# 3.2.1 Conservation of Momentum

### The Principle of Conservation of Momentum

• The principle of conservation of momentum is:
• The total momentum of a system remains constant provided no external force acts on it
• For example if two objects collide:

the total momentum before the collision = the total momentum after the collision

• Remember momentum is a vector quantity. This allows oppositely-directed vectors to cancel out so the momentum of the system as a whole is zero
• Momentum is always conserved over time

#### External and Internal Forces

• External forces are forces that act on a structure from outside e.g. friction and weight
• Internal forces are forces exchanged by the particles in the system e.g. tension in a string
• Which forces are internal or external will depend on the system itself, as shown in the diagram below:

Internal and external forces on a mass on a spring

• You may also come across a system with no external forces being described as a ‘closed’ or ‘isolated’ system
• These all still refer to a system that is not affected by external forces
• For example, a swimmer diving from a boat:
• The diver will move forward, and, to conserve momentum, the boat will move backwards
• This is because the momentum beforehand was zero and no external forces are present to affect the motion of the diver or the boat

### Collisions in One & Two Dimensions

#### One-dimensional momentum problems

• Momentum (p) is equal to: p = m × v
• Using the conversation of linear momentum, it is possible to calculate missing velocities and masses of components in the system. This is shown in the example below

• To find out whether a collision is elastic or inelastic, compare the kinetic energy before and after the collision
• If the kinetic energy is conserved, it is an elastic collision
• If the kinetic energy is not conserved, it is an inelastic collision
• Elastic collisions are commonly those where objects colliding do not stick together and then move in opposite directions
• Inelastic collision are where objects collide and stick together after the collision

#### Two-dimensional momentum problems

• Since momentum is a vector, in 2D it can be split up into its x and y components
• Review revision notes 1.3 Scalars & Vectors on how to resolve vectors

#### Exam Tip

If an object is stationary or at rest, it’s velocity equals 0, therefore, the momentum and kinetic energy are also equal to 0.

When a collision occurs in which two objects are stuck together, treat the final object as a single object with a mass equal to the sum of the two individual objects.

In 2D problems, make sure you’re confident resolving vectors. Here is a small trick to remember which component is cosine or sine of the angle for a vector R:

Resolving vectors with sine and cosine

### Author: Ashika

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.
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