You set a small ball of mass m atop a large ball of mass M ≫m and drop the pair from height h .

You set a small ball of mass m atop a large ball of mass M...

You set a small ball of mass $m$ atop a large ball of mass $M \gg m$ and drop the pair from height $h .$ Assuming the balls are perfectly elastic, show that the smaller ball rebounds to height $9 h$.

Key Concepts

Elastic Collision

An elastic collision is one where no kinetic energy is lost and both momentum and kinetic energy are conserved during the interaction. This principle is critical for determining the velocities of colliding objects after impact, especially when one object's motion is reversed relative to the other.

Conservation of Momentum and Energy

In any collision, especially in elastic ones, the total momentum and total kinetic energy of the system remain constant. These conservation laws allow us to set up equations relating the pre- and post-collision velocities, which are essential for solving the dynamics of the problem.

Relative Motion and Frame of Reference Analysis

Analyzing the collision in a suitable reference frame, such as the center-of-mass frame, simplifies the understanding of how relative velocities are transformed during an elastic collision. In this frame, the reversal of relative velocity is a key concept that directly influences the rebound height of the smaller object.

Mass Ratio Approximation

When one mass is significantly larger than the other (M >> m), the motion of the larger mass can be approximated as nearly unchanged during the collision. This approximation simplifies the problem and allows for the derivation of results, such as an enhanced bounce height for the smaller object.

Transcript

00:01 Okay, in this problem, you have a small ball and a large ball all dropping, and we want to show that smaller ball goes to a height 9 -h.

00:10 So it's just kind of interesting.

00:12 I hope it doesn't get too complicated, but we'll see.

00:17 So let's draw this problem out, and then m is bigger, much bigger than this m.

00:32 Okay, and oh yeah, let's distinguish this.

00:35 There's our big old m.

00:37 So their center of mass, let's say, is at a height h from the ground.

00:46 So, and then when they drop this one, so when this one hits the ground, this one's going to rebound and hit the upper one.

00:59 And then it's going to impart momentum to send the other mass to height 9h.

01:05 That's kind of crazy.

01:05 It's 9h, and you don't even have to know anything about the masses.

01:08 This is an interesting problem.

01:14 So let's think about what's going on.

01:16 So as this mask gets down here and it rebounds, let's pause to think how we can get its final velocity.

01:31 So the balls are perfectly elastic.

01:34 You know, i'm inclined to say that this, as they go, they're, you know, we can figure out their speed by doing, we can do energy conservation.

01:49 So for either ball, mgh is one half mb squared.

01:53 The m's cancel.

01:54 So you get b is going to be the square root of 2gh.

01:59 And i'm assuming since they don't give us any geometric thing, so we can neglect the size of the ball and say that they're all on average about the same height.

02:07 So anyway, that's going to be their velocities when they get to the bottom.

02:11 Another question to me is, so how do i prove that this ball is going to bounce back up with the same velocity as it once had? i think that's kind of intuitive.

02:27 I don't see how you could model the problem any differently if you didn't know anything about the ball or sorry, if you didn't know anything about the wall.

02:36 So maybe i'll come back to that and see if i can find any way to prove it...

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