A perfectly elastic collision: no kinetic energy is lost.

What happens to the spheres when e=0?

They move together after impact (common velocity).

State conservation of momentum for two spheres.

m1u1+m2u2=m1v1+m2v2, with velocities signed.

Why must velocities be signed in the momentum equation?

Momentum is a vector; opposite directions have opposite signs.

How do you find the kinetic energy lost in a collision?

Total KE before minus total KE after.

How do you find the two unknown velocities after a collision?

Solve conservation of momentum together with Newton's law of restitution.

Direct impact and Newton's law of restitutionEdexcel A-Level Further Maths: Flashcards

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State Newton's law of restitution.

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State Newton's law of restitution.: Speed of separation $=e\times$ speed of approach.
What range of values can the coefficient of restitution $e$ take?: $0\le e\le1$ .
What does $e=1$ mean?: A perfectly elastic collision: no kinetic energy is lost.
What happens to the spheres when $e=0$ ?: They move together after impact (common velocity).
State conservation of momentum for two spheres.: $m_1u_1+m_2u_2=m_1v_1+m_2v_2$ , with velocities signed.
What is the speed of approach for two spheres moving towards each other at $u_1$ and $u_2$ ?: $u_1+u_2$ (sum of the speeds).
A sphere hits a fixed wall at speed $u$ at right angles. What is its rebound speed?: $eu$
What is the impulse of a wall on a sphere of mass $m$ hitting it at $u$ with coefficient $e$ ?: $m(1+e)u$ .
Why must velocities be signed in the momentum equation?: Momentum is a vector; opposite directions have opposite signs.
How do you find the kinetic energy lost in a collision?: Total KE before minus total KE after.
When is kinetic energy conserved in a collision?: Only when $e=1$ .
How do you find the two unknown velocities after a collision?: Solve conservation of momentum together with Newton's law of restitution.

Exam questions on Direct impact and Newton's law of restitution

Two smooth spheres $A$ and $B$ of equal radii, with masses $2$ kg and $3$ kg, move towards each other along the same straight line on a smooth horizontal surface. $A$ has speed $5$ m s $^{-1}$ and $B$ has speed $2$ m s $^{-1}$ . They collide directly. The coefficient of restitution between the spheres is $0.5$ .
Hence find the velocity of $B$ after the collision, stating its direction.2 marks
A smooth ball of mass $0.3$ kg strikes a fixed smooth vertical wall at right angles with speed $8$ m s $^{-1}$ . The coefficient of restitution between the ball and the wall is $0.5$ .
Find the kinetic energy lost by the ball in the impact.2 marks
Two smooth spheres $A$ and $B$ of equal radii have masses $m$ and $3m$ . They lie on a smooth horizontal surface. $A$ moves with speed $6$ m s $^{-1}$ and collides directly with $B$ , which is at rest. The coefficient of restitution between the spheres is $\frac12$ .
Find the speed of $A$ and the speed of $B$ immediately after the collision.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).