Successive direct impactsEdexcel A-Level Further Maths: Subtopic test

1. 1
   A small ball is dropped from rest at a height of $2.5$ m above a smooth horizontal floor. The coefficient of restitution between the ball and the floor is $0.8$. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed of the ball just before its first impact with the floor.

   [1 mark]

   • A$49$ m s$^{-1}$
   • B$7$ m s$^{-1}$
   • C$24.5$ m s$^{-1}$
   • D$3.5$ m s$^{-1}$

   (b)

   Find the greatest height reached by the ball after its first bounce.

   [1 mark]

   • A$2.0$ m
   • B$2.5$ m
   • C$1.28$ m
   • D$1.6$ m

   (c)

   Find the greatest height reached by the ball after its second bounce.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two smooth spheres $P$ and $Q$ of equal radii and equal mass $1$ kg lie on a smooth horizontal surface, in a straight line perpendicular to a fixed smooth vertical wall, with $Q$ between $P$ and the wall. $Q$ is at rest and $P$ moves towards $Q$ at $4$ m s$^{-1}$. The coefficient of restitution between $P$ and $Q$ is $0.5$, and between $Q$ and the wall is $\frac23$.

   (a)

   Find the speed of $Q$ immediately after the first collision between $P$ and $Q$.

   [1 mark]

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

   (b)

   Find the speed of $Q$ immediately after it hits the wall.

   [1 mark]

   • A$3$ m s$^{-1}$
   • B$1.5$ m s$^{-1}$
   • C$2$ m s$^{-1}$
   • D$4.5$ m s$^{-1}$

   (c)

   Explain why $P$ and $Q$ collide a second time.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Three smooth spheres $A$, $B$ and $C$ of equal radii lie in a straight line on a smooth horizontal surface, with $B$ between $A$ and $C$. $A$ has mass $2m$, and $B$ and $C$ each have mass $m$. $B$ and $C$ are at rest. $A$ moves towards $B$ at $6$ m s$^{-1}$. The coefficient of restitution between any two of the spheres is $\frac12$.

   (a)

   Find the speed of $B$ immediately after the first collision between $A$ and $B$.

   [3 marks]

   (b)

   $B$ then collides directly with $C$. Show that there is a further collision between $A$ and $B$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two smooth spheres $A$ and $B$ of equal radii, with masses $3$ kg and $1$ kg, lie on a smooth horizontal surface. $B$ is between $A$ and a fixed smooth vertical wall, in a line perpendicular to the wall, and $B$ is at rest. $A$ moves towards $B$ at $8$ m s$^{-1}$. The coefficient of restitution between $A$ and $B$ is $\frac12$, and between $B$ and the wall is $\frac13$.

   (a)

   (i) Find the speeds of $A$ and $B$ immediately after their first collision.
   (ii) Find the speed of $B$ immediately after it hits the wall.
   (iii) Explain why $A$ and $B$ collide again.

   [6 marks]

   (b)

   Find the speeds of $A$ and $B$ immediately after their second collision, and show that there is a third collision between $A$ and $B$.

   [6 marks]

   Total for question 4: 12 marks

End of questions