Energy transfer in Newton's cradle

[Rya_Sly]

Homework Statement

Three balls of masses m1, m2 and m3 are suspended in a horizontal line by light wires and are almost touching. The mass m1 is given a horizontal velocity v so that it collides head-on with the mass m2.
Find an expression for the final kinetic energy of m3?
What value of m2 results in the maximum energy transfer to the mass m3?

Homework Equations

The Attempt at a Solution

I considered the problem as two successive two body elastic collisions, with one body initially stationary.
For a collision between mass m1 with initial velocity v and stationary mass m2, I transformed in the zero momentum frame (velocity m1*v/(m1*m2) ), found the respective velocities before and after the collision (the balls keep their velocities form before the collision but swap directions) and then transformed back into the lab frame. So after the collision I got:
v1 = (m1-m2)*v/(m1+m2), for the first ball
v2 = 2m1*v/(m1+m2) for the second ball
So the second ball receives a fraction of 4*m1*m2/((m1+m2)**2) kinetic energy from the first ball

Similarly, the third ball receives a fraction of 4*m2*m3/((m2+m3)**2) from the second ball, so the kinetic energy of the third ball at the end is:
8*(m1**2)*(m2**2)*m3*(v**2)/(((m1+m2)**2)*(m2+m3)**2))

And to get m2 for the maximum energy transfer I differentiated this expression with respect to m2 and got m2=sqrt(m1*m3).

Now I am told neither of these results is correct, and I'm not sure where I've gone wrong, or if the answer I'm comparing to is wrong. Also, apologies for the messy maths, I'm not sure how to make the equations display nicer.

 

[mfb]

And to get m2 for the maximum energy transfer I differentiated this expression with respect to m2 and got m2=sqrt(m1*m3).

I get the same result for your assumptions, and it was also my first guess before calculating it.
It could be a trick question, but I would be surprised to see that: a small mass m₂ for a large m₃ will lead to multiple collisions, and might allow to transfer more energy to ball 3.

To format expressions, you can use LaTeX.

 

[gneill]

If you do a web search for "Newton's Cradle analysis", you'll should find this short article, Newton's Cradle by Donald Simanek, which sheds light on some of the shortcomings of simple analysis attempts. Enjoy.