Conservation of linear momentum, impulse as change in momentum, the coefficient of restitution and Newton's experimental law, direct and oblique impacts, and successive collisions.

What this dot point is asking

AQA wants you to apply conservation of linear momentum to collisions, use impulse as the change in momentum, apply Newton's experimental law with the coefficient of restitution, handle direct and oblique impacts including impacts with a wall, and work through successive collisions.

Conservation of momentum and impulse

The coefficient of restitution

The two equations you always use together for a direct collision are conservation of momentum and Newton's experimental law. Momentum gives one equation linking the two unknown final velocities; restitution gives a second. With two equations in two unknowns the velocities are determined uniquely, so the method is mechanical once you have fixed a positive direction and written the speed of approach and speed of separation carefully.

Oblique impacts

When a sphere strikes a smooth wall obliquely, resolve the velocity into a component parallel to the wall and a component perpendicular to it. Because the wall is smooth it exerts no force along its own surface, so the parallel component is unchanged. The perpendicular component is reversed in direction and multiplied by $e$, exactly as in a direct impact against a fixed surface. Treat the two components separately, then recombine with Pythagoras for the new speed and with an inverse tangent for the new angle. For an oblique impact between two smooth spheres the same idea applies along the line of centres (where momentum and restitution act) while the components perpendicular to that line are unchanged for each sphere.

Successive collisions

A ball bouncing vertically on a fixed floor loses a factor of $e$ in speed at each bounce, because the floor is the fixed surface in Newton's law. If it leaves the floor at speed $u$, it returns at $eu$, then $e^2 u$, and so on, so the rebound speeds form a geometric sequence with ratio $e$. The rebound heights, proportional to the square of the speed, form a geometric sequence with ratio $e^2$. Because $0 < e < 1$ these sequences converge, so the total distance travelled and the total time until the ball comes to rest are both finite and can be summed as geometric series, a standard exam application that ties this topic to sequences.