What happens to the component perpendicular to the wall?

It is reversed and multiplied by e.

How do you treat successive impacts?

One at a time: the velocity after one impact is the velocity before the next.

Two perpendicular walls with the same e: what is the final velocity?

-e times the initial velocity (parallel to the original path).

When is the final path not parallel to the original?

When the two coefficients of restitution are different.

How is the angle with a wall changed by an impact?

\tan\theta{after}=e\tan\theta{before}

How do you find the angle between the path and a wall?

\tan\theta=\frac{perpendicular component}{parallel component}

How do you find the speed after impacts?

Pythagoras on the final components.

How do you find total kinetic energy lost over several impacts?

\frac12m(u^2-v^2) with the initial and final speeds.

Does the order of impacts at two perpendicular walls change the final velocity?

No; each wall changes a different component.

Successive oblique impactsEdexcel A-Level Further Maths: Flashcards

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At a smooth wall, which velocity component is unchanged?

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At a smooth wall, which velocity component is unchanged?: The component parallel to the wall.
What happens to the component perpendicular to the wall?: It is reversed and multiplied by $e$ .
A wall parallel to $\mathbf{i}$ changes which component?: Only the $\mathbf{j}$ -component.
How do you treat successive impacts?: One at a time: the velocity after one impact is the velocity before the next.
Two perpendicular walls with the same $e$ : what is the final velocity?: $-e$ times the initial velocity (parallel to the original path).
When is the final path not parallel to the original?: When the two coefficients of restitution are different.
Component perpendicular to two parallel walls after $n$ impacts?: $e^n$ times the original.
How is the angle with a wall changed by an impact?: $\tan\theta_{\text{after}}=e\tan\theta_{\text{before}}$
How do you find the angle between the path and a wall?: $\tan\theta=\frac{\text{perpendicular component}}{\text{parallel component}}$
How do you find the speed after impacts?: Pythagoras on the final components.
How do you find total kinetic energy lost over several impacts?: $\frac12m(u^2-v^2)$ with the initial and final speeds.
Does the order of impacts at two perpendicular walls change the final velocity?: No; each wall changes a different component.

Exam questions on Successive oblique impacts

A smooth particle of mass $0.5$ kg moves on a smooth horizontal table. Two fixed smooth vertical walls $W_1$ and $W_2$ meet at right angles: $W_1$ is parallel to $\mathbf{i}$ and $W_2$ is parallel to $\mathbf{j}$ . The particle has velocity $(8\mathbf{i}+6\mathbf{j})$ m s $^{-1}$ , hits $W_1$ and then hits $W_2$ . The coefficient of restitution between the particle and each wall is $0.5$ .
Find the total kinetic energy lost by the particle in the two impacts.2 marks
A smooth sphere of mass $0.2$ kg moves on a smooth horizontal surface between two fixed parallel smooth vertical walls $W_1$ and $W_2$ . It hits $W_1$ with speed $12$ m s $^{-1}$ , its direction of motion making an angle of $60^\circ$ with the wall, and then crosses to hit $W_2$ . The coefficient of restitution between the sphere and each wall is $\frac23$ .
Find the total kinetic energy lost by the sphere in the two impacts.2 marks
A smooth particle $P$ of mass $0.4$ kg moves on a smooth horizontal surface with velocity $(6\mathbf{i}+8\mathbf{j})$ m s $^{-1}$ . It hits a fixed smooth vertical wall $W_1$ that is parallel to $\mathbf{j}$ , and then hits a fixed smooth vertical wall $W_2$ that is parallel to $\mathbf{i}$ . The coefficient of restitution between $P$ and $W_1$ is $\frac13$ , and between $P$ and $W_2$ is $\frac12$ .
Find the speed of $P$ immediately after it hits $W_1$ , and the angle between its path and $W_1$ .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).