0:00
All right, hello.
00:01
In this question we're given this setup where we have two balls elevated a height m.
00:06
We're going to ignore the radii of the balls and we drop them from that height.
00:10
They're going to go down and hit the ground.
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Obviously the bottom ball, the bigger ball, will hit the ground first and we're told it bounces back perfectly elastic.
00:18
So it's going to have the same speed after it bounces off the ground that it had before it hit the ground.
00:24
I've labeled that as va.
00:27
Also note our little ball is going to be moving at the same speed because when we drop objects it doesn't matter what their mass is for how they're going to accelerate.
00:37
Right? and then we're told that the big ball is going up, the little ball is going down, they collide perfectly elastic, and then both balls are going to end up going up after that fact.
00:48
And we're asked what is the ratio of m over m such that the larger ball stops upon hitting the smaller ball.
00:57
So we're going to say vc here equals zero.
00:59
So the larger ball doesn't actually continue going up, it just stops dead.
01:04
What is the ratio of the masses that allows that to occur? so in order to do this we're going to look at these two situations as our initial and final for a conservation of momentum problem.
01:17
So initially what do we have? i'm going to call up as positive y here.
01:23
Initially we have the little mass moving down at some speed va, and in the opposite direction we have the big mass moving up at some speed va.
01:34
And then after the fact, after they collide, we say that we have the little mass moving up at some speed vb, and then we have zero speed for the big mass.
01:44
So we're going to have zero there.
01:46
Okay, great.
01:47
So we have an equation that looks like va times big m minus little m equals little m times vb, but we don't actually know what any of these values are here.
01:58
So in order to do that we're going to look at a conservation of energy equation for when they fall here.
02:04
So our final initial energy are equal.
02:06
Initially we have gravitational potential, and at the end we have kinetic energy.
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So the speed that either ball is going to be traveling at the bottom is going to be 2gh.
02:17
Again it doesn't depend on their mass here.
02:21
So we know what va is going to be now.
02:23
So awesome.
02:24
So we have root 2gh now times big m minus little m equals m times, well vb we still don't know.
02:32
So how are we going to go find that? we're going to use the fact that this collision here is perfectly elastic.
02:38
So that means that our kinetic energy is going to be conserved.
02:42
So initially what is our kinetic energy? well we have one half little m times va squared, and we also have one half big m times va squared.
02:51
And then at the end the only thing that's moving after the collision is one half little m times vb squared.
02:59
So now we have two, oh i forgot to write vb.
03:02
Now we have two equations and only one unknown.
03:08
That unknown being vb.
03:10
So we can go ahead and get rid of that.
03:11
So i'm going to just plug in va, which we found to be root 2gh into this equation.
03:16
And then i'm going to go ahead and just solve this.
03:18
I'm going to solve this equation here on the left for vb, and then plug that into the equation on the right.
03:24
So that's what vb will equal, and then i'm going to go down here and plug that into this equation here.
03:29
Just going to give myself lots of space here so i can do the manipulations...