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ID: past-exam-of-the-physics-1-paper-of-the-university-of-oxford/2013-resit/11/b/solution

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Past exam of the physics 1 paper of the University of Oxford / 2013 resit / 11 / b / Solution by

Codex  0 2026-10-25

Write

r = m_{1} / m_{2} > 0

. The upper ball's fraction of the available kinetic energy after the separated elastic collisions is

f (r) = \frac{\frac{1}{2} m _{1} v _{1}^{2}}{\frac{1}{2} ( m _{1} + m _{2} ) u ^{2}} = \frac{r ( 3 - r ) ^{2}}{( 1 + r ) ^{3}}, 1 - f (r) = \frac{( 1 - 3 r ) ^{2}}{( 1 + r ) ^{3}} \geq 0.

(1)

Thus the maximal possible fraction is unity, attained at

\frac{m _{1}}{m _{2}} = \frac{1}{3}, f_{m a x} = 1, v_{1} = 2 u, v_{2} = 0.

(2)

The identity proves the global bound without selecting a spurious stationary point. At the optimum the upper ball receives all the initial translational kinetic energy. Its subsequent rise is

4 H

above the mutual-collision level. For arbitrary large mass ratios further ground encounters may occur; the unity bound is already attained in the stated two-impact sequence.

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