How Does the Velocity of Two Colliding Balls Change After Impact?

[Samyuerux]

Homework Statement

4. The double-ball bounce
Consider 2 balls on top of another that are being dropped from a height h = 2 m,
see the Figure below. They will hit the ground and bounce back and we want to
compute the velocity of the upper ball after the bounce.
(a) (b)
We can model this as a sequence of two collisions. Let us assume that the upper ball
has a mass of 100 g, and that the lower ball has a mass of 10 kg. We rst consider
the collision between the lower ball and the ground.
(a) What is the velocity of the balls just before the lower ball hits the ground?

(b) Assume that the collision between the ball and the ground is perfectly elastic,
what is the velocity of the lower ball just after it hits the ground?
Now let us consider the collision of the upper and the lower ball.
(c) Assume that the collision is elastic with a coecient of restitution e = 0:9.
Compute the velocity of the upper ball after the collision.
(d) What is the maximum height the upper ball reaches after the collision?

The Attempt at a Solution

For this question for part a where two balls are falling and you work out the velocity, because its on top to i count it as one system? or do i calculate the velocities individually?
Just need to clarify this so i can do it correctly :)

 

[Samyuerux]

Just realized they will both have same velocity.. lol
and velocity will be about 2.8 m/s i believe

 

[Samyuerux]

Ok so far i have velocity of balls before hitting the ground using d = 1/2at^2 + Vit
this works out as 2= 5t^2
therefor t = 0.28 seconds
And then using V = a.t i get V at 2.8 seconds (g = 10m/s^2) for this example

Then for velocity after collision i find the Ep (potential energy) = m.g.h = 200
And then use that for Ek = 1/2mv^2
which means 200 = 1/2 10. V^2
so V^2 = 200/5
V = Sqaure root of 40
= 6.325 m/s
Is that correct so far?

 

[rcgldr]

This problem is more complicated in the real world. One issue is if the upper ball bounces off the lower ball before, at the same time, or after the lower ball bounces off the ground. You're probably supposed to assume that the upper ball bounces off the lower ball at the same time the lower ball bounces off the ground. The upper ball is going to end up with much more velocity than the lower ball, and may end up bouncing higher than the initial height it was dropped from, since some of the energy is transferred from the lower ball to the upper ball during a bounce.

 

[asteriatic]

I would approach this problem by first defining the variables and assumptions, such as the mass of the balls, the height they are dropped from, and the coefficient of restitution. I would then use the principles of conservation of energy and momentum to solve for the velocity of the lower ball just before it hits the ground, and the velocity of the lower ball just after it hits the ground assuming a perfectly elastic collision.

For part c, I would use the coefficient of restitution to calculate the velocity of the upper ball after the collision. And for part d, I would use the principles of conservation of energy to calculate the maximum height the upper ball reaches after the collision.

In terms of calculating the velocities, I would treat the two balls as a single system since they are connected and will collide with each other. This will allow for a more accurate calculation of the velocities after the collision. However, if the problem specifically asks for the individual velocities of each ball, then they should be calculated separately.