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

Section 1

Direct impact

In a direct impact two smooth spheres of equal radii (or a sphere and a smooth wall) collide so that the velocities before and after the collision are along the line of centres. The spheres are modelled as particles on a smooth horizontal surface. During the impact the two spheres exert equal and opposite impulses on each other along the line of motion.

Key termsdirect impactline of centres

Exam tip

Draw a before-and-after diagram, mark one direction as positive and give every velocity a sign.

Section 2

Conservation of momentum

With no external horizontal force, total momentum along the line of motion is unchanged. For masses $m_1$ , $m_2$ with velocities $u_1,u_2$ before and $v_1,v_2$ after: $m_1u_1+m_2u_2=m_1v_1+m_2v_2.$ Velocities are signed: a sphere moving against the positive direction has a negative velocity. Example: $A$ ( $2$ kg) at $5$ m s $^{-1}$ and $B$ ( $3$ kg) at $-2$ m s $^{-1}$ have total momentum $10-6=4$ kg m s $^{-1}$ .

Key termsconservation of momentum

Common mistake

Adding speeds instead of signed velocities when the spheres move in opposite directions.

Section 3

Newton's law of restitution

Newton's law of restitution states that the speed of separation is $e$ times the speed of approach: $v_2-v_1=e\,(u_1-u_2),$ for sphere 1 behind sphere 2. The coefficient of restitution $e$ satisfies $0\le e\le1$ . If $e=1$ the spheres are perfectly elastic. If $e=0$ they move together after impact. Combining the law with conservation of momentum gives two simultaneous equations for $v_1$ and $v_2$ . Example: $m_1=4$ kg at $6$ m s $^{-1}$ hits $m_2=2$ kg at rest with $e=0.5$ . Then $24=4v_1+2v_2$ and $v_2-v_1=3$ , so $v_1=3$ m s $^{-1}$ and $v_2=6$ m s $^{-1}$ .

Key termscoefficient of restitutionspeed of approachspeed of separation

Common mistake

Applying $e$ to a single sphere's speed instead of to the difference of velocities of the two spheres.

Section 4

Impact with a fixed surface

A sphere hitting a fixed smooth wall at right angles with speed $u$ rebounds with speed $v=eu$ in the opposite direction. The wall exerts an impulse of magnitude $I=m(v+u)=m(1+e)u$ , because the velocity reverses. Example: $m=0.3$ kg, $u=8$ m s $^{-1}$ , $e=0.5$ gives $v=4$ m s $^{-1}$ and $I=3.6$ N s.

Key termsimpulse

Section 5

Loss of kinetic energy and the range of e

Kinetic energy lost $=$ total KE before $-$ total KE after. For the example above, before $=\frac12(4)(36)=72$ J and after $=\frac12(4)(9)+\frac12(2)(36)=54$ J, so $18$ J is lost. Energy is lost whenever $e<1$ and conserved when $e=1$ . A result with $e>1$ or $e<0$ is impossible. Exam questions often set an inequality, such as the values of $e$ for which a sphere changes direction, and combine it with $0\le e\le1$ .

Key termsperfectly elastic

Exam tip

Check your answer by recomputing the momentum after the collision, and make sure the speed of separation is not bigger than the speed of approach.

That's the notes covered.

Carry on to the next subtopic.

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).