Consider two particles of masses and approaching each other head-on with equal speeds…

PHYSICS · Multi-part · AP · Physics 1: Algebra-based · Linear Momentum · Linear Momentum

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. (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

. (ii) By treating

K_{\text{final}}(e)

as a quadratic function, determine the value of

e

for which the kinetic energy loss is maximum. (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.

Related theory

Collisions
Collisions in one dimension
Work, Energy and Power
Linear momentum
Types of collisions (elastic and inelastic)
Elastic and inelastic collisions

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