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Problem 27 Medium Difficulty

Two ice skaters, Daniel (mass 65.0 kg) and Rebecca (mass 45.0 kg), are practicing. Daniel stops to tie his shoelace and, while at rest, is struck by Rebecca, who is moving at 13.0 m/s before she collides with him. After the collision, Rebecca has a velocity of magnitude 8.00 m/s at an angle of 53.1$^\circ$ from her initial direction. Both skaters move on the frictionless, horizontal 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. $7.20 \mathrm{m} / \mathrm{s}$
B. $-680 \mathrm{J}$

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Video Transcript

problem. 8.27. So we have Ah, Rebecca and Daniel. Yeah, Daniel, Uh, where ice skaters were looking down on them on an ice rink. Daniel is stopped. We're told he's tying his shoe and, ah, Rebecca is coming towards him at 13 meters per second and unfortunately, she she runs into him and then we're told that told her initial speed and the direction is sort of arbitrary. We're going to put it in the we're going to call that positive ex direction. And so I will be up there with their masses. And so they were told her final speed and the angle that she makes with her initial direction. And then we need to find Daniels, uh, velocity after the collision, this part A and then part B. We want to find what is the change in kinetic energy. So all of the components of the velocity and the plane of the skating rink have to be conserved because it's friction, this ice. So our initial momentum is just Rebecca's mass spends her initial velocity, and then this has to be equal to her. Mass time is her final velocity, plus Daniel's mass times his my velocity. So then solving this for his final velocity, we find that this is equal to the ratio of their masses times the difference between her initial speed and her final speed. So now her initial speed is going to be, you know, the 13 meters per second in the extraction and her final speed is gonna be that eight meters per second times the coastline of the angle of the exact direction and the sine of the angle and the wider action. What? So then we find that this is going to be five point six paid meters per second and the next direction and negative for points for three meters per second in the wider action. And if we had chosen because we're not really told this in the problem. So we get to pick what we decided that she was going to be going downward in this figure, and he's going to be upward. And obviously, you know, this would be positive because this would have been negative. Okay, so now we want, uh, you know, this gives us the direction in the magnitude, but it's a little bit easier to think about if we have at them separately. So just like for any vector, this is just the square root of the some of the components squared, and we find that this is 7.20 meters per second and then his angle. I guess it was stated in the drawing. There it's going to be the Arc tangent. Uh, the two components, which, if we draw little pictures here, is the ex component is the lie component. Here's the opposite. Over the adjacent is the tangent of that angle. And so take the arc tangent on both sides and you get that those those numbers into the Arc Tangent and we find that his angle is 38 suffer part B. We want to know what the change in the kinetic energy is. So the initial kinetic energy is only due to Rebecca moving because these candles initially at rest and then the final is going to be because will is going to be the sun. Was it there kinetic energies? So Delta K is obviously just going to be your finalize initial kinetic energies in plugging in all the numbers and they're doing the subtraction. We discovered that the kinetic energy of the whole system has decreased by 680. Jules, it's good that we end up with the negative sign because otherwise energy would have had to have been added from somewhere. Since there's nothing to add any energy that would, uh, I would have kept us up at night. So there we go, this user, his desert. This is Daniel's final velocity, and this is the amount of energy that was lost. We concede it was not in the last clip him.