Physics Help Forum Help with frame of reference question

 Kinematics and Dynamics Kinematics and Dynamics Physics Help Forum

 Nov 4th 2014, 01:58 PM #1 Junior Member   Join Date: Oct 2014 Posts: 8 Help with frame of reference question A superball of mass M and a much lighter superball of mass m are dropped together vertically, with the light ball directly above the heavy ball, but not in contact. Show that the lighter ball will fly just over 8 times the height from which it was dropped. I know that I need to use momentum and different frames of reference, but I'm very confused as to how... Thank you
Nov 4th 2014, 06:10 PM   #2

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 Originally Posted by PhysicsKidd A superball of mass M and a much lighter superball of mass m are dropped together vertically, with the light ball directly above the heavy ball, but not in contact. Show that the lighter ball will fly just over 8 times the height from which it was dropped. I know that I need to use momentum and different frames of reference, but I'm very confused as to how... Thank you
I'm getting closer to 9x but that will all depend on how the masses are chosen.

Here are some highlights to get you started. You will need to verify the steps on your own.

I am going to assume all collisions are elastic. That means when the larger ball hits the ground it is perfectly reflected. Let v1 = sqrt(2gh) downward be the velocity of the large ball when it hits the floor. (Prove this.) So immediately after it hits the floor it has a velocity v1 upward. Additionally the small ball with also have a velocity of v1 downward just before the collision with the large ball. (Why?)

Calling +y upward, we have from momentum conservation:
Mv1 - mv1 = MV + mv
where V and v are unknowns.

Likewise energy is conserved in elastic collisions, so
(1/2)Mv1^2 + (1/2)mv1^2 = (1/2)MV^2 + (1/2)mv^2

So we have two equations in the unknowns v and V. We wish to find v.

First solve the momentum equation for V:
V = (1/M)(Mv1 - mv1 - mv)

Now put that value of V into the energy equation. Skipping along a bit I get the following result. (See the attachment below.)

As ugly as this looks, it's a quadratic. Now, I'm not going to do it, but you can get some numbers in to do an approximation. That'll make it easier to work with. I don't know if your instructor would like it that way. You'll have to make a judgement call.

Okay, solving the quadratic equations is a bear, but possible. Just do it one step at a time. I get two possible results:
1) v = -v1. This implies the smaller ball as passed right through the larger one. Not possible.

2) v = (3M/(M + m))v1 - m/(M + m)v1 As this gives an upward velocity for the small ball we are good with this.

Now, to see how high (H) the smaller ball is after the collision we get H = v^2/(2g). I'm going to use M = 100 kg and m = 1 kg. A little extreme I suppose. You can use any mass ratio you like. I get v = 3*v1 (approximately). This gives H = 9h.

-Dan
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 Nov 5th 2014, 01:02 PM #3 Junior Member   Join Date: Oct 2014 Posts: 8 Thank you, that was very helpful, but do you know how to do that with using the instantaneous rest frame of the large ball just before it hits the smaller one? Thank you!
 Nov 5th 2014, 01:13 PM #4 Forum Admin     Join Date: Apr 2008 Location: On the dance floor, baby! Posts: 2,454 I think what you are looking for is something called a "Breit Wall" frame. (Better known as the "brick wall.") In this frame we would set the smaller ball to hit the larger one like a "brick wall." ie We are talking about a reference frame where the smaller ball hits the larger ball as if the larger ball doesn't move at all. So the smaller ball rebounds with exactly the same speed as when it hit. Of course, if you calculate V by the solution I gave you, you will find that the larger ball mostly sort of just stays in contact with the floor anyway, so the Breit and lab frames are essentially the same anyway. It's hard to find a good reference for the Breit frame. I did come across this lecture but I have not had time to review it to see how good it is. Feel free to take a look....It probably has some good collision explanations in there as well. -Dan __________________ Do not meddle in the affairs of dragons for you are crunchy and taste good with ketchup. See the forum rules here.

 Tags frame, question, reference