Conservation of Momentum

The principle of conservation of momentum states that the total momentum of a system of objects is conserved (stays the same) provided no external resultant force acts on the system.

$\text{total momentum before} = \text{total momentum after}$

Collisions

In a collision between two objects, momentum is conserved. The objects may bounce apart or stick together.

Objects stick together (perfectly inelastic):

$m_1 v_1 + m_2 v_2 = (m_1 + m_2) v$

Objects bounce apart:

$m_1 u_1 + m_2 u_2 = m_1 v_1 + m_2 v_2$

Explosions

Before an explosion, the total momentum is zero (both parts are at rest). After the explosion, the total momentum must still be zero, so the parts move in opposite directions with equal and opposite momenta.

$0 = m_1 v_1 + m_2 v_2$

Worked example: collision

A 2.0 kg trolley moving at 3.0 m/s collides with a stationary 1.0 kg trolley. They stick together. Find the velocity after the collision.

$2.0 \times 3.0 + 1.0 \times 0 = (2.0 + 1.0) \times v$ $6.0 = 3.0v$ $v = 2.0\text{ m/s}$

Worked example: explosion

A 3.0 kg gun fires a 0.020 kg bullet at 400 m/s. Find the recoil velocity of the gun.

$0 = 3.0 \times v_g + 0.020 \times 400$ $0 = 3.0v_g + 8.0$ $v_g = -2.7\text{ m/s}$

The gun recoils at 2.7 m/s in the opposite direction to the bullet.

Common errors and how to correct them

Ignoring direction and adding all momenta as positive. Assign positive and negative directions. Objects moving left when right is positive have negative velocity.

Using the wrong mass after collision. When objects stick together, the combined mass is $m_1 + m_2$. When they bounce, each object keeps its own mass.

How to apply this in an exam

Write the conservation equation: total momentum before = total momentum after. Show all substitutions clearly. State the direction of the final velocity.