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Momentum: Ice skater throws a ball

CON‑5.D.1.1 (LO)
A simple conservation of momentum problem involving an ice skater and a ball. Created by Sal Khan.

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Video transcript

Welcome back. I'll now do a couple of more momentum problems. So this first problem, I have this ice skater and she's on an ice skating rink. And what she's doing is she's holding a ball. And this ball-- let me draw the ball-- this is a 0.15 kilogram ball. And she throws it. Let's just say she throws it directly straight forward in front of her, although she's staring at us. She's actually forward for her body. So she throws it exactly straight forward. And I understand it is hard to throw something straight forward, but let's assume that she can. So she throws it exactly straight forward with a speed-- or since we're going to give the direction as well, it's a velocity, right, cause speed is just a magnitude while a velocity is a magnitude and a direction-- so she throws the ball at 35 meters per second, and this ball is 0.15 kilograms. Now, what the problem says is that their combined mass, her plus the ball, is 50 kilograms. So they're both stationary before she does anything, and then she throws this ball, and the question is, after throwing this ball, what is her recoil velocity? Or essentially, well how much, by throwing the ball, does she push herself backwards? So what is her velocity in the backward direction? And if you're not familiar with the term recoil, it's often applied to when someone, I guess, not that we want to think about violent things, but if you shoot a gun, your shoulder recoils back, because once again momentum is conserved. So there's a certain amount of momentum going into that bullet, which is very light and fast going forward. But since momentum is conserved, your shoulder has velocity backwards. But we'll do another problem with that. So let's get back to this problem. So like I just said, momentum is conserved. So what's the momentum at the start of the problem, the initial momentum? Let me do a different color. So this is the initial momentum. Initially, the mass is 50 kilograms, right, cause her and the ball combined are 50 kilograms, times the velocity. Well the velocity is 0. So initially, there is 0 velocity in the system. So the momentum is 0. The P initial is equal to 0. And since we start with a net 0 momentum, we have to finish with a net 0 momentum. So what's momentum later? Well we have a ball moving at 35 meters per second and the ball has a mass of 0.15 kilograms. I'll ignore the units for now just to save space. Times the velocity of the ball. Times 35 meters per second. So this is the momentum of the ball plus the new momentum of the figure skater. So what's her mass? Well her mass is going to be 50 minus this. It actually won't matter a ton, but let's say it's 49-- what is that-- 49.85 kilograms, times her new velocity. Times velocity. Let's call that the velocity of the skater. So let me get my trusty calculator out. OK, so let's see. 0.15 times 35 is equal to 5.25. So that equals 5.25. plus 49.85 times the skater's velocity, the final velocity. And of course, this equals 0 because the initial velocity was 0. So let's, I don't know, subtract 5.25 from both sides and then the equation becomes minus 5.25 is equal to 49.85 times the velocity of the skater. So we're essentially saying that the momentum of just the ball is 5.25. And since the combined system has to have 0 net momentum, we're saying that the momentum of the skater has to be 5.25 in the other direction, going backwards, or has a momentum of minus 5.25. And to figure out the velocity, we just divide her momentum by her mass. And so divide both sides by 49.85 and you get the velocity of the skater. So let's see. Let's make this a negative number divided by 49.85 equals minus 0.105. So minus 0.105 meters per second. So that's interesting. When she throws this ball out at 35 meters per second, which is pretty fast, she will recoil back at about 10 centimeters, yeah, roughly 10 centimeters per second. So she will recoil a lot slower, although she will move back. And if you think about it, this is a form of propulsion. This is how rockets work. They eject something that maybe has less mass, but super fast. And that, since we have a conservation of momentum, it makes the rocket move in the other direction. Well anyway, let's see if we could fit another problem in. Actually, it's probably better to leave this problem done and then I'll have more time for the next problem, which will be slightly more difficult. See you soon.