How do you handle successive impacts?

One at a time: velocities after one impact are the velocities before the next.

Velocities after impact for equal masses, one initially at rest at speed u?

\frac{u(1-e)}{2} and \frac{u(1+e)}{2}

When will two spheres on a line collide again?

When they approach each other, or the rear sphere moves faster than the front sphere.

When is there no further collision?

When the spheres separate and no sphere is moving towards a wall.

How do you show a further collision in an exam?

State the velocities and directions, and the reason they approach.

Which law applies between two spheres?

Conservation of momentum and Newton's law of restitution.

Which coefficient do you use when a sphere hits a wall?

The coefficient of restitution between the sphere and the wall.

Why keep one positive direction?

Rebounds change the sign of a velocity, so signs prevent mistakes.

Successive direct impactsEdexcel A-Level Further Maths: Flashcards

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What happens to a ball's speed at each impact with a smooth floor?

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What happens to a ball's speed at each impact with a smooth floor?: It is multiplied by $e$ .
By what factor does the bounce height change at each impact?: $e^2$
How do you handle successive impacts?: One at a time: velocities after one impact are the velocities before the next.
Rebound speed of a sphere hitting a fixed wall at right angles at speed $u$ ?: $eu$
Velocities after impact for equal masses, one initially at rest at speed $u$ ?: $\frac{u(1-e)}{2}$ and $\frac{u(1+e)}{2}$
When will two spheres on a line collide again?: When they approach each other, or the rear sphere moves faster than the front sphere.
When is there no further collision?: When the spheres separate and no sphere is moving towards a wall.
How do you show a further collision in an exam?: State the velocities and directions, and the reason they approach.
Speed of a ball after the third impact with the floor, starting at speed $u$ ?: $e^3u$
Which law applies between two spheres?: Conservation of momentum and Newton's law of restitution.
Which coefficient do you use when a sphere hits a wall?: The coefficient of restitution between the sphere and the wall.
Why keep one positive direction?: Rebounds change the sign of a velocity, so signs prevent mistakes.

Exam questions on Successive direct impacts

A small ball is dropped from rest at a height of $2.5$ m above a smooth horizontal floor. The coefficient of restitution between the ball and the floor is $0.8$ . Take $g=9.8$ m s $^{-2}$ .
Find the greatest height reached by the ball after its second bounce.2 marks
Two smooth spheres $P$ and $Q$ of equal radii and equal mass $1$ kg lie on a smooth horizontal surface, in a straight line perpendicular to a fixed smooth vertical wall, with $Q$ between $P$ and the wall. $Q$ is at rest and $P$ moves towards $Q$ at $4$ m s $^{-1}$ . The coefficient of restitution between $P$ and $Q$ is $0.5$ , and between $Q$ and the wall is $\frac23$ .
Explain why $P$ and $Q$ collide a second time.2 marks
Three smooth spheres $A$ , $B$ and $C$ of equal radii lie in a straight line on a smooth horizontal surface, with $B$ between $A$ and $C$ . $A$ has mass $2m$ , and $B$ and $C$ each have mass $m$ . $B$ and $C$ are at rest. $A$ moves towards $B$ at $6$ m s $^{-1}$ . The coefficient of restitution between any two of the spheres is $\frac12$ .
Find the speed of $B$ immediately after the first collision between $A$ and $B$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).