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Oblique Collision

Show that kin...

Question

Show that kinetic energy is always lost in inelastic collision.

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Solution

From conservation of momentum
$m_{1} v_{1} = (m_{1} + m_{2}) v_{2} \to v_{2} = \frac{m_{1}}{m_{1} + m_{2}} v_{1}$
The ratio of kinetic energies before & after collision is

$\frac{K E_{f}}{K E_{i}} = \frac{\frac{1}{2} (m_{1} + m_{2}) {[\frac{m_{1}}{m_{1} + m_{2}} v_{1}]}_{1}^{2}}{\frac{1}{2} m_{1} v_{1}^{2}} = \frac{m_{1}}{m_{1} + m_{2}}$

The fraction of kinetic energy lost is
$\frac{K E_{i} - K E_{f}}{K E_{i}} = \frac{[1 - \frac{m_{1}}{m_{1} + m_{2}}]}{K E_{i}} K E_{i} = \frac{m_{2}}{m_{1} + m_{2}}$
Hence energy always loss in inelastic collision.

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