Q.A simple pendulum has a bob A of mass $m$ on a string of length 1 m. The bob A is pulled aside until the string is horizontal (level with the pivot) and released from rest. At the lowest point of its swing, A strikes a second bob B of the same mass $m$ that rests on a table exactly at that lowest point. The collision is elastic and the sizes of the bobs may be neglected. Taking $g = 9.8\ \text{m s}^{-2}$, calculate

Elastic Collision – From Intuition to Precision

Imagine two billiard balls on a table. They smash into each other, bounce apart, and keep moving. Now imagine two lumps of clay that collide and stick together — they stop moving as separate objects. The first case feels like nothing is lost; the second clearly loses motion. That feeling is the seed of the idea.

An elastic collision is the idealised version of that first case: a collision where no kinetic energy is turned into heat, sound, or permanent deformation. All the motion-energy that went in comes out again as motion-energy of the same total amount. The objects bounce perfectly.

The precise statement

For any collision between two bodies (call them $A$ and $B$), two things are always true if no external force acts:

1. Total linear momentum is conserved — this is a law of physics, always.
2. Total kinetic energy may or may not be conserved — that depends on the nature of the collision.

An elastic collision is defined by the second condition: both total momentum and total kinetic energy are conserved.

Let the masses be $m_1$, $m_2$ and the velocities before collision be $u_1$, $u_2$; after collision, $v_1$, $v_2$. Then:

These two equations together define an elastic collision.

What this means physically

In an elastic collision, the objects do not get dented, heated, or stuck. They exchange energy and momentum purely through reversible deformation — like two perfect springs that compress and then fully recover. Real collisions are never perfectly elastic (some energy always leaks into sound or heat), but many are close enough: billiard balls, steel ball bearings, gas molecules.

A useful derived result

From the two conservation equations, you can derive a neat relation for one-dimensional collisions:

That is, the relative speed of approach equals the relative speed of separation. This is often easier to use than the full energy equation in problems. …