Further Mechanics 1: Momentum and collisionsAQA A-Level Further Maths: Topic test

1. 1
   Railway truck $A$ of mass $6000$ kg is moving at $3$ m s⁻¹ along a straight horizontal track when it collides with truck $B$ of mass $4000$ kg, which is at rest. The trucks couple together on impact. Resistances to motion may be ignored.

   (a)

   Find the speed of the coupled trucks immediately after the collision, in m s⁻¹.

   [1 mark]

   • A$1.2$
   • B$1.5$
   • C$1.8$
   • D$3.0$

   (b)

   Find the magnitude of the impulse exerted by $A$ on $B$ in the collision, in N s.

   [1 mark]

   • A$7200$
   • B$10\,800$
   • C$12\,000$
   • D$18\,000$

   (c)

   Find the kinetic energy lost in the collision.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A ball of mass $0.25$ kg is released from rest at a height of $1.6$ m above a horizontal floor and falls vertically. It rebounds to a height of $0.4$ m. Air resistance may be ignored and $g=9.8$ m s⁻².

   (a)

   Find the speed of the ball just before it hits the floor, in m s⁻¹.

   [1 mark]

   • A$3.96$
   • B$31.4$
   • C$2.8$
   • D$5.6$

   (b)

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

   [1 mark]

   • A$0.25$
   • B$0.5$
   • C$2.0$
   • D$0.4$

   (c)

   Find the magnitude and direction of the impulse exerted by the floor on the ball.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A trolley $A$ of mass $4$ kg is at rest on a smooth horizontal track. For $0\le t\le4$ seconds a horizontal force of magnitude $(24t-6t^{2})$ newtons acts on it along the track.

   (a)

   Find the magnitude of the impulse of the force over the $4$ seconds and hence the speed of the trolley when $t=4$.

   [3 marks]

   (b)

   At $t=4$ the trolley $A$ collides directly with a second trolley $B$ of mass $12$ kg, which is at rest on the track. The coefficient of restitution between the trolleys is $0.5$. Find the speed of each trolley after the collision and state the direction of motion of $A$ relative to its original direction.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A smooth ball of mass $0.15$ kg strikes a fixed smooth horizontal floor. Just before impact its velocity is $(20\mathbf{i}-15\mathbf{j})$ m s⁻¹, where $\mathbf{i}$ is horizontal and $\mathbf{j}$ is vertically upwards. The coefficient of restitution between the ball and the floor is $0.6$. The effect of gravity during the impact may be ignored.

   (a)

   Find the velocity of the ball immediately after the impact, and hence its speed and the angle its path then makes with the horizontal.

   [6 marks]

   (b)

   Find the magnitude and direction of the impulse exerted by the floor on the ball. The impact lasts $0.02$ s. Find the average force exerted by the floor on the ball during the impact and explain why it is reasonable to ignore the weight of the ball during the impact. Take $g=9.8$ m s⁻².

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A man of mass $80$ kg stands on a raft of mass $120$ kg which is at rest on still water. He leaps horizontally off the raft. Resistance from the water may be ignored.

   (a)

   He leaves the raft with a horizontal speed of $1.5$ m s⁻¹ relative to the water. Find the speed of the raft immediately afterwards, in m s⁻¹.

   [1 mark]

   • A$2.25$
   • B$1.0$
   • C$0.6$
   • D$1.5$

   (b)

   Find the magnitude of the impulse exerted by the raft on the man in the situation in part (a), in N s.

   [1 mark]

   • A$180$
   • B$300$
   • C$80$
   • D$120$

   (c)

   Instead, the man leaps off with a horizontal speed of $1.5$ m s⁻¹ relative to the raft. Find the speed of the raft immediately afterwards.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   Smooth sphere $A$ of mass $0.5$ kg moving at $4$ m s⁻¹ collides directly with smooth sphere $B$ of mass $1.5$ kg moving in the same direction at $1$ m s⁻¹ on a smooth horizontal surface. The coefficient of restitution between the spheres is $0.4$.

   (a)

   Find the speed of $B$ immediately after the collision, in m s⁻¹.

   [1 mark]

   • A$2.05$
   • B$2.15$
   • C$1.2$
   • D$0.85$

   (b)

   Find the magnitude of the impulse exerted by $A$ on $B$, in N s.

   [1 mark]

   • A$3.075$
   • B$0.425$
   • C$1.575$
   • D$1.8$

   (c)

   Find the kinetic energy lost in the collision.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A smooth sphere of mass $0.3$ kg moves along a smooth horizontal floor between two fixed, smooth, parallel vertical walls $P$ and $Q$. It moves at $12$ m s⁻¹ perpendicular to the walls, towards $P$. It hits $P$, rebounds, and then hits $Q$. The coefficient of restitution between the sphere and each wall is $0.5$.

   (a)

   Find the speed of the sphere immediately after it hits $P$ and immediately after it hits $Q$.

   [3 marks]

   (b)

   Find the magnitude of the impulse exerted by $Q$ on the sphere, and the total kinetic energy lost by the sphere in the two impacts.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two smooth spheres $A$ and $B$, of masses $3$ kg and $5$ kg, move on a smooth horizontal plane with velocities $(4\mathbf{i}-3\mathbf{j})$ m s⁻¹ and $(-2\mathbf{i}+3\mathbf{j})$ m s⁻¹ respectively. They collide and coalesce.

   (a)

   Find the velocity of the combined particle immediately after the collision, and the kinetic energy lost.

   [6 marks]

   (b)

   Find the impulse exerted by $B$ on $A$ as a vector, and its magnitude to 3 significant figures. Show that the impulse exerted by $A$ on $B$ is equal and opposite. The collision lasts $0.05$ s; find the average magnitude of the force acting on $A$ during the collision.

   [6 marks]

   Total for question 8: 12 marks

End of questions