Consider a particle of mass m acted on by a force F in an inertial frame. Prove that dK = F. dr, (1) where dr is the change in the position vector of the particle in a time dt, and dK is the corresponding change in the kinetic energy K = mv². %3D
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- a. What is vav,x for the interval from t = 0 to t = 3.60 s? answer in m/s b. What is vav,x for the interval from t = 0 to t = 3.50 s? anser in m/sThe kinetic energy of a mass moving in the x-y plane is: T = mx² + my² Using the transformation = rcos (0) and y = rsin (0), derive a general expression for kinetic energy of the mass in the r, basis.You and your secret celebrity crush pass each other in rockets that are each traveling 0.56c relative to the other. Your secret celebrity crush takes a nap and you decide to time how long the nap takes. In your frame, the nap takes 1.7 h. How long was the nap in your secret celebrity crush's frame? Unbeknownst to you, your secret celebrity crush's agent is onboard the celebrity's rocket and saw you timing the nap. Concerned, the agent decides to time how long you were timing the nap. In the agent's frame, how long were you timing your secret celebrity crush's nap?
- 2. For this series of 4 questions, a man sits in the chair which is pin-connected to the 10-m frame BC. Frame BC rotates counterclockwise. Your final goal is to determine the magnitude of the support force N exerted by the chair on the man at the instant 0 = 30°. But before that, you need to answer some other questions. Please pay attention: the numbers may change from problem to problem since they are randomized. 2) If at this instance he has a speed of 6.5 m/s, which is increasing at 0.4 m/s², determine the magnitude of his total acceleration in m/s². Your answer must include 2 places after the decimal point. Your Answer: Answer 10 m BA cylindrical spaceship of length 35.0 m and diameter 8.35 m is traveling in the direction of its cylindrical axis (length). It passes by the Earth at a relative speed of 2.44××1088 m/s. a) What is the length of the ship, as measured by an Earth observer? b) What is the diameter of the ship, as measured by an Earth observer? c) How long does it take the spaceship to travel a distance of 10.0 km according to the ship's pilot? d) How long does it take the spaceship to travel a distance of 10.0 km according to an Earth-based observer?In certain physical models, the nonhomogeneous term, or forcing term, g(t) in the equation ay" + by' + cy = g(t) may not be continuous, but have a jump discontinuity. If this occurs, a reasonable solution can still be obtained using the following procedure. Consider the following initial value problem. 240 if 0sts7T/6 y" + 4y' + 40y = g(t); y(0) = 0, y'(0) = 0, where g(t) = . if t> 7x/6 Complete parts (a) through (c) below. (a) Find a solution to the initial value problem for 0 sts 71/6. The solution for 0sts 7n/6 is y(t) = - 6 e -21 *cos 6t - 2 e - 2t sin 6t + 6 (Type an equation.) (b) Find a general solution for t> 71/6. The general solution for t> 7x/6 is y(t) = C, e-2 cos 6t + C, e -2t sin 6t (Type an equation. Do not use d. D. e, E, i, or I as arbitrary constants since these letters already have defined meanings.) (c) Now choose the constants in the general solution from part (b) so that the solution from part (a) and the solution from part (b) agree, together with their first…
- 1. Taking north as positive and south as negative for all reference frames, suppose Mayor Adams is riding a southbound train to New York City traveling at 40 m/s relative to the ground, while Governor Hochul is riding a northbound train to Albany at 33 m/s relative to the ground. A) What is the velocity of Mayor Adams in Governor Hochul's reference frame (i.e. in the reference frame where Governor Hochul isn't moving)? B) What is the velocity of Governor Hochul in Mayor Adams reference frame? C) To his surprise Mayor Adams observes a groundhog running along the aisle towards the back of the train with a speed of 2 m/s relative to the train. What is the velocity of the groundhog in Governor Hochul's reference frame? D) What is the groundhog's velocity in the reference frame of someone standing at the train station? E) What is its velocity in Mayor Adams's reference frame? F) What is its velocity in the groundhog's frame? 30 10 ft. 5 ft. 15 ft.An experimentalist in a laboratory finds that a particle has a helical path. The position of this particle in the laboratory frme is given by r(t)= R cos(wt)i + R sin(wt)j + vztk R,vz, and w are constants. A moving frame has velocity (Vm)L= vzk relative to the laboratory frame. In vector form: A)What is the path of the partical in the moving frame? B)what is the velocity of the particle as a function of time relative to the moving frame? C)What is the acceleration of the particle in each frame? D)How should the accelerartion in each frame be realted?Does your answer to part c make sense?dx 4x 3. In so-called "natural units" (which is just a sneaky way to let us ignore a bunch of constants), the relativistic kinetic energy of a rigid body is given by the formula 1 КЕ — т V1 – v2 where m is the rest mass of the body and v is its relative speed. Alien scientists on a space station are observing an object falling into a black hole. As the object falls, it is disintegrating, losing mass at a rate of 3 (so its mass is changing at a rate of -3). How fast is the kinetic energy of the main part of the object changing when its mass is 20, its velocity is .7, and it is accelerating at a rate of .1 (remember that acceleration is the derivative of velocity with respect to time: a = dt 1Note that this formula does not make sense when v > 1. That is because in natural units, a speed of 1 corresponds to the speed of light, and nothing with positive rest mass can go that fast.
- q3A person is traveling at 28.3 m/s due west on the highway and sees a tractor ahead of him moving ahead of him perpendicularly away from the road (North) at 6.50m/s. (Both these motions are relative to a frame at rest on the ground.) What is the car's velocity in the tractor's frame (magnitude and direction)?Chapter 37, Problem 015 The center of our Milky Way galaxy is about 24000 ly away. (a) To eight significant figures, at what constant speed parameter would you need to travel exactly 24000 ly (measured in the Galaxy frame) in exactly 27 y (measured in your frame)? (b) Measured in your frame and in lightyears, what length of the Galaxy would pass by you during the trip? (a) Number Units (b) Number Units