E. Thus, he made the educated guess that r=kEabtcfor some dimensionless constant kand some unknown exponents a,b, and c. Given that [E]=ML2T-2 , determine the values of the exponents necessary to make this equation dimensionally consistent. (Hint: Notice the equation implies that k=rEabtcand that [k]=1 ). (b) By analyzing data from high-energy conventional explosives, Taylor found the formula he derived seemed to be valid as long as the constant khad the value 1.03. From the film reel, he was able to determine many values of rand the corresponding values of t. For example, he found that after 25.0 ms, the fireball had a radius of 130.0 m. Use these values, along with an average air density of 1.25kg/m3 , to calculate the initial energy release of the Trinity detonation in joules (J). (Hint: To get energy in joules, you need to make sure all the numbers you substitute in are expressed in terms of SI base units.) (c) The energy released in large explosions is often cited in units of “tons of TNT” (abbreviated “t TNT”), where 1 t TNT is about 4.2 GJ. Convert yow answer to (b) into kilotons of TNT (that is, kt TNT). Compare your answer with the quick-and-duty estimate of 10 kt TNT made by physicist Enrico Fermi shortly after witnessing the explosion from what was thought to be a safe distance. (Reportedly, Fermi made his estimate by dropping some shredded bits of paper right before the remnants of the shock wave hit him and looked to see how far they were carried by it.)The purpose of this problem is to show the entire concept of dimensional consistency can be summarized but the old saying “You can’t add apples and oranges.” It you have studied power series expansions in a calculus course, you know the standard mathematical funstions such as trigonometric functions, logarithms, and exponential function can be expressed as infinite sums of the form where the an are dimensionless constants for all n = 0, 1, 2, … and x is the argument of the function. (If you have not studied power series in calculus yet, just trust us.) Use this fact to explain why the requirement that all terms in an equation have the same dimensions is sufficient as a definition of dimensional consistency. That is, it actually implies the arguments of standard mathematical funstions must be dimensional consistency. That is, it actually implies the arguments of standard mathematical functions must be dimensionless, so it is not really necessary to make this latter condition a separate requirement of the definition of dimensional consistency as we have done in this section.Check Your Understanding Two motorboats named Alice and Bob are moving on a lake. Given the information about their velocity vectors in each of the following situations, Indicate whether their velocity vectors are equal or otherwise. (a) Alice moves north at 6 knots and Bob moves west at 6 knots. (b) Alice moves west at 6 knots and Bob moves west a 3 knots. (C) Alice moves northeast at 6 knots and Bob moves south at 3 knots. (d) Alice moves northeast at 6 knots and Bob moves southwest a 6 knots. (e) Alice moves northeast at 2 knots and Bob moves closer to the shore northeast at 2 knots.Check Your Understanding A cave diver enters a tong underwater tunnel. When her displacement with respect to the entry point is 20 m, she accidentally drops her camera, but she doesn’t notice it missing until she is some 6 m farther into the tunnel. She swims back 10 m but cannot find the camera, so she decides to end the dive. How far from the entry point Is she? Taking the positive direction out of the tunnel, what is her displacement vector relative to the entry point?Check Your understanding Using the three displacement vectors A , B , and F in Figure 2.13, choose a convenient scale, and use a ruler and a protractor to find vector G given by the vector equation G=A+2BF . The three displacement vectors A , B , and C in Figure are specified by their magnitudes A=10.0,B=7.0 and C=8.0 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =30 . The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) R=A+B , (b) D=AB , and (c) S=A3B+C .Check your Understanding A blue fly lands on a sheet of graph paper a point located 10.0 cm to the right of its left edge and 8.0 cm above its bottom edge and walks slowly to a point located 5.0 cm from the left edge and 5.0 cm from the bottom edge. Choose the rectangular coordinate system with the origin at the lower left-side comet of the paper and find the displacement vector of the fly. Illustrate your solution by graphing.Check Your Understanding If the displacement vector of a blue fly walking on a sheet of graph paper is D=(5.00 i3.00 j)cm , find Its magnitude and direction.Check Your Understanding If Trooper runs 20 m west before taking a rest, what is his displacement vector?Check Your Understanding If the average velocity vector of the drone In the displacement in Example 2.7 is u=(15.0i+31.7j +2.5k)m/s , what is the magnitude of the drone’s velocity vector?Check Your understanding Three displacement vectors A , B , and F in (Figure 2.13) are specified by their magnitudes A=10.0,B=7.0 and F=20.00 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =110 . The physical units of the magnitudes are centimeters. Use the analytical method to find vector G=A+2BF . Verify that G=28.15cm and that G=68.65 . The three displacement vectors A , B , and C in Figure are specified by their magnitudes A=10.0,B=7.0 and C=8.0 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =30 . The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) R=A+B , (b) D=AB , and (c) S=A3B+C .Check Your Understanding Suppose that Balto in Example 2.10 leaves the game to attend to more important matter, but Astro Clifford and Dug continue playing. Astro and Clifford’s pull on the toy does not change, but Dug rims around and bites on the toy in a different place. With how big a force and in what direction must Dug pull on the toy now to balance out the combined pulls from Clifford and Astro? Illustrate this situation by drawing a vector diagram indicating all forces involved.Check Your Understanding Verify that vector v V obtained in Example 2.14 is indeed a unit vector by computing its magnitude. If the convoy in Example 2.8 was moving across a desert flatland—that is, it the third component of its velocity was zero—what is the unit vector of its direction of motion? Which geographic direction does it represent?Check Your Understanding For the vectors given in Figure 2.13, find the scalar products AB and FC . The three displacement vectors A , B , and C in Figure 2.13 are specified by their magnitudes A=10.0,B=7.0 and C=8.0 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =30 . The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) R=A+B , (b) D=AB , and (c) S=A3B+C .Check Your Understanding For vector in a rectangular coordinate system, use Equation 2.29 through Equation 2.32 to show that and .Check Your Understanding Find the angle between forces F1and F3in Example 2.16.Check Your Understanding How much work is done by the first dog and by the second dog In Example 2.16 on the displacement in Example 2.17?Check Your Understanding For the vectors given in Figure 2.13, find the scalar products AB and CF . The three displacement vectors A , B , and C in Figure 2.13 are specified by their magnitudes A=10.0,B=7.0 and C=8.0 , respectively, and by their respective direction angles with the horizontal direction =35,=110 , and =30 . The physical units of the magnitudes are centimeters. Choose a convenient scale and use a ruler and a protractor to find the following vector sums: (a) R=A+B , (b) D=AB , and (c) S=A3B+C .Check Your Understanding Given two vectors and , find (a) (b) (c) the angle between and , and (d) the angle between and vector .Scalars and Vectors A weather forecast states the temperature is predicted to be 5Cthe following day. Is this temperature a vector or a scalar quantity? Explain.Which of the following is a vector a person’s height, the altitude on Mt. Everest, the velocity of a fly, the age of Earth, the boiling point of water, the cost of a book, Earth’s population, or the acceleration of gravity?Give a specific example of a vector, stating its magnitude, units, and direction.What do vectors and scalars have in common? How do they differ?Suppose you add two vectors A and B . What relative direction between them produces the resultant with the greatest magnitude? What is the maximum magnitude? What relative direction between them produces the resultant with the smallest magnitude? What is the minimum magnitude?Is it possible to add a scalar quantity to a vector quantity?Is it possible for two vectors of different magnitudes to add to zero? Is it possible for three vectors of different magnitudes to add to zero? Explain.Does the odometer in an automobile indicate a scalar or a vector quantity?When a 10,000-rn runner competing on a 400-rn track crosses the finish line, what is the runner’s net displacement? Can this displacement be zero? Explain.A vector has zero magnitude. Is it necessary to specify its direction? Explain.Can a magnitude of a vector be negative?Can the magnitude of a particle’s displacement be greater that the distance traveled?If two vectors are equal, what can you say about their components? What can you say about their magnitudes? What can you say about their directions?If three vectors sum up to zero, si1at geometric condition do they satisfy?Give an example of a nonzero vector that has a component of zero.Explain why a vector cannot have a component greater than its own magnitude.If two vectors are equal, what can you say about their components?If vectors A and B are orthogonal, what is the component of B along the direction of A ? What Is the component of A along the direction of B ?If one of the two components of a vector is not zero, can the magnitude of the other vector component of this vector be zero?If two vectors have the same magnitude, do their components have to be the same?What is wrong with the following expressions? How can you correct them? (a) C=AB , (b) C=AB , (c) C=AB , (d) C=AB , (e) C+2A=B , (f) C=AB , (g) AB=AB , (h) C=2AB , (i) C=A/B , and (j) C=A/B .If the cross product of two vectors vanishes, what can you say about their directions?If the dot product of two vectors vanishes, what can you say about their directions?What is the dot product of a vector with the cross product that this vector has with another vector?A scuba diver makes a slow descent into the depths of the ocean. His vertical position with respect to a boat on the surface changes several times. He makes the first stop 9.0 m from the boat but has a problem with equalizing the pressure, so he ascends 3.0 m and then continues descending for another 12.0 m to the second stop. From there, he ascends 4 m and then descends for 18.0 m, ascends again for 7 m and descends again for 24.0 m, where he makes a stop, waiting for his buddy. Assuming the positive direction up to the surface, express his net vertical displacement vector in terms of the unit vector. What is his distance to the boat?In a tug-of-war game on one campus, 15 students pull on a rope at both ends in an effort to displace the central knot to one side or the other. Two students pull with force 196 N each to the light, four students pull with force 98 N each to the left, five students pull with force 62 N each to the left, three students pull with force 150 N each to the right, and one student pulls with force 250 N to the left. Assuming the positive direction to the tight, express the net pull on the knot in terms of the unit vector. How big is the net pull on the knot? In what direction?Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point and what is the compass direction of a line connecting your starting point to your final position? Use a graphical method.For the vectors given in the following figure, use a graphical method to find the following resultants: (a) A+B , (b) C+B , (c) D+F , (d) AB , (e) DF , (f) A+2F , (g) C+2D ; and (h) A4D+2F .A delivery man starts at the post office, chives 40 km north, then 20 km west, then 60 km northeast, and finally 50 km north to stop for lunch. Use a graphical method to find his net displacement vector.An adventurous dog strays from home, runs three blocks east, two blocks north, one block east, two block north, and two blocks west. Assuming that each block is about 100 m, how far from home and in what direction is the dog? Use a graphical method.In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The wind shifts a great deal during the day and he is blown along the following directions: 2.50 km and 45.0 north of west, then 4.70 km and 60.0 south of east, then 1.30 km and 25.0 south of west, then 5.10 km straight east, then 1.70 km and 5.00 east of north, then 7.20 km and 55.0 south of west, and finally 2.80 km and 10.0 north of east. Use a graphical method to find the castaway’s final position relative to the island.A small plane flies 40.0 km in a direction 60 north of east and then files 30.0 km in a direction 15 north of east. Use a graphical method to find the total distance the plane covers from the starting point and the direction of the path to the final position.A trapper walks a 5.0-km straigt4ine distance from his cabin to the lake, as shown in the following figure. Use a graphical method (the parallelogram rule) to determine the trapper’s displacement directly to the east and displacement directly to the north that sum up to his resultant displacement vector. If the trapper walked only in directions east and north, zigzagging his way to the lake, how many kilometers would he have to walk to get to the lake?A surveyor measures the distance across a river that flows straight north by the following method. Starting directly across from a tree on the opposite bank, the surveyor walks 100 m along the river to establish a baseline. She then sights across to the tree and reads that the angle from the baseline to the tree is 35 . How wide is the river?A pedestrian walks 6.0 km east and then 13.0 km north. Use a graphical method to find the pedestrian’s resultant displacement and geographic direction.The magnitudes of two displacement vectors are A=20mand B=6m . What are the largest and the smallest values of the magnitude of the resultant.Assuming the +x -axis is horizontal and points to the tight, resolve the vectors given In the following figure to their scalar components and express them in vector component form.Suppose you walk 18.0 m straight west and then 25.0 m straight north. How far are you from your starting point? What is your displacement vector? What is the direction of your displacement? Assume the +x -axis is horizontal to the right.You drive 7.50 km in a straight line in a direction 15 east of north. (a) Find the distance you would have to drive straight east and then straight north to arrive at the same point. (b) Show that you still arrive at the some point if the east and north legs are reversed in order. Assume the +x -axis is to the east.A sledge is being pulled by two horses on a flat terrain. The net force on the sledge can be expressed in the Cartesian coordinate system as vector , where and denote directions to the east and north, respectively. Find the magnitude and direction of the pull.A trapper walks a 5.0-lan straight-line distance from her cabin to the lake, as shown in the following figure. Determine the east and north components of her displacement vector. How many more kilometers would she have to walk if she walked along the component displacements? What is her displacement vector?The polar coordinates of a point are 4/3and 5.50 m. What are its Cartesian coordinates?Two points in a plane have polar coordinates P1(2.500m,/6) and P2(3.800m,2/3). Determine their Cartesian coordinates and the distance between them in the Cartesian coordinate system. Round the distance to a nearest centimeter.A chameleon is resting quietly on a lanai screen, waiting for an insect to come by. Assume the origin of a Cartesian coordinate system at the lower left-hand corner of the screen and the horizontal direction to the right as the +x -direction. If its coordinates are (2.000 m, 1.000 m), (a) how far is it from the corner of the screen? (b) What is its location in polar coordinates?Two points in the Cartesian plane are A(2.00m,4.00m) and B(-3.00m,3.00m) . Find the distance between them and their polar coordinates.A fly enters through an open window and zooms around the room. In a Cartesian coordinate system with three axes along three edges of the room, the fly changes its position from point b(4.0m,1.5m,2.5m) to point e(1.0m,4.5in,0.5m) . Find the scalar components of the fly’s displacement vector and express its displacement vector in vector component form. What is its magnitude?For vectors and , calculate (a) and its magnitude and direction angle, and (b) and its magnitude and direction angle.A particle undergoes three consecutive displacements given by vectors , and . (a) Find the resultant displacement vector of the particle. (b) What is the magnitude of the resultant displacement? (c) If all displacements were along one line, how far would the particle travel?Given two displacement vectors A=(3.00i-4.00j-4.00K)m and B=(2.00i+3.00j-7.00K)m , find the displacements and their magnitudes for (a) C=A+B and (b) D=2AB .A small plane flies 40.0 km ma direction 60 north of east and then flies 30.0 km in a direction 15 north of east. Use the analytical method to find the total distance the plane covers from the starting point, and the geographic direction of its displacement vector. What is its displacement vector?In an attempt to escape a desert island, a castaway builds a raft and sets out to sea. The wind shifts a great deal during the day, and she is blown along the following straight lines: 2.50 km and 45.0 north of west, then 4.70 km and 60.0 south of east, then 130 km and 25.0 of west, then 5.10 km due east, then 1.70 km and 5.00 east of north, then 7.20 km and 55.0 south of west, and finally 2.80 km and 10.0 north of east. Use the analytical method to find the resultant vector of all her displacement vectors. What is its magnitude and direction?Assuming the +x -axis is horizontal to the right for the vectors given in the following figure, use the analytical method to find the following resultants: (a) A+B , (b) C+B , (c) D+F , (d) AB , (e) DF , (f) A2F , (g) C2D+3F , and (h) A4D+2F .Given the vectors in the preceding figure, find vector Rthat solves equations (a) D+R=Fand (b) D-2D+5R=3F . Assume the +x -axis is horizontal to the right.A delivery man starts at the post office, drives 40 km north, then 20 km west, then 60 km northeast, and finally 50 km north to stop for lunch. Use the analytical method to determine the following: (a) Find his net displacement vector. (b) How far is the restaurant from the post office? (c) If he returns directly from the restaurant to the post office, what is his displacement vector on the return trip? (d) What Is his compass heading on the return trip? Assume the +x-axis is to the east.An adventurous dog strays from home, runs three blocks east, two blocks north, and one block east, one block north, and two blocks west. Assuming that each block Is about a 100 yd, use the analytical method to find the dog’s net displacement vector, its magnitude, and Its direction. Assume the +x-axis is to the east. How would your answer be affected if each block was about 100 m?If , and , find the unknown constants a and b such that .Given the displacement vector , find the displacement vector so that .Find the unit vector of direction for the following vector quantities: (a) Force , displacement , and (c) velocity .At one point in space, the direction of the electric field vector Is given In the Cartesian system by the unit vector . If the magnitude of the electric field vector is E=400.0V/m , what are the scalar components , and of the electric field vector at this point? What is the direction angle of the electric field vector at this point?A barge is pulled by the two tugboats shown in the following figure. One tugboat pulls on the barge with a force of magnitude 4000 units of force at 15 above the line AB (see the figure and the other tugboat pulls on the barge with a force of magnitude 5000 units of force at 12 below the line AB. Resolve the pulling forces to their scalar components and find the components of the resultant force pulling on the barge. What is the magnitude of the resultant pull? What is its direction relative to the line AB?In the control tower at a regional airport, an air traffic controller monitors two aircraft as their positions change with respect to the control tower. One plane is a cargo carrier Boeing 747 and the other plane is a Douglas DC-3. The Boeing is at an altitude of 2500 m, climbing at 100 above the horizontal, and moving 30 north of west. The DC-3 is at an altitude of 3000 m, climbing at 50 above the horizontal, and cruising directly west. (a) Find the position vectors of the planes relative to the control tower. (b) What is the distance between the planes at the moment the air traffic controller makes a note about their positions?Assuming the axis is horizontal to the right for the vectors in the following figure, find the following scalar products (a) (b) (c) (d) (e) (g) and (h)Assuming the +x-axis is horizontal to the right for the vectors in the preceding figure, find (a) the component of vector Aalong vector C , (b) the component of vector Calong vector A , (c) the component of vector i along vector F , and (d) the component of vector Falong vector i .Find the angle between vectors for (a) D=(-3.0i-4.0j)m and A=(-3.0i+4.0j)m and (b) D=(2.0i+4.0j+K)m and B=(-2.0i+3.0j+2.0K)m .Find the angles that vector makes with the and axes.Show that the force vector is orthogonal to the force vector Assuming the +x-axis is horizontal to the right for the vectors in the previous figure, find the following vector products: AC , (b) AF , (c) DC , (d) A(F+2C) , (e) iB , (f) jB , (g)Find the cross product for (a) and (b) and (c) and and (d) and For the vectors in the earlier figure, find (a) (AF)D , (b) (AF)(DB) , and (c) (AF)(DB) .(a) If AF=BF , can we conclude A=B ? (b) If AF=BF , can we conclude A=B ? (c) If FA=BF , can we conclude A=B ? Why or why not?You fly 32.0 km in a straight line in still air in the direction 35.0 south of west. (a) Find the distances you would have to fly due south and then due west to arrive at the same point. (b) Find the distances you would have to fly first in a direction 45.0south of west and then in a direction 45.0 west of north. Note these are the components of the displacement along a different set of axes-namely, the one rotated by 45 with respect to the axes in (a).Rectangular coordinates of a point are given by (2,y) and its polar coordinates are given by (r,/6) . Find y y and r .If the polar coordinates of a point are (r,)and its rectangular coordinates are (x,y) , determine the polar coordinates of the following points: (a) (x,y) , (b) (2x,2y) , and (C) (3x,3y) .74AP Starting at the island of Moi in an unknown archipelago, a fishing boat makes a round trip with two stops at the islands of Noi and Poi. It sails from Moi for 4.76 nautical miles (nmi) in a direction 37 north of east to Noi. From Noi, it sails 69 west of north to Poi. On its return leg from Poi, it sails 28 east of south. What distance does the boat sail between Noi and Poi? What distance does it sail between Moi and Poi? Express your answer both in nautical miles and in kilometei. Note: 1 nmi = 1852m.An air traffic controller notices two signals from two planes on the radar monitor. One plane is at altitude 800 m and in a 19.2-km horizontal distance to the tower in a direction 25 south of west. The second plane is at altitude 1100 m and its horizontal distance is 17.6 km and 20 south of west. What is the distance between these planes?Show that when A+B=C then A2+B2+2ABcos , where is the angle between vectors A and B .Four force vectors each have the same magnitude f.What is the largest magnitude the resultant force vector may have when these forces are added? What is the smallest magnitude of the resultant? Make a graph of both situations.A skater glides along a circular path of radius 5.00 m in clockwise direction. ‘When he coasts around one- half of the circle, starting from the west point, find (a) the magnitude of his displacement vector and (b) how far he actually skated. (c) What is the magnitude of his displacement vector when he skates all the way around the circle and comes back to the west point?A stubborn dog is being walked on a leash by its owner. At one point, the dog encounters an interesting scent at some spot on the ground and wants to explore It in detail, but the owner gets impatient and pulls on the leash with force along the leash. (a) What is the magnitude of the pulling force? (b) What angle does the leash make with the vertical?If the velocity vector of a polar bear is u=(-18.0 i -13.0 j)km/h , how fast and in what geographic direction Is it heading? Here, and are directions to geographic east and north, respectively.Find the scalar components of three-dimensional vectors G and H in the following figure and write the vectors in vector component form in terms of the unit vectors of the axes.A diver explores a shallow reef off the coast of Belize. She initially swims 90.0 m north, makes a turn to the east and continues for 200.0 m, then follows a big grouper for 80.0 m in the direction 30 north of east. In the meantime, a local current displaces her by 150.0 m south. Assuming the current is no longer present, in what direction and how far should she now swim to come back to the point where she started?A force vector A has x and y components, respectively, of -8.80 units of force and 15.00 units of force. The x and y components of force vector B are, respectively, 13.20 units of force and -6.60 units of force. Find the components of force vector C that satisfies the vector equation AB+3C=0 .Vectors A and B are two orthogonal vectors in the xy -plane and they have identical magnitudes. If A=3.0 i+4.0 j , find B .For the three-dimensional vectors in the following figure, find (a) GH , (b) GH , and (c) GH .Show that (BC)A is the volume of the parallelepiped, with edges formed by the three vectors in the following figure.Vector B is 5.0 cm long and vector A is 4.0 cm long. Find the angle between these two vectors when |A+B|=3.0cm and |AB|=3.0cm .What is the component of the force vector G=(3.0 i+4.0 j+10.0 k)N along the force vector H=(1.0 i+4.0 j)N ?The following figure shows a triangle formed by the three vectors A , B , and C . If vector C is drawn between the midpoints of vectors A and B , show that C=C/2 between points in a plane do not change when a coordinate system is rotated In other words, the magnitude of a vector Is invariant under rotations of the coordinate system. Suppose a coordinate system S is rotated about its origin by angle to become a new coordinate system S’, as shown in the following figure. A point in a plane has coordinates (x,y)in S and coordinates (x,y) in S’. (a) Show that, during the transformation of rotation, the coordinates in S are expressed in terms of the coordinates in S by the following relations: {x=xcos+ysiny=xsin+ycos (b) Show that the distance of point Pto the origin is invariant under rotations of the coordinate system. Here, you have to show that x2+y2=x2+y2. (c) Show that the distance between points Pand Q is invariant under rotations of the coordinate system. Here, you have to show that ( xP =xQ )2+( yP yQ )2=( xP =xQ )2+( yP yQ )2 .Check your Understanding A cyclist rides 3 km west and then tune around and tides 2 kin east. (a) What Is his displacement? (b) What is the distance traveled? (c) What is the magnitude of his displacement?Check your Understanding The position of an object as a function of time is x(t)=3t2m . (a) What is the velocity of the object as a function of time? (b) Is the velocity ever positive? (c) What are the velocity and speed at t=1.0s ?Check Your Understanding Protons in a linear accelerator are accelerated from rest to 2.0107m/sin10-4s . What is the average acceleration of the protons?Check Your Understanding An airplane lands on a runway traveling east. Describe its acceleration.Check Your Understanding A manned rocket accelerates at a rate of 20m/s2 during launch. How long does it take the rocket to teach a velocity of 400 m/s?Check Your Understanding A bicycle has a constant velocity of 10 m/s. A person starts from rest and runs to catch up to the bicycle in 30 s. What is the acceleration of the person?Check Your Understanding A chunk of ice beaks off a glacier and falls 30.0 m before it hits the water. Assuming it falls freely (there is no air resistance), how bag does it take to hit the water? Which quantity increases faster, the speed of the Ice chunk or its distance traveled?Check Your Understanding A particle starts from rest and has an acceleration function 510tm/s2 . (a) What is the velocity function? (b) What is the position function? (c) When is the velocity zero?Position, Displacement, and Average Velocity Give an example in which there are clear distinctions among distance traveled, displacement, and magnitude of displacement. Identify each quantity in your example specifically.Under what circumstances does distance traveled equal magnitude of displacement? What is the only case in which magnitude of displacement and displacement are exactly the same?Bacteria move back and forth using their flagella (structures that look like little tails). Speeds of up to 50m/s(50106m/s) have been observed. The total distance traveled by a bacterium is large for its size, whereas its displacement is small. Why is this?Give an example of a device used to measure time and identify what change in that device indicates a change in time.Does a car’s odometer measure distance traveled or displacement?During a given time interval the average velocity of an object is zero. What can you say conclude about its displacement over the time interval?There is a distinction between average speed and the magnitude of average velocity. Give an example that illustrates the difference between these two quantities.Does the speedometer of a car measure speed or velocity?If you divide the total distance traveled on a car trip (as determined by the odometer) by the elapsed time of the trip, are you calculating average speed or magnitude of average velocity? Under what circumstances are these two quantities the same?How are instantaneous velocity and instantaneous speed related to one another? How do they differ?Is it possible for speed to be constant while acceleration is not zero?Is it possible for velocity to be constant while acceleration is not zero? Explain.Give an example in which velocity is zero yet acceleration is not.If a subway train is moving to the left (has a negative velocity) and then comes to a stop, what is the direction of its acceleration? Is the acceleration positive or negative?Plus and minus signs are used in one-dimensional motion to indicate direction. What is the sign of an acceleration that reduces the magnitude of a negative velocity? Of a positive velocity?Motion with Constant Acceleration When analyzing the motion of a single object, what is the required number of known physical variables that are needed to solve for the unknown quantities using the kinematic equations?State two scenarios of the kinematics of single object where three known quantities require two kinematic equations to solve for the unknowns.What is the acceleration of a rock thrown straight upward on the way up? At the top of its flight? On the way down? Assume there is no air resistance.An object that is thrown straight up falls back to Earth. This is one-dimensional motion. (a) When is its velocity zero? (b) Does its velocity change direction? (c) Does the acceleration have the same sign on the way up as on the way down?Suppose you throw a rock nearly straight up at a coconut in a palm tree and the rock just misses the coconut on the way up but hits the coconut on the way down. Neglecting air resistance and the slight horizontal variation in motion to account for the hit and miss of the coconut, how does the speed of the rock when it hits the coconut on the way down compare with what it would have been if it had hit the coconut on the way up? Is it more likely to dislodge the coconut on the way up or down? Explain.The severity of a fall depends on your speed when you strike the ground. All factors but the acceleration from gravity being the same, how many times higher could a safe fall on the Moon than on Earth (gravitational acceleration on the Moon is about one-sixth that of the Earth)?How many times higher could an astronaut jump on the Moon than on Earth if her takeoff speed is the same in both locations (gravitational acceleration on the Moon is about on-sixth of that on Earth)?Finding Velocity and Displacement from Acceleration When given the acceleration function, what additional information is needed to find the velocity function and position function?Position, Displacement, and Average Velocity Consider a coordinate system in which the positive x axis is directed upward vertically. What are the positions of a particle (a) 5.0 m directly above the origin and (b) 2.0 m below the origin?A car is 2.0 km west of a traffic light at t=0 and 5.0 km east of the light at t=6.0min . Assume the origin of the coordinate system is the light and the positive x direction is eastward. (a) What are the car’s position vectors at these two times? (b) What is the car’s displacement between 0 mm and 6.0 mm?The Shanghai maglev train connects Longyang Road to Pudong International Airport, a distance of 30 km. The journey takes 8 minutes on average. What is the maglev train’s average velocity?The position of a particle moving along the x -axis is given by x(t)=4.02.0tm . (a) At what time does the particle cross the origin? (b) What is the displacement of the particle between t=3.0s and t=6.0s ?A cyclist rides 8.0 km east for 20 minutes, then he turns and heads west for 8 minutes and 3.2 km. Finally, he rides east for 16 km, which takes 40 minutes. (a) What is the final displacement of the cyclist? (b) What is his average velocity?On February 15, 2013, a superbolide meteor (brighter than the Sun) entered Earth’s atmosphere over Chelyabinsk, Russia, and exploded at an altitude of 23.5 km. Eyewitnesses could feel the intense heat from the fireball, and the blast wave from the explosion blew out windows in buildings. The blast wave took approximately 2 minutes 30 seconds to reach ground level. (a) What was the average velocity of the blast wave? b) Compare this with the speed of sound, which is 343 m/s at sea level.A woodchuck runs 20 m to the right in 5 s, then rums and runs 10 m to the left in 3 s. (a) What is the average velocity of the woodchuck? (b) What is its average speed?Sketch the velocity-versus-time graph from the following position-versus-time graph.Sketch the velocity-versus-time graph from the following position-versus-time graph.Given the following velocity-versus-time graph, sketch the position-versus-time graph.An object has a position function x(t)=5tm . (a) What is the velocity as a function of time? (b) Graph the position function and the velocity function.A particle moves along the x -axis according to x(t)=10t2m . (a) What is the instantaneous velocity at t=2s and t=3s ? (b) What is the instantaneous speed at these time? (c) What is the average velocity between t=2s and t=3s ?Unreasonable results. A particle moves along the x -axis according to x(t)=3t3+5t . At what time is the velocity of the particle equal to zero? Is this reasonable?Average and Instantaneous Acceleration A cheetah can accelerate from rest to a speed of 30.0 m/s in 7.00 s. What is its acceleration?Dr. John Paul Stapp was U.S. Air Force officer who studied the effect of extreme acceleration on the human body. On December 10, 1954, Stapp rode a rocket sled, accelerating form rest to a top speed of 282 m/s (1015 km/h) in 5.00 s and was brought jarringly back to rest an only 1.40 s. Calculate his (a) acceleration in his direction of motion and (b) acceleration opposite to his direction of motion. Express each in multiples of g(9.80m/s2) by taking its ratio to the acceleration of gravity.Sketch the acceleration-versus-time graph from the following velocity-versus-time graph.A commuter backs her car out of her garage with an acceleration of 1.40m/s2 . (a) How long does it take her to reach a speed of 2.00m/s2 ? (b) If she then brakes to a stop in 0.800 s, what is her acceleration?Assume an intercontinental ballistic goes from rest to a suborbital speed of 6.50 km/s 60.0 s (the actual speeds and time are classified). What is its average acceleration in meters per second and in multiples of g(9.80m/s2)An airplane, starting from rest, move down the runway at constant for 18 s and then takes off at a speed of 60 m/s. What is the average acceleration of the plane?Motion with Constant Acceleration A particle moves in a straight line at constant velocity of 30 m/s. What is its displacement between t=0 and t=5.0s ?A particle moves in a straight line with an initial velocity of 0 m/s and a constant acceleration of 30m/s2 . If t=0 at x=0 , what is the particle’s position at t=5s .A particle moves in a straight line with an initial velocity of 30 m/s and constant acceleration 30m/s2 . (a) What is its displacement at t=5s ? (b) What is its velocity at this same time?(a) Sketch a graph of velocity versus time corresponding to the graph of displacement versus time given in the following figure. (b) Identify the time or times (ta,tb,tcetc.) at which the instantaneous velocity has the greatest positive value. (c) At which times is it zero? (d) At which times is it negative?(a) Sketch a graph of acceleration versus time corresponding to the graph of velocity versus time given in the following figure. (b) Identify the time or times (ta,tb,tcetc.) at which the acceleration has the greatest positive value (c) At which times is zero? (d) At times is it negative?A particle has a contant acceleration of 6.0m/s2 . (a) If its initial velocity is 2.0 m/s, at what time is its displacement 5.0 m? (b) What is its velocity at that time?At t=10s , a particle is moving from left to right with a speed of 5.0 m/s. At t=20s , the particle is moving right to left with a speed of 8.0 m/s. Assuming the particle’s acceleration is constant, determine (a) its acceleration, (b) its initial velocity, and (c) the instant when its velocity is zero.A well-thrown ball is caught in a well-padded mitt. If the acceleration of the ball is 2.10104m/s2 , and 1.85 ms (1ms=103s) elapses from the time the ball first touches the mitt unit it stops, what is the initial velocity of the ball?A bullet in a gun is accelerated from the firing chamber to the end of the barrel at an average rate of 6.20105m/s2 for 8.10104s . What is its muzzle velocity (that is, its final velocity)(a) A light-rail commuter train accelerates at a rate of 1.35m/s2 . How long does it take to reach its top speed of 80.0 km/h, starting from rest? (b) The same train ordinarily decelerates at a rate of 1.65m/s2 . How long does it take to came to a stop from its top speed? (c) In emergencies, the train can decelerate more rapidly, coming to rest from 80.0 km/h in 8.30 s. What is its emergency acceleration in meters per second sqquared?While entering a freeway, a car accelerates from rest at a rate of 2.40m/s2 for 12.0 s. (a) Draw a sketch of the situation. (b) List the knowns in this problem. (c) How far does the car travel in those 12.0 s? To solve this part, first indentify the unknown, then indicate how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, check your units, and discuss whether the answer is reasonable. (d) What is the car’s final velocity? Solve for this unknown in the same manner as in (c), showing all steps explicitly.Unreasonable results At the end of a race, a runner decelerates from a velocity of 9.00 m/s at a rate of 2.00m/s2 . (a) How far does she travel in the next 5.00 s? (b) What is her final velocity? (c) Evaluate the result. Does it make sense?Blood is accelerated from rest to 30.0 cm/s in a distance of 1.80 cm left ventricle of the heart. (a) Make a sketch of the situation. (b) List the knowns in this problem. (c) How long does the acceleration take? To solve this part, first identify the unknown, then discuss how you chose the appropriate equation to solve for it. After choosing the equation show your steps in solving fir the unknown, checking your units. (d) Is the answer reasonable when compared with the time for a heartbeat?During a slap shot, a hockey player accelerates the puck from a velocity of 8.00 m/s to 40.0 m/s in the same direction. If this shot take 3.33102s , what is the distance over which the puck accelerates?A powerful motocycle can accelerate from rest to 26.8 m/s (100 km/h) in only 3.90 s. (a) What is its average acceeration? (b) How far does it travel in that time?Freight trains can product only relatively small accelerations. (a) What is the final velocity of a freight train that accelerates at a rate of 0.0500m/s2 for 8.00 min, starting with an intial velocity of 4.00 m/s? (b) If the train can slow down at a rate of 0.550m/s2 , how long will it take to come to a stop from this velocity? (c) How far will it travel in each case?A fireworks shell is accelerated from rest to a velocity of 65.0 m/s over a distance of 0.250 m. (a) Calculate the acceleration. (b) How long did theacceleration last?A swan on a lake gets airborne by flapping its wings and running on top of the water. (a) If the swan must reach a velocity of 6.00 m/s to take off and it accelerates from rest at an average rate of 0.35m/s2 , how far will it travel before becoming airborne? (b) How long does this take?A woodpecker’s brain is specially protected from large accelerations by tendon-like attachments inside the skull. While pecking on a tree, the woodpecker’s head comes to a stop from an initial velocity of 0.600 m/s in a distance of only 2.00 mm. (a) Find the acceleration in meters per second squared and in multiples of g , where g=9.80m/s2 . (b) Calculate the stopping time. (c) The tendons cradling the brain stretch, making its stopping distance 4.50 mm (greater then the head and, hence, less acceleration of the brain). What is the brain’s acceleration, expressed in multiples of g ?An unwary football player collides with a padded goalpost while running at a velocity of 7.50 m/s and comes to a full stop after compressing the padding and his body 0.350 m. (a) What is his acceleration? (b) How long does the collision last?A care package is dropped out of a cargo plane and lands in the forest. If we assume the care package speed on impact is 54 m/s (123 mph), then what is its acceleration? Assume the trees and snow stops it over a distance of 3.0 m.An express train passes through a station. It enters with an initial velocity of 22.0 m/s and decelerates at a rate of 0.150m/s2 as it goes through. The station in 210.0 m long. (a) How fast is it going when the nose leaves the station? (b) How long is the nose of the train in the station? (c) If the train is 130 m long, what is the velocity of the end of the train as it leaves? (d) When does the end of the train leave the station?Unreasonable results Dragsters can actually reach a top speed of 145.0 m/s in only 4.45 s. (a) Calculate the average acceleration for such a dragster. (b) Find the final velocity of this dragster starting from rest and accelerating at the rate found in (a) for 402.0 m (a quarter mile) without using any information on time. (c) Why is the final velocity greater than that used to find the average acceleration? (Hint: Consider whether the assumption of constant acceleration is valid for a dragster. If not, discuss whether the acceleration would be greater at the beginning or end of the run and what effect that would have on the final velocity.)Calculate the displacement and velocity at times of (a) 0.500 s, (b) 1.00 s, (c) 1.50 s, and (d) 2.00 s for a ball thro straight up with an initial velocity of 15.0 m/s. Take the point of release to be y0=0.Calculate the displacement and velocity at times of (a) 0.500 s, (b) 1.00 s, (c) 1.50 s. (d) 2.00 s, and (e) 2.50 s for a rock thro straight down with an initial velocity of 14.0 m/s from the Verrazano Narrows Bridge in New York City. The roadway of this bridge is 70.0 m above the water.A basketball referee tosses the ball straight up for the starting tip-off. At what velocity must a basketball player leave the ground to rise 1.25 m above the floor in an attempt to get the ball?A rescue helicopter is hovering over a person whose boat has sunk. One of the rescuers throws a life preserver straight down to the victim with an initial velocity of 1.40 m/s and observes that it takes 1.8 s to reach the water. (a) List the knowns in this problem. (b) How high above the water was the preserver released? Note that the downdraft of the helicopter reduces the effects of air resistance on the falling life preserver, so that an acceleration equal to that of gravity is reasonable.Unreasonable results A dolphin in an aquatic show jumps straight up out of the water at a velocity of 15.0 m/s. (a) List the knowns in this problem. (b) How high does his body rise above the water? To solve this part, first note that the final velocity is now a known, and identify its value. Then, identify the unknown and discuss how you chose the appropriate equation to solve for ft. After choosing the equation, show your steps in solving for the unknown, checking units, and discuss whether the answer is reasonable. (c) How long a time is the dolphin in the air? Neglect any effects resulting from his size or orientation.A diver bounces straight up from a diving board, avoiding the diving board on the way down, and falls feet first into a pool. She starts with a velocity of 4.00 m/s and her takeoff point is 1.80 m above the pool. (a) What is her highest point above the board? (b) How long a time are her feet in the air? (c) What is her velocity when her feet hit the water?(a) Calculate the height of a cliff if it takes 2.35 s for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of 8.00 m/s. (b) How long a time would it take to reach the ground if it is thrown straight down with the same speed?A very strong, but inept, shot putter puts the shot straight up vertically with an initial velocity of 11.0 m/s. How long a time does he have to get out of the way if the shot was released at a height of 2.20 m and he is 1.80 m tall?You throw a ball straight up with an initial velocity of 15.0 m/s. It passes a tree branch on the way up at a height of 7.0 m. How much additional time elapses before the ball passes the tree branch on the way back down?A kangaroo can jump over an object 2.50 m high. (a) Considering just its vertical motion, calculate its vertical speed when it leaves the ground. (b) How long a time is it in the air?Standing at the base of one of the cliffs of Mt. Arapiles in Victoria, Australia, a hiker hears a rock break loose from a height of 105.0 in. He can’t see the rock right away, but then does, 1.50 s later. (a) How far above the hiker is the rock when he can hear it? (b) How much time does he have to move before the rock hits his head?There is a 25O-m-high cliff at Half Dome in Yosemite National Pad in California. Suppose a boulder breaks loose from the top of this cliff. (a) How fast will It be going when It strikes the ground? (b) Assuming a reaction time of 0.300 s, how long a time will a tourist at the bottom have to get out of the way after hearing the sound of the rock breaking loose (neglecting the height of the tourist, which would become negligible anyway if hit)? The speed of sound is 335.0 m/s on this day.The acceleration of a particle varies with time according to the equation a(t)=pt2qt3 . Initially, the velocity and position are zero. (a) What is the velocity as a function of time? (b) What is the position as a function of time?Between t=0 and t=t0 , a rocket moves straight upward with an acceleration given by a(t)=ABt1/2 , where A and B are constants. (a) If x is in meters and t is in seconds, what are the units of A and B ? (b) If the rocket starts from rest, how does the velocity vary between t=0 and t=t0 ? (c) If its initial position is zero, what is the rocket’s position as a function of time during this same time Interval?The velocity of a particle moving along the x -axis varies with time according to v(t)=A+Bt1 , where A=2m/s , B=0.25m , and 1.0st8.0s . Determine the acceleration and position of the particle at t=2.0s and t=5.0s . Assume that x(t=1s)=0 .A particle at rest leaves the origin with its velocity Increasing with time according to v(t)=3.2tm/s . At 5.0 s, the particles velocity stats decreasing according to [16.01.5(t5.0)]m/s . This decrease continues until t=11.0s , after which the particle’s velocity remains constant at 7.0 m/s. (a) What is the acceleration of the particle as a function of time? (b) What Is the position of the particle at t=2.0s , t=7.0s , and t=12.0s ?Professional baseball player Nolan Ryan could pitch a baseball at approximately 160.0 km/h. At that average velocity, how long did it take a ball thrown by Ryan to reach home plate, which is 18.4 m from the pitcher’s mound? Compare this with the average reaction time of a human to a visual stimulus, which is 0.25 s.An airplane leaves Chicago and makes the 3000-km trip Los Angeles in 5.0 h. A second plane leaves Chicago one-half hour later and arrives in Los Angeles at the same time. Compare the average velocities of the two planes. Ignore the curvature of Earth and the difference in altitude between the two cities.Unreasonable Results A cyclist rides 16.0 km east, then 8.0 km west, then 8.0 km east, then 32.0 km west, and finally 11.2 km east. If his average velocity is 24 km/h, how long did it take him to complete the trip? Is this a reasonable time?An object has an acceleration of +1.2cm/s2 . At t=4.0sits velocity is -3.4cm/s . Determine the objects’ velocities at t=1.0s and t=6.0s .A particle moves along the x -axis according to the equation x(t)=2.04.0t2 . What are the velocity and acceleration at t=2.0s and t=5.0s ?A particle moving at constant acceleration has velocities of 2.0m/s at t=2.0s and -7.6m/s at t=5.2s . What is the acceleration of the particle?A train is mowing up a steep grade at constant velocity (see following figure) when its caboose breaks loose and starts rolling freely along the track. After 5.0 s, the caboose is 30 m behind the train. What is the acceleration of the caboose?

Page: [1][2][3][4][5][6][7]