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Problem 15 Medium Difficulty

$\bullet$ Two ice skaters, Daniel (mass 65.0 $\mathrm{kg}$ ) and Rebecca (mass
$45.0 \mathrm{kg} ),$ are practicing. Daniel stops to tie his shoelace and,
while at rest, is struck by Rebecca, who is moving at 13.0 $\mathrm{m} / \mathrm{s}$
before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 $\mathrm{m} / \mathrm{s}$ at an angle of $53.1^{\circ}$ from her
initial direction. Both skaters move on the frictionless, hori-
zontal surface of the rink. (a) What are the magnitude and
direction of Daniel's velocity after the collision? (b) What is
the change in total kinetic energy of the two skaters as a result
of the collision?

Answer

(a) $v_{D 2}=7.20 \mathrm{m} / \mathrm{s}, 38.0^{\circ}$ below $+x$ -axis
(b) $\Delta K=-678.5 \mathrm{J}$

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Video Transcript

this question. We have to ice skaters Daniel or D, with a mass of 65 kilograms, Rebecca, or are with a massive 45 kilograms. Danielle's initially at rest, while Rebecca is traveling at 13 meters per second towards Daniel. On day after the collision, Rebecca travels at eight meters per second at an angle of 53.1 degrees from her initial velocity. So in part A, we want to find Daniel's final philosophy. And in order to do that, we're gonna first set some sign conventions. So we will call Thebes Action of Rebecca's Initial Velocity Positive X. So her final philosophy is 53.1 degrees away from X. So we are going to do this just using conservation of momentum. So initially, only Rebecca has momentum, and it's all in the X direction. So the in the X direction the mass of Rebecca Times three Velocity of Rebecca is equal to know in the final condition after the collision. Presumably, both of them are moving in to some degree. Some of their movement is in the X direction for both skaters, so it'll includes Rebecca's mass times. The final the X component of Rebecca's final of philosophy, plus Daniels Mass. Times the X component of his final philosophy. So we want toe isolate. Four Daniels Final X component Velocity on its own Circling disobey tracked this first term from both sides on Divide by Daniels Mess. So that works out to be V. F. D. X is equal to Mass of Rebecca Times Rebecca's initial velocity plus b f R X over Daniels, Mass. Now we don't actually know exactly Theo X component of Rebecca's final velocity, but we do know the angle we do. You know the angle of her velocity from the X direction. So using trigonometry, we have we using trigonometry. We know that the F R X of the X component of her of her final velocity is equal to her total final speed Times co sign of, um, 53.1 degrees. Since that's the That's the angle from the extraction. So fucking in some Constance here we have mass of Rebecca is 45 kilograms. Three. The initial velocity is 13.0 meters per second, and then Theo X component of her final philosophy is eight meters per second Times co sign of 53.1 degrees. The mess of Daniel, meanwhile, is 65.0 kilograms. Plugging all of this into a calculator. We find that then the X component of Danielle's final velocity is five 0.68 meters per second. Now we need to do the same thing in the Y direction. Eso Initially, neither of them have any velocity in the Y direction, so zero is equal to In the final case, it's just going to be the white components of reckons. Final velocity. So using trigonometry again, that's 8.0 meters per second times signed this time since we're looking for the component perpendicular toothy X direction 53.1 Secrets, um, ends. Daniel also contributes to the white components final momentum. So in this case, his final moment, um in the y direction is simply his mass times the f d. Why So now we want Teoh subtract the first term from both sides as well as divide by Daniels Mass. To get V F. D. Y is equal to negative eight meters per second times sign of 53.1 degrees over his mass, which is 65.0 kilograms plugging this into a calculator, refines that his final velocity in the Y direction is negative 4.43 meters per second. So finally, we want to get an angle from the from these two numbers. So we have the X and Y components of Daniel's final velocity. We want to find the total final velocity as well as its angle from the X direction. So first of all, we can finds the magnitude of the velocity. So the magnitude of the velocity just using Pythagorean theorem is the square root of 5.68 meters per second squared bus. Negative 4.43 meters per second also squared and putting that into a calculator, we finds that hit The magnitude of his final velocity is 7.20 meters per second, and that is the That is half of this answer. So now we want to find the angle. So, first of all, since we're defining the angle of Rebecca's final velocity from the X Direction since defining her angle to be positive, his angle has to be negative because that's the only way way have conservation of momentum. They have to go in opposite X directions. So using our knowledge of trigonometry 10 of his initial our tent, the tangent of Daniels final, um, angle from the X direction, which I'm calling fate a D is equal to negative 4.43 meters per second over 5.68 meters per second on this is, of course, um, the final speed in the Y direction over the final speed in the extraction. Sorry, which is the F. D. X? So, um, taking to the inverse tangent of both sides refines that he travels at an angle of negative 38 degrees relative to the X axis with again the X axis being Rebecca's initial the direction of Rebecca's initial velocity and with Rebecca's final angle from that direction being a positive angle. So that's it for part A. And in Part B. We want to finds the difference in kinetic energy over the course of this collision. So, um, the difference in kinetic energy, which I'm calling Delta K is equal to the final. Kinetic energy finest the initial kinetic energy. So the final kinetic energy is, in turn, the kinetic energy of Rebecca, plus the final kinetic energy of Daniel, which is 1/2 times Rebecca's mass times. Rebecca's final velocity squared. Same for Daniel. Sorry. Dfd for Daniel Squared. And in the initial case, only Rebecca is moving. So only she has kinetic energy. Now we determined all of the's. Constance previously in the case of the 1st 2 terms, re calculated them there. The answers to the previous questions the previous part of this question in the third term were given these. So it should be fairly straightforward to plug this into a calculator and finds that the difference in kinetic energy of the system after versus before the collision is negative. 678.5 Jules. And that is the answer for this part, and that's the question.

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