What happens to the velocity component perpendicular to the line of centres?

It is unchanged for each sphere.

For a smooth wall, which component is unchanged?

The component parallel to the wall.

What is the rebound rule for the normal component at a fixed wall?

It is reversed and multiplied by e.

Which two laws apply along the line of centres?

Conservation of momentum and Newton's law of restitution.

How do you find the speed after impact?

Combine the two components with Pythagoras.

How are the angles before and after impact at a wall related?

\tan\theta{after}=e\tan\theta{before}, with \theta measured to the wall.

How do you handle vector form?

Take the line of centres along \mathbf{i}; apply the laws to \mathbf{i}-components; \mathbf{j}-components stay the same.

How do you find the impulse on a sphere?

Change in its momentum (mass times change in velocity).

How do you find kinetic energy lost?

KE before minus KE after, using the full speeds.

Oblique impactEdexcel A-Level Further Maths: Flashcards

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Along which line does the impulse act in an oblique impact of smooth spheres?

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Along which line does the impulse act in an oblique impact of smooth spheres?: The line of centres.
What happens to the velocity component perpendicular to the line of centres?: It is unchanged for each sphere.
For a smooth wall, which component is unchanged?: The component parallel to the wall.
What is the rebound rule for the normal component at a fixed wall?: It is reversed and multiplied by $e$ .
Which two laws apply along the line of centres?: Conservation of momentum and Newton's law of restitution.
How do you find the speed after impact?: Combine the two components with Pythagoras.
How are the angles before and after impact at a wall related?: $\tan\theta_{\text{after}}=e\tan\theta_{\text{before}}$ , with $\theta$ measured to the wall.
How do you handle vector form?: Take the line of centres along $\mathbf{i}$ ; apply the laws to $\mathbf{i}$ -components; $\mathbf{j}$ -components stay the same.
How do you find the impulse on a sphere?: Change in its momentum (mass times change in velocity).
What is the direction of the impulse between two smooth spheres?: Along the line of centres.
How do you find kinetic energy lost?: KE before minus KE after, using the full speeds.
When is no kinetic energy lost?: When $e=1$ .

Exam questions on Oblique impact

A smooth particle of mass $0.4$ kg moves on a smooth horizontal surface and strikes a fixed smooth vertical wall with speed $10$ m s $^{-1}$ . Its direction of motion makes an angle of $60^\circ$ with the wall. The coefficient of restitution between the particle and the wall is $0.5$ .
Find the kinetic energy lost by the particle in the impact.2 marks
A smooth ball of mass $0.2$ kg hits a smooth horizontal floor. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertically upwards respectively. Immediately before the impact the velocity of the ball is $(6\mathbf{i}-8\mathbf{j})$ m s $^{-1}$ . The coefficient of restitution between the ball and the floor is $0.75$ .
Find the speed of the ball immediately after impact, and the angle between its direction of motion and the floor.2 marks
Two smooth spheres $A$ and $B$ of equal radii and equal mass $m$ lie on a smooth horizontal surface, with $B$ at rest. $A$ moves with speed $10$ m s $^{-1}$ and strikes $B$ . At the instant of impact the direction of motion of $A$ makes an angle $\alpha$ with the line of centres, where $\cos\alpha=0.8$ . The coefficient of restitution between the spheres is $0.5$ .
Find the speed of $B$ immediately after the impact.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).