Energy in 1D Collisions

How might energy be involved in collision problems?

A question may require the change in kinetic energy to be calculated due to a collision, or to an impulse being applied
Remember that although total energy will be conserved, there may be a change in the kinetic energy of the objects involved in the collision
The kinetic energy of an object can be calculated using $\frac{1}{2} m v^{2}$
If a particle with mass $m_{1}$ and velocity $u_{1}$ collides with a particle of mass $m_{2}$ and velocity $u_{2}$ , then the loss in kinetic energy would be:
- $(\frac{1}{2} m_{1} {u_{1}}^{2} + \frac{1}{2} m_{2} {u_{2}}^{2}) - (\frac{1}{2} m_{1} {v_{1}}^{2} + \frac{1}{2} m_{2} {v_{2}}^{2})$
- This is essentially the difference between the total kinetic energy before the collision, and the total kinetic energy after the collision

When is kinetic energy conserved in collisions?

When $e = 1$ , which would be a perfectly elastic collision, kinetic energy will be conserved, and no energy is lost due to the impact
When $e < 1$ , some energy will be lost due to the collision
- In reality, all collisions will have a coefficient of restitution of less than 1, but we may still choose to model some scenarios as perfectly elastic
- It is also important to understand that energy is not "lost", it is simply transferred to other forms such as heat and sound
There can also be situations where the kinetic energy of a system increases
- For example when a cannon is fired, the cannonball and cannon itself start with zero velocity, and hence zero kinetic energy
- When fired, the cannon ball moves forward, and the cannon recoils backwards, so they now both have velocities, and hence kinetic energy
- In this scenario, the energy has been converted from chemical energy stored in the gunpowder

Exam Tip

As $v^{2}$ is used when finding kinetic energy, it will always be positive (and hence a scalar), so you do not need to enter the negative signs in your calculator when finding the kinetic energy

Worked example

A small smooth sphere A of mass 3 kg moves at 12 ms^-1 on a smooth horizontal table. It collides directly with a second small smooth sphere B of mass 5 kg, which is moving in the opposite direction with a speed of 4 ms^-1. The spheres coalesce and move with velocity $v$ after the collision.

Find the loss of kinetic energy due to the impact.

worked example for a collisions problem finding the loss in kinetic energy due to the impact

Energy in 1D Collisions (Edexcel A Level Further Maths: Further Mechanics 1)

Revision Note

Energy in 1D Collisions

How might energy be involved in collision problems?

When is kinetic energy conserved in collisions?

Exam Tip

Worked example

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Author: Jamie W