Coefficient of restitution and collisionsAQA A-Level Further Maths: Subtopic test

1. 1
   A ball is moving at $8$ m s⁻¹ directly towards a fixed smooth vertical wall. It hits the wall and rebounds directly with speed $5$ m s⁻¹.

   (a)

   Find the coefficient of restitution between the ball and the wall.

   [1 mark]

   • A$\frac85$
   • B$\frac38$
   • C$\frac58$
   • D$\frac{13}{8}$

   (b)

   What fraction of the ball's kinetic energy is lost in the impact?

   [1 mark]

   • A$\frac{25}{64}$
   • B$\frac{39}{64}$
   • C$\frac38$
   • D$\frac{9}{64}$

   (c)

   After rebounding, the ball hits a second fixed smooth wall, parallel to the first and made of the same material, directly. Find the speed of the ball after it rebounds from the second wall.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two smooth spheres $A$ and $B$, each of mass $m$, move in the same direction along a straight line on a smooth horizontal surface. $A$ is moving at $5$ m s⁻¹ and $B$ is ahead of it, moving at $1$ m s⁻¹. The spheres collide directly. The coefficient of restitution between them is $0.5$.

   (a)

   Find the speed of separation of the two spheres after the collision.

   [1 mark]

   • A$4$ m s⁻¹
   • B$6$ m s⁻¹
   • C$2.5$ m s⁻¹
   • D$2$ m s⁻¹

   (b)

   Find the speed of $B$ after the collision.

   [1 mark]

   • A$4$ m s⁻¹
   • B$2$ m s⁻¹
   • C$3$ m s⁻¹
   • D$6$ m s⁻¹

   (c)

   Find, in terms of $m$, the kinetic energy lost in the collision.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle $P$ of mass $2m$, moving at $4u$, collides directly with a particle $Q$ of mass $m$ moving at $u$ in the opposite direction on a smooth horizontal surface. The coefficient of restitution between $P$ and $Q$ is $\frac15$.

   (a)

   Find the speeds of $P$ and $Q$ after the collision.

   [3 marks]

   (b)

   $Q$ then hits a fixed smooth wall, perpendicular to its direction of motion, for which the coefficient of restitution is $\frac12$. $P$ has not yet reached the wall. Find the speed of $Q$ after it hits the wall, and explain why a second collision between $P$ and $Q$ must occur.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Three identical smooth spheres $A$, $B$ and $C$, each of mass $m$, lie at rest on a straight line on a smooth horizontal table, with $B$ between $A$ and $C$. The coefficient of restitution between any two of the spheres is $e$, where $0<e\le1$. Sphere $A$ is projected directly towards $B$ with speed $u$ and collides with it. $B$ then moves on and collides with $C$.

   (a)

   (i) Find, in terms of $u$ and $e$, the velocities of $A$ and $B$ after their collision.
   (ii) Find, in terms of $u$ and $e$, the velocities of $B$ and $C$ after the collision between $B$ and $C$.

   [6 marks]

   (b)

   Show that $A$ collides with $B$ again if $e<1$, and state what happens if $e=1$.

   [6 marks]

   Total for question 4: 12 marks

End of questions