Physics

Problem

Consider two particles of masses

m

and

2m

approaching each other head-on with equal speeds

u

on a frictionless line. The coefficient of restitution is variable. a. (i) Using algebraic reasoning, express the total kinetic energy of the system after collision,

K_{\text{final}}

, as a function of the coefficient of restitution

e

. b. (ii) By treating

K_{\text{final}}(e)

as a quadratic function, determine the value of

e

for which the kinetic energy loss is maximum. c. (iii) Prove that

K_{\text{final}}\le K_{\text{initial}}

for every real value of

e

in the interval

0\le e\le1

, and hence show that the kinetic energy of an isolated two-body system cannot increase merely because of collision.

Intuitive explanation

When two objects collide, momentum is always conserved, but energy loss depends on how "bouncy" the collision is—that's what the coefficient of restitution $e$ controls. Here, the equal speeds and mass ratio create a special symmetry: you can use momentum conservation plus the restitution equation to find both final velocities for any value of $e$ . The key insight is that $e$ ranges from 0 (perfectly inelastic, they stick) to 1 (perfectly elastic, maximum bounce), and each value produces different outcomes worth exploring.

Intuitive explanation

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