Momentum and impulse as vectorsEdexcel A-Level Further Maths: Subtopic test

1. 1
   A ball of mass $0.5$ kg moves on a smooth horizontal surface with velocity $(4\mathbf{i}-3\mathbf{j})$ m s$^{-1}$ when it is struck by a club. Immediately afterwards its velocity is $(-2\mathbf{i}+5\mathbf{j})$ m s$^{-1}$. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal and perpendicular. The club is in contact with the ball for $0.1$ s.

   (a)

   Find the impulse exerted by the club on the ball.

   [1 mark]

   • A$(-3\mathbf{i}+4\mathbf{j})$ N s
   • B$(3\mathbf{i}-4\mathbf{j})$ N s
   • C$(-6\mathbf{i}+8\mathbf{j})$ N s
   • D$(\mathbf{i}+\mathbf{j})$ N s

   (b)

   Find the magnitude of the impulse exerted by the club on the ball.

   [1 mark]

   • A$7$ N s
   • B$25$ N s
   • C$5$ N s
   • D$1$ N s

   (c)

   Find the magnitude of the average force exerted by the club on the ball.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Particles $A$ and $B$, of masses $2$ kg and $3$ kg, move on a smooth horizontal surface and collide. Before the collision $A$ has velocity $(3\mathbf{i}+\mathbf{j})$ m s$^{-1}$ and $B$ has velocity $(-2\mathbf{i}+4\mathbf{j})$ m s$^{-1}$. After the collision $A$ has velocity $(-3\mathbf{i}+4\mathbf{j})$ m s$^{-1}$.

   (a)

   Find the total momentum of the two particles before the collision.

   [1 mark]

   • A$(12\mathbf{i}+14\mathbf{j})$ N s
   • B$14\mathbf{j}$ N s
   • C$(\mathbf{i}+5\mathbf{j})$ N s
   • D$(4\mathbf{i}+6\mathbf{j})$ N s

   (b)

   Find the velocity of $B$ after the collision.

   [1 mark]

   • A$(6\mathbf{i}+6\mathbf{j})$ m s$^{-1}$
   • B$(3\mathbf{i}+3\mathbf{j})$ m s$^{-1}$
   • C$(-2\mathbf{i}+4\mathbf{j})$ m s$^{-1}$
   • D$(2\mathbf{i}+2\mathbf{j})$ m s$^{-1}$

   (c)

   Find the impulse exerted by $A$ on $B$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ball of mass $0.4$ kg moves on a smooth horizontal surface and strikes a smooth fixed vertical wall that is parallel to the unit vector $\mathbf{i}$. Just before the impact its velocity is $(6\mathbf{i}-8\mathbf{j})$ m s$^{-1}$ and just after it is $(6\mathbf{i}+4\mathbf{j})$ m s$^{-1}$. The ball is in contact with the wall for $0.02$ s.

   (a)

   Find the impulse exerted by the wall on the ball, and state the direction of this impulse relative to the wall.

   [3 marks]

   (b)

   Find the magnitude of the average force exerted by the wall on the ball, and the speed of the ball immediately after the impact.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Particles $P$ and $Q$, of masses $4$ kg and $2$ kg, move on a smooth horizontal surface and collide. Just before the collision $P$ has velocity $(5\mathbf{i}-2\mathbf{j})$ m s$^{-1}$ and $Q$ has velocity $(-4\mathbf{i}+6\mathbf{j})$ m s$^{-1}$. Just after the collision $P$ has velocity $(2\mathbf{i}+\mathbf{j})$ m s$^{-1}$.

   (a)

   (i) Find the velocity of $Q$ immediately after the collision.
   (ii) Find the magnitude of the impulse exerted on $Q$ by $P$.

   [6 marks]

   (b)

   (i) Find the impulse exerted on $P$ by $Q$, and explain how it confirms your answer to (a)(ii).
   (ii) The collision lasts $0.05$ s. Find the magnitude of the average force exerted on $P$ by $Q$.

   [6 marks]

   Total for question 4: 12 marks

End of questions