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## Homework Statement

An executive toy consists of three suspended steel balls of masses M, n and m arranged in order with their centres in a horizontal line. The ball of mass M is drawn aside in their common plane until its centre has been raised by h and is then released. if M ≠ m and all collisions are elastic, how must n be chosen so that the ball of mass m rises to the gratest possible height? What is that height? (Neglect multiple collisions)

## Homework Equations

velocity after a perfectly elastic collision

v2' = (m2 - m1)v2/(m1+m2) + 2m1v1/(m1+m2)

## The Attempt at a Solution

After releasing M, its vellocity immediately before the first collision is

[itex]V = \sqrt{2gh}[/itex]

Then, M collides with n, and n's velocity immediately after collision is

[itex]v = \frac{2M}{M+n}\sqrt{2gh}[/itex]

Analogously, m's velocity immediately after n colides

[itex]u = \frac{2n}{n+m}\frac{2M}{M+n}\sqrt{2gh}[/itex]

m shall rises H

[itex]mu^{2}/2 = mgH[/itex]

[itex]H = 16h\frac{M^{2}n^{2}}{(M+n)^{2}(m+n)^{2}} [/itex]

How am I supposed to maximize H from it? What should be done to solve it?

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