Restitution – A Level Further Mathematics OCR Revision

Introduction to Restitution

Restitution refers to the process in a collision where energy is conserved and objects return to their initial shape and size.
It is associated with the coefficient of restitution, a value that describes how much kinetic energy is preserved in a collision.
When the coefficient of restitution equals one, the collision is perfectly elastic. This means kinetic energy is conserved, and the objects return to their original shapes and sizes after collision.
In contrast, when the coefficient of restitution equals zero, the collision is perfectly inelastic. Kinetic energy is not conserved.

The coefficient of restitution is given by the relative speed after the collision divided by the relative speed before the collision. Mathematically, this is shown as e = (v2 - v1) / (u1 - u2) where ‘e’ is the coefficient of restitution, ‘v’ values are velocities after the collision, and ‘u’ values are velocities before collision.
This equation holds in one-dimensional collisions but can be expanded to multiple dimensions by considering the velocities in each direction separately.
The coefficient of restitution can also be determined experimentally, by measuring velocities before and after a collision, and solving the above formula for ‘e’.

Restitution is a critical concept in many areas of physics and engineering, including automotive safety, sports equipment design, and material science testing.
For example, the design of helmets or car bumpers often considers the materials’ coefficients of restitution to maximise energy absorption and minimise damage.
Similarly, engineers designing sports balls or racquet strings will consider restitution to provide optimal performance and energy transfer.

Higher coefficients of restitution generally mean more efficient energy transfer. This is because less kinetic energy is lost to deformation or heat.
On the other hand, a lower coefficient of restitution means more energy is absorbed by the object, which can be beneficial in certain applications, such as safety devices.

Understanding restitution allows us to analyse and predict the outcomes of collisions, be they between two pool balls, cars in an accident, or particles in a gas.
By taking into account the objects’ initial velocities and their coefficients of restitution, we can predict their final velocities after the collision. This is crucial in many branches of physics and engineering.