Successive impactsEdexcel International A Level Further Maths: Subtopic test

1. 1
   A particle of mass $0.4$ kg moves with speed $12$ m s$^{-1}$ along smooth horizontal ground directly towards a fixed smooth vertical wall, hitting the wall at right angles. The coefficient of restitution between the particle and the wall is $\frac23$.

   (a)

   Find the speed of the particle immediately after it hits the wall.

   [1 mark]

   • A$18$ m s$^{-1}$
   • B$4$ m s$^{-1}$
   • C$6$ m s$^{-1}$
   • D$8$ m s$^{-1}$

   (b)

   Find the kinetic energy lost in the impact with the wall.

   [1 mark]

   • A$28.8$ J
   • B$16$ J
   • C$12.8$ J
   • D$32$ J

   (c)

   Find the magnitude of the impulse exerted by the wall on the particle.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Particles $A$ and $B$ have masses $2$ kg and $1$ kg. They lie on smooth horizontal ground in a straight line perpendicular to a fixed smooth vertical wall, with $B$ between $A$ and the wall. Particle $A$ is projected towards $B$ with speed $6$ m s$^{-1}$. The coefficient of restitution between $A$ and $B$ is $\frac12$, and between $B$ and the wall is $\frac13$.

   (a)

   Find the speed of $B$ immediately after $A$ first collides with it.

   [1 mark]

   • A$6$ m s$^{-1}$
   • B$3$ m s$^{-1}$
   • C$4$ m s$^{-1}$
   • D$12$ m s$^{-1}$

   (b)

   Find the speed of $B$ immediately after it rebounds from the wall.

   [1 mark]

   • A$3$ m s$^{-1}$
   • B$18$ m s$^{-1}$
   • C$2$ m s$^{-1}$
   • D$6$ m s$^{-1}$

   (c)

   Explain why there must be a second collision between $A$ and $B$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three particles $A$, $B$ and $C$, of masses $4$ kg, $2$ kg and $3$ kg, lie at rest in a straight line on smooth horizontal ground, with $B$ between $A$ and $C$. Particle $A$ is projected towards $B$ with speed $6$ m s$^{-1}$. The coefficient of restitution between any two of the particles is $\frac12$.

   (a)

   Show that, immediately after the first collision, $A$ has speed $3$ m s$^{-1}$ and $B$ has speed $6$ m s$^{-1}$.

   [3 marks]

   (b)

   $B$ then collides with $C$. Find the speeds of $B$ and $C$ immediately after this collision, and show that there is a further collision between $A$ and $B$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Particles $P$ and $Q$, of masses $m$ and $3m$, lie at rest on smooth horizontal ground in a straight line perpendicular to a fixed smooth vertical wall, with $Q$ between $P$ and the wall. Particle $P$ is projected towards $Q$ with speed $u$. The coefficient of restitution between $P$ and $Q$ is $\frac13$, and between $Q$ and the wall is $\frac12$.

   (a)

   (i) Find, in terms of $u$, the speeds of $P$ and $Q$ immediately after they first collide.
   (ii) Find the speed of $Q$ immediately after it hits the wall.

   [6 marks]

   (b)

   Show that $P$ and $Q$ collide a second time. Find their speeds after the second collision and deduce that there are no further collisions.

   [6 marks]

   Total for question 4: 12 marks

End of questions