Prove that kinetic energy is not conserved in inelastic collision

Question

A ball of mass $m$ moving with velocity $u$ makes a perfectly inelastic collision with a stationary ball of equal mass $m$ . Prove that exactly half the initial kinetic energy is lost.

Solution — Step by Step

Before collision:

Ball 1: mass $m$ , velocity $u$
Ball 2: mass $m$ , velocity 0

In a perfectly inelastic collision, the two bodies stick together and move with a common velocity $v$ after the collision.

By conservation of momentum (momentum is always conserved in all collisions):

mu + m \times 0 = (m + m)v

mu = 2mv

v = \frac{u}{2}

KE_{before} = \frac{1}{2}mu^2 + \frac{1}{2}m(0)^2 = \frac{1}{2}mu^2

After collision, total mass = $2m$ , velocity = $\frac{u}{2}$ :

KE_{after} = \frac{1}{2}(2m)\left(\frac{u}{2}\right)^2 = \frac{1}{2}(2m) \cdot \frac{u^2}{4} = \frac{mu^2}{4}

KE_{lost} = KE_{before} - KE_{after} = \frac{1}{2}mu^2 - \frac{1}{4}mu^2 = \frac{1}{4}mu^2

\frac{KE_{lost}}{KE_{before}} = \frac{\frac{1}{4}mu^2}{\frac{1}{2}mu^2} = \frac{1}{2} = 50\%

Exactly half the kinetic energy is lost in a perfectly inelastic collision between equal masses. Since $KE_{after} < KE_{before}$ , kinetic energy is NOT conserved.

Why This Works

Kinetic energy is lost to deformation, heat, and sound during the collision. When two equal masses collide and stick, the centre of mass frame shows that both masses were initially moving toward the centre of mass, and after sticking, there’s no relative motion — all kinetic energy in the CM frame is gone.

Momentum is conserved because Newton’s third law ensures the forces between the two balls are equal and opposite — the impulse they exchange cancels.

Alternative Method — Using the General Formula

For a general perfectly inelastic collision of mass $m_1$ (velocity $u$ ) with $m_2$ (at rest):

v = \frac{m_1 u}{m_1 + m_2}

\frac{KE_{lost}}{KE_{initial}} = \frac{m_2}{m_1 + m_2}

For $m_1 = m_2 = m$ : fraction lost = $\frac{m}{2m} = \frac{1}{2}$ . Same result, derived generally.

Common Mistake

Students sometimes apply conservation of kinetic energy directly (writing $\frac{1}{2}mu^2 = \frac{1}{2}(2m)v^2$ ), which gives $v = \frac{u}{\sqrt{2}}$ . This is WRONG — kinetic energy is NOT conserved in an inelastic collision. Momentum must be conserved to find $v$ . Energy conservation applies only to elastic collisions.

Question

Solution — Step by Step

Why This Works

Alternative Method — Using the General Formula

Common Mistake

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