Elastic and Inelastic Collisions

In all collisions (and explosions), total momentum is conserved provided no external resultant force acts. The distinction between elastic and inelastic is about kinetic energy.

Elastic collisions

Total kinetic energy before = total kinetic energy after.

No kinetic energy is converted to other forms. Truly elastic collisions are rare at the macroscopic scale (bouncing steel balls come close).

Inelastic collisions

Total kinetic energy after $<$ total kinetic energy before.

Some kinetic energy is converted to thermal energy, sound, or deformation energy. Most real collisions are inelastic.

Perfectly inelastic collisions

The objects stick together and move as one mass. Maximum kinetic energy is lost (consistent with conservation of momentum).

$m_1 u_1 + m_2 u_2 = (m_1 + m_2)v$

Example: A 2 kg trolley at 3 m/s collides with a stationary 4 kg trolley. They stick together.

$2(3) + 4(0) = (2 + 4)v$ $v = 1 \text{ m/s}$

KE before $= \frac{1}{2}(2)(3^2) = 9$ J

KE after $= \frac{1}{2}(6)(1^2) = 3$ J

KE lost $= 6$ J (converted to thermal energy and sound)

Common errors and how to correct them

• Thinking momentum is not conserved in inelastic collisions. Momentum is ALWAYS conserved (no external force). Only kinetic energy differs.
• Forgetting to assign negative velocity to objects moving in the opposite direction.

How to apply this in an exam

Calculate total momentum before and after (they must be equal). Then calculate KE before and after to determine if elastic or inelastic.