HW11 - Impulse and Momentum

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 1/23 HW11 - Impulse and Momentum Due: 11:59pm on Friday, December 9, 2022 To understand how points are awarded, read the Grading Policy for this assignment. Impulse on a Baseball Learning Goal: To understand the relationship between force, impulse, and momentum. The effect of a net force acting on an object is related both to the force and to the total time the force acts on the object. The physical quantity impulse is a measure of both these effects. For a constant net force, the impulse is given by . The impulse is a vector pointing in the same direction as the force vector. The units of are or . Recall that when a net force acts on an object, the object will accelerate, causing a change in its velocity. Hence the object's momentum ( ) will also change. The impulse-momentum theorem describes the effect that an impulse has on an object's motion: . So the change in momentum of an object equals the net impulse, that is, the net force multiplied by the time over which the force acts. A given change in momentum can result from a large force over a short time or a smaller force over a longer time. In Parts A, B, C consider the following situation. In a baseball game the batter swings and gets a good solid hit. His swing applies a force of 12,000 to the ball for a time of . Part A Assuming that this force is constant, what is the magnitude of the impulse on the ball? Enter your answer numerically in newton seconds using two significant figures. ANSWER: Correct We often visualize the impulse by drawing a graph showing the force versus time. For a constant net force such as that used in the previous part, the graph showing the magnitude of the force versus time will look like the one shown in . = 8.4

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 2/23 Part B The magnitude of the net force versus time graph has a rectangular shape. Often in physics geometric properties of graphs have physical meaning. ANSWER: Correct The assumption of a constant net force is idealized to make the problem easier to solve. A real force, especially in a case like the one presented in Parts A and B, where a large force is applied for a short time, is not likely to be constant. A more realistic graph showing the magnitude of the force that the swinging bat applies to the baseball will show the force building up to a maximum value as the bat comes into full contact with the ball. Then as the ball loses contact with the bat, the graph will show the magnitude of the force decaying to zero. It will look like the graph in . Part C For this graph, the length of the rectangle corresponds to the impulse. For this graph, the height of the rectangle corresponds to the impulse. For this graph, the area of the rectangle corresponds to the impulse. For this graph, the slope of the rectangle corresponds to the impulse.

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 3/23 If both the graph representing the constant net force and the graph representing the variable net force represent the same impulse acting on the baseball, which geometric properties must the two graphs have in common? ANSWER: Correct When the net force varies over time, as in the case of the real net force acting on the baseball, you can simplify the problem by finding the average net force acting on the baseball during time . This average net force is treated as a constant force that acts on the ball for time . The impulse on the ball can then be found as . Graphically, this method states that the impulse of the baseball can be represented by either the area under the net force versus time curve or the area under the average net force versus time curve. These areas are represented in the figure as the areas shaded in red and blue respectively. The impulse of an object is also related to its change in momentum. Once the impulse is known, it can be used to find the change in momentum, or if either the initial or final momentum is known, the other momentum can be found. Keep in mind that . Because both impulse and momentum are vectors, it is essential to account for the direction of each vector, even in a one-dimensional problem. Part D Assume that a pitcher throws a baseball so that it travels in a straight line parallel to the ground. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. Define the direction the pitcher originally throws the ball as the + x direction. The impulse on the ball caused by the bat will be in the ANSWER: maximum force area slope positive x direction. negative x direction.

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 4/23 Correct Part E Now assume that the pitcher in Part D throws a 0.145- baseball parallel to the ground with a speed of 32 in the + x direction. The batter then hits the ball so it goes directly back to the pitcher along the same straight line. What is the ball's x -component of velocity just after leaving the bat if the bat applies an impulse of to the baseball? Enter your answer numerically in meters per second using two significant figures. ANSWER: Correct The negative sign in the answer indicates that after the bat hits the ball, the ball travels in the opposite direction to that defined to be positive. Problem 11.5 - Enhanced - with Hints and Feedback Force-versus-time graph is shown in . Part A What value of gives an impulse of 6.0 ? Express your answer with the appropriate units. Hint 1. How to approach the problem Impulse is defined as a time integral of force or, in other words, as the area under a force-versus-time graph. ANSWER: = -26 = 1000

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 5/23 Correct Problem 11.22 - Enhanced - with Hints and Feedback A proton is traveling to the right at . It has a head-on perfectly elastic collision with a carbon atom. The mass of the carbon atom is 12 times the mass of the proton. Part A What is the speed of the proton after the collision? Express your answer with the appropriate units. Hint 1. How to determine the speed of the proton This is a case of a perfectly elastic collision between a proton and a carbon atom, so both momentum and energy are conserved. Solving two equations simultaneously, you can find the speed of each particle. ANSWER: Correct Part B What is the speed of the carbon atom after the collision? Express your answer with the appropriate units. Hint 1. How to determine the speed of the carbon atom This is a case of a perfectly elastic collision between a proton and a carbon atom, so both momentum and energy are conserved. Solving two equations simultaneously, you can find the speed of each particle. ANSWER: Correct Part C What is the direction of the proton after the collision? Hint 1. How to determine the direction of the proton = 1.69×10 7 = 3.08×10 6

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 6/23 The collision is one-dimensional. After the collision, both particles move along the horizontal axis. ANSWER: Correct Part D What is the direction of the carbon atom after the collision? Hint 1. How to determine the direction of the carbon atom The collision is one-dimensional. After the collision, both particles move along the horizontal axis. ANSWER: Correct Problem 11.34 - Enhanced - with Expanded Hints A 20 ball of clay traveling east at 4.0 collides with a 40 ball of clay traveling north at 2.0 . Part A What is the speed of the resulting 60 ball of clay? Express your answer with the appropriate units. Hint 1. How to approach the problem Make a pictorial representation of the problem with the - and -axes defined. Find the - and -components of the total momentum of the system before and after the collision. Remember that the collision is inelastic. Write upward downward to the right to the left upward downward to the right to the left

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 7/23 the law of conservation of momentum in component form. Now you can find the - and -components of the velocity of the resulting ball. To find the speed, you have to find the magnitude of the resulting velocity. Hint 2. Simplify: -component Let the -axis point east. Find the -component of the velocity of the resulting ball using the law of conservation of momentum. Express your answer with the appropriate units. ANSWER: Hint 3. Simplify: -component Let the -axis point north. Find the -component of the velocity of the resulting ball using the law of conservation of momentum. Express your answer with the appropriate units. ANSWER: ANSWER: = 1.3 = 1.3 = 1.89

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 8/23 Correct MODEL: This problem deals with the conservation of momentum in two dimensions in an inelastic collision. VISUALIZE: SOLVE: The conservation of momentum equation gives Substituting in the given values, we find Part B What is the direction of the resulting ball of clay? Express your answer in degrees. Hint 1. How to approach the problem Let the -axis point east and the -axis point north. Then the angle is measured counterclockwise from the - axis. Use the values of the - and -components of the resulting velocity and trigonometric identities to find the angle. Hint 2. Simplify: direction angle Choose the correct equation for the direction angle. ANSWER:

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 9/23 ANSWER: Correct SOLVE: Using the results from Part A and the fact that and , we get MCAT (R) Prep: Compression of human bone While unrealistic, we will examine the forces on a leg when one falls from a height by approximating the leg as a uniform cylinder of bone with a diameter of 2.3 and ignoring any shear forces. Human bone can be compressed with approximately 1.7 10 before breaking. A man with a mass of 100 falls from a height of 7 . Assume his acceleration once he hits the ground is constant. For these calculations, = 10 . Part A What is his speed just before he hits the ground? ANSWER: Correct Part B With how much force can the "leg" be compressed before breaking? = 45 north of east 1200 8.4 12 140

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 10/23 ANSWER: Correct Part C If he lands "stiff legged" and his shoes only compress 1 , what is the magnitude of the average force he experiences as he slows to a rest? ANSWER: Correct Part D If he bends his legs as he lands, he can increase the distance over which he slows down to 50 . What would be the average force he experiences in this scenario? ANSWER: Correct Part E Dyne is also a unit of force and 1 = 10 . What is the maximum a bone can be compressed in ? ANSWER: 7.1 10 2.8 10 1.7 10 7.1 10 7.0×10 5 1200 7000 1.4×10 6 2.8×10 4 1.4×10 4 140 1200

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 11/23 Correct Part F Which of the following is the reason that we would recommend that the man bend his legs while landing from such a fall? ANSWER: Correct Problem 11.17 - Enhanced - with Hints and Feedback Three identical train cars, coupled together, are rolling east at speed . A fourth car traveling east at 2 catches up with the three and couples to make a four-car train. A moment later, the train cars hit a fifth car that was at rest on the tracks, and it couples to make a five-car train. Part A What is the speed of the five-car train? Express your answer in terms of . Hint 1. How to approach the problem Since the train cars couple together, this is a case of perfectly inelastic collisions. Momentum is conserved in these collisions in the impulse approximation, in which the external forces during the time of the collision are ignored. 1.7 10 1.7 10 1.7 10 1.7 10 Bending his legs decreases his overall change in momentum, thus decreasing the force applied by the ground. Bending his legs decreases the speed at which he hits the ground, thus decreasing the force applied by the ground. Bending his legs increases the time over which the ground applies force, thus decreasing the force applied by the ground. Bending his legs allows him to push back up on the ground and negate some of the effects of the force applied by the ground.

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 12/23 ANSWER: Correct A One-Dimensional Inelastic Collision Block 1, of mass = 5.90 , moves along a frictionless air track with speed = 17.0 . It collides with block 2, of mass = 31.0 , which was initially at rest. The blocks stick together after the collision. Part A Find the magnitude of the total initial momentum of the two-block system. Express your answer numerically. Hint 1. How to approach the problem Find the initial momentum of each block separately; then add the two momenta together. Recall that momentum is equal to mass times velocity: . ANSWER: Correct Part B Find , the magnitude of the final velocity of the two-block system. Express your answer numerically. = = 100

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 13/23 Hint 1. How to approach the problem Apply conservation of momentum, keeping in mind that the mass after the collision is the sum of the individual blocks' masses. ANSWER: Correct Part C What is the change in the two-block system's kinetic energy due to the collision? Express your answer numerically in joules. Hint 1. Find the initial kinetic energy Find , the initial kinetic energy of the two-block system. Express your answer numerically in joules. ANSWER: ANSWER: Correct Elastic Collision in One Dimension Block 1, of mass , moves across a frictionless surface with speed . It collides elastically with block 2, of mass , which is at rest ( ). After the collision, block 1 moves with speed , while block 2 moves with speed . Assume that , so that after the collision, the two objects move off in the direction of the first object before the collision. = 2.72 = 853 = -716 J

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 14/23 Part A This collision is elastic. What quantities, if any, are conserved in this collision? Hint 1. What to think about If no net external force acts on a system of colliding objects, then the total momentum of the system is conserved. This holds for both elastic and inelastic collisions. In an elastic collision, is kinetic energy conserved? ANSWER: Correct Part B What is the final speed of block 1? Express in terms of , , and . Hint 1. How to approach the problem The first step in solving most collision problems is to apply conservation of momentum. This is done by writing a vector equation that captures the fact that the initial and the final momenta of the system are equal. In this problem, because the collision is elastic, kinetic energy is also conserved. This yields a second equation, this one capturing the fact that the initial and the final kinetic energies of the system are equal. The result is two equations (conservation of momentum and conservation of energy) and two unknowns ( and ). Use algebra to solve the equations and find the unknowns. Hint 2. Apply conservation of momentum Conservation of momentum states that the total momentum before the collision is equal to the total momentum after the collision: . Use this equation to find an expression for . kinetic energy only momentum only kinetic energy and momentum

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 15/23 Express in terms of , , and . Hint 1. Find the final momentum of the system The magnitude of the initial momentum of this system (consisting of the two blocks), , is equal to . Find an expression for the magnitude of the final momentum, , of this system. Note that since , both objects will move in the positive direction after the collision. Express in terms of , , , and . ANSWER: ANSWER: Hint 3. Apply conservation of energy Conservation of energy states that the total kinetic energy before the collision is equal to the total kinetic energy after the collision: . Use this equation to find an expression for . Express in terms of , , and . Hint 1. Find the final kinetic energy of the system The initial kinetic energy of the system is . Find an expression for the system's final kinetic energy . Express in terms of , , , and . ANSWER: ANSWER: Hint 4. Putting it together Now that you have obtained an equation from each of the two conservation laws (conservation of energy and conservation of momentum), it becomes a matter of algebra to solve for in terms of , , and . If you find the algebra tricky, start by manipulating the two equations to find an expression for in terms of and . Your answer should depend only on and . Hint 1. Detailed help with the math In the equations derived from both conservation laws, the term appears (you will need to factor the difference of squares to see this term in the equation derived from conservation of energy). Use = = = =

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 16/23 the conservation of momentum equation to find an expression for , and then substitute this expression into the conservation of energy equation. Finally, solve for . ANSWER: ANSWER: Correct Part C What is the final speed of block 2? Express in terms of , , and . Hint 1. Using the result from the previous part If you solved the previous part, you already have an expression for in terms of , , and . If you subsitute this expression into the conservation of momentum equation, you can solve for . ANSWER: Correct Problem 11.24 - Enhanced - with Expanded Hints A package of mass is released from rest at a warehouse loading dock and slides down the 3.5- -high, frictionless chute to a waiting truck. Unfortunately, the truck driver went on a break without having removed the previous package, of mass , from the bottom of the chute. = = =

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 17/23 Part A Suppose the packages stick together. What is their common speed after the collision? Express your answer with the appropriate units. Hint 1. How to approach the problem Use the energy conservation to determine the speed of the upper package just before the collision, then use the momentum conservation to determine the speed of the packages after the inelastic collision. Hint 2. Simplify: the upper package before the collision Determine the velocity of the upper package just before the collision, at the bottom of the chute. Express your answer with the appropriate units. Enter positive value for the motion down the chute and negative value for the motion up the chute. ANSWER: ANSWER: 8.3 = 2.8

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 18/23 Correct MODEL: Model the two packages as particles. Momentum is conserved in inelastic collisions. Visualize: SOLVE: For a package with mass the conservation of energy equation is Using and , For the perfectly inelastic collision the conservation of momentum equation is Using , we get Part B Suppose the collision between the packages is perfectly elastic. To what height does the package of mass rebound? Express your answer with the appropriate units. Hint 1. How to approach the problem. Use the energy conservation to determine the speed of the upper package just before the collision, then use the momentum conservation to determine the speed of the upper package after the elastic collision, and then use the energy conservation again to determine the height reached by the upper package. Hint 2. Simplify: the upper package after the collision Determine the velocity of the upper package just after the collision, at the bottom of the chute. Express your answer with the appropriate units. Enter positive value for the motion down the chute and negative value for the motion up the chute. ANSWER:

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 19/23 ANSWER: Correct MODEL: Model the two packages as particles. Momentum and kinetic energy is conserved in elastic collisions. Visualize: SOLVE: For a package with mass the conservation of energy equation is Using and , For the elastic collision, the mass package rebounds with velocity The negative sign with shows that the package with mass rebounds and goes to the position . We can determine by applying the conservation of energy equation as follows. For a package of mass : Using , and , we get ± Catching a Ball on Ice -2.8 = 39

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 20/23 Olaf is standing on a sheet of ice that covers the football stadium parking lot in Buffalo, New York; there is negligible friction between his feet and the ice. A friend throws Olaf a ball of mass 0.400 that is traveling horizontally at 12.0 . Olaf's mass is 68.3 . Part A If Olaf catches the ball, with what speed do Olaf and the ball move afterward? Express your answer numerically in meters per second. Hint 1. How to approach the problem Using conservation of momentum and the fact that Olaf's initial momentum is zero, set the initial momentum of the ball equal to the final momentum of Olaf and the ball, then solve for the final velocity. Hint 2. Find the ball's initial momentum What is , the initial momentum of the ball? Express your answer numerically in kilogram meters per second. ANSWER: ANSWER: Correct Part B If the ball hits Olaf and bounces off his chest horizontally at 7.70 in the opposite direction, what is his speed after the collision? Express your answer numerically in meters per second. Hint 1. How to approach the problem = 4.80 = 6.99×10 −2

12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 21/23 The initial momentum of the ball is the same as in Part A. Apply conservation of momentum, keeping in mind that both Olaf and the ball have a nonzero final momentum. Hint 2. Find the ball's final momentum Taking the direction in which the ball was initially traveling to be positive, what is , the ball's final momentum? Express your answer numerically in kilogram meters per second. ANSWER: ANSWER: Correct Problem 11.28 - Enhanced - with Video Solution Dan is gliding on his skateboard at 3.00 . He suddenly jumps backward off the skateboard, kicking the skateboard forward at 7.00 . Dan's mass is 60.0 and the skateboard's mass is 6.00 . For general problem-solving tips and strategies for this topic, you may want to view a Video Tutor Solution of Softball toss . Part A How fast is Dan going as his feet hit the ground? Express your answer with the appropriate units. ANSWER: Correct Here we learn how to calculate the speed of a system after some of its mass is ejected, using momentum conservation. Colliding Cars In this problem we will consider the collision of two cars initially moving at right angles. We assume that after the collision the cars stick together and travel off as a single unit. The collision is therefore completely inelastic. = -3.08 = 0.115 2.60

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 22/23 Two cars of masses and collide at an intersection. Before the collision, car 1 was traveling eastward at a speed of , and car 2 was traveling northward at a speed of . After the collision, the two cars stick together and travel off in the direction shown. Part A First, find the magnitude of , that is, the speed of the two-car unit after the collision. Express in terms of , , and the cars' initial speeds and . Hint 1. Conservation of momentum Recall that conservation of linear momentum may be expressed as a vector equation, . Each vector component of linear momentum is conserved separately. Hint 2. x and y components of momentum The momentum of the two-car system immediately after the collision may be written as , where the x and y directions are the eastward and northward directions, respectively. Find and Express the two components, separated by a comma, in terms of , , and . ANSWER: Hint 3. A vector and its components Recall that the square of the magnitude of a vector is given by the Pythagorean formula: . Hint 4. Velocity and momentum Find , the magnitude of the final velocity. Express in terms of the magnitude of the final momentum and the masses and . ANSWER: , = , =

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12/19/22, 12:16 PM HW11 - Impulse and Momentum https://session.masteringphysics.com/myct/assignmentPrintView?assignmentID=10072426 23/23 ANSWER: Correct Part B Find the tangent of the angle . Express your answer in terms of the momenta of the two cars, and . ANSWER: Correct Part C Suppose that after the collision, ; in other words, is . This means that before the collision: ANSWER: Correct Score Summary: Your score on this assignment is 99.6%. You received 19.92 out of a possible total of 20 points. = = The magnitudes of the momenta of the cars were equal. The masses of the cars were equal. The velocities of the cars were equal.

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