Result

Perfectly inelastic collision

v_{f} = \frac{m _{1} v _{1 i} + m _{2} v _{2 i}}{m _{1} + m _{2}}

Statement

The common final velocity of two objects that collide and stick together.

Symbols and units

$v_{f}$	common final velocity	m/s
$m_{1}, m_{2}$	masses	kg
$v_{1 i}, v_{2 i}$	initial velocities	m/s

When it applies

Two objects that collide and move together afterward, in the absence of external horizontal force.

Don't use when

Only for objects that move together after impact — an elastic or partially elastic collision needs separate momentum and energy equations.

Derivation

Momentum is conserved even though kinetic energy is not:

m_{1} v_{1 i} + m_{2} v_{2 i} = (m_{1} + m_{2}) v_{f}

Solve for the single final velocity:

v_{f} = \frac{m _{1} v _{1 i} + m _{2} v _{2 i}}{m _{1} + m _{2}}

Worked examples

Coupling train cars

A 20,000 kg car at 3.0 m/s couples with an identical stationary car:

v_{f} = \frac{( 20000 ) ( 3.0 ) + ( 20000 ) ( 0 )}{40000} = 1.5 m/s

Half the initial kinetic energy is lost to the coupling — characteristic of a perfectly inelastic collision between equal masses.

Watch out for

Kinetic energy is always lost in a perfectly inelastic collision (except in the trivial case of equal and opposite momenta cancelling to rest) — never assume it is conserved here.

Connects to

Conservation of momentum