Ratio of masses for maximum energy transfer in elastic collision?

[gaobo9109]

Homework Statement

Two elastic balls of mass m₁ and m₂ are placed on top of each other with a small gap between them and then dropped onto the ground. What is ratio m₁/m₂, for which the upper ball m₁ ultimately receives the largest fraction of the total energy? (m₁<m₂)

Homework Equations

The Attempt at a Solution

At the instant when lower ball(m₂) hit the ground, both balls will have the same velocity u₁. And when the lower ball bounces up, it will have a velocity of -u₁. However, the problem is that there is a gap between the two balls, and if I introduce another variable h, I don't think this variable can be eliminated at the end to give the answer m₁/m₂ = 1/3.

 

[kuruman]

Don't worry about the gap. The approximation here is that the collision is over very fast, i.e. fast enough so that gravity does not affect the motion during the collision. Then we can say that momentum is conserved because no (appreciable) external forces act on the system. Treat the problem so that the lower ball hits the ground first, reverses direction and collides with the upper ball while it is still moving down.

 

[gaobo9109]

Then I think it is correct to say that downward velocity of the upper ball is u¹ and upward velocity of lower ball is -u₁. Then,

m₁u₁-m₂u₁=m₁v₁ + m₂v₂

u₁-u₁=v₂-v₁
Since we are interested only on the velocity of the upper ball,so
v₂=2u₁+v₁
Substitute this equation into previous equation
u₁ (m₁-m₂) = m₁v₁ + m₂(2u₁+v₁)
u₁ (m₁-3m₂)=v₁(m₁+m₂)
v₁=u₁(m₁-3m₂)/(m₁+m₂)
But I don't know how to continue from here to find maximum kinetic energy gained?

 

[kuruman]

Then I think it is correct to say that downward velocity of the upper ball is u¹ and upward velocity of lower ball is -u₁.

I like to think of "down" as negative and "up" as positive. Please humor me.

m₁u₁-m₂u₁=m₁v₁ + m₂v₂

u₁-u₁=v₂-v₁

The first equation is OK. The second equation does not follow because the masses are not equal so you cannot cancel them. Besides, isn't u₁-u₁ always zero?

Look, you have two unknowns, the final velocities of each ball. Momentum conservation is only one equation relating these unknowns. You need a second equation. What is that equation?

 

[gaobo9109]

Sorry that is a typing error. The actual equation is supposed to be like this
u₁-(-u₁)=v₂-v₁

Initial relative velocity = final relative velocity

 

[kuruman]

v₁=u₁(m₁-3m₂)/(m₁+m₂)
But I don't know how to continue from here to find maximum kinetic energy gained?

Correct so far for the velocity of the top ball. Now you need to find the kinetic energy gained by it after the collision, then worry about its maximum. Can you find an expression for the KE gained? If "yes", write it in terms of the ratio of the masses μ = m₁/m₂, then see for what value of μ your expression is maximum.