Conservation of momentum

In a closed system:

total momentum before an event = total momentum after the event

A 'closed system' is something that is not affected by external forces. This is called the principle of conservation of momentum. Momentum is conserved in collisions and explosions.

Conservation of momentum explains why a gun or cannon recoils backwards when it is fired. When a cannon is fired, the cannon ball gains forward momentum and the cannon gains backward momentum. Before the cannon is fired (the 'event'), the total momentum is zero. This is because neither object is moving. The total momentum of the cannon and the cannon ball after being fired is also zero, with the cannon and cannon ball moving in opposite directions.

Calculations involving collisions

Collisions are often investigated using small trolleys. The diagrams show an example.

There are two trolleys, red and blue, The blue trolley is heading towards the stationary red one. There is an arrow above the trolley to indicate motion and direction.

Two trolleys have collided and are shown as being together. Combined weights of the trolleys are shown.

Before collision

After collision

You can use the principle of conservation of momentum to calculate the velocity of the combined trolleys after the collision.

Example calculation

Calculate the velocity of the trolleys after the collision in the example above.

First calculate the momentum of both trolleys before the collision:

2 kg trolley = 2 × 3 = 6 kg m/s

8 kg trolley = 8 × 0 = 0 kg m/s

Total momentum before collision = 6 + 0 = 6 kg m/s

Total momentum (p) after collision = 6 kg m/s (because momentum is conserved)

Mass (m) after collision = 10 kg

Next, rearrange p = m v to find v:

  v = \frac{p}{m}

v = 6 \div 10

v = 0.6 \ m/s

Note that the 2 kg trolley is travelling to the right before the collision. As its velocity and the calculated velocity after the collision are both positive values, the combined trolleys must also be moving to the right after the collision.

Calculations involving explosions

The principle of conservation of momentum can be used to calculate the velocity of objects after an explosion.

Example calculation

A cannon ball of mass 4.0 kg is fired from a stationary 96 kg cannon at 120 m/s. Calculate the velocity of the cannon immediately after firing.

Total momentum of cannon and cannon ball before = 0 kg m/s (because neither object is moving)

Total momentum of cannon and cannon ball after collision = 0 kg m/s (because momentum is conserved)

Momentum of cannon ball after firing = 4.0 × 120 = 480 kg m/s

Momentum of cannon after firing = -480 kg m/s (because it recoils in the opposite direction)

Rearrange p = m v to find v:

  v = \frac{p}{m}

v = 480 \div 96

v = -5.0 \ m/s

Note that the forward velocity of the cannon ball was given a positive value. The negative value for the cannon's velocity shows that it moved in the opposite direction.

curriculum-key-fact
Momentum is a vector quantity, so it is important to consider both magnitude and direction. For example, if travelling east is given a positive value, travelling west is given a negative value.