Momentum, impulse and collisions requiring resolvingAQA A-Level Further Maths: Subtopic test

1. 1
   A smooth ball of mass $0.2$ kg hits a fixed smooth vertical wall with speed $10$ m s⁻¹. Its direction of motion before the impact makes an angle of $60^\circ$ with the wall. The coefficient of restitution between the ball and the wall is $0.5$.

   (a)

   Find the component of the ball's velocity parallel to the wall immediately after the impact.

   [1 mark]

   • A$5\sqrt3$ m s⁻¹
   • B$5$ m s⁻¹
   • C$2.5\sqrt3$ m s⁻¹
   • D$10$ m s⁻¹

   (b)

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

   [1 mark]

   • A$\frac{\sqrt3}{2}$ N s
   • B$2\sqrt3$ N s
   • C$1$ N s
   • D$\frac{3\sqrt3}{2}$ N s

   (c)

   Find the speed of the ball immediately after the impact.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A smooth ball of mass $0.3$ kg moves on a smooth horizontal floor with velocity $(8\mathbf{i}-6\mathbf{j})$ m s⁻¹ and hits a fixed smooth vertical wall that is parallel to the vector $\mathbf{i}$. The coefficient of restitution between the ball and the wall is $0.75$. Here $\mathbf{i}$ and $\mathbf{j}$ are perpendicular horizontal unit vectors.

   (a)

   Find the velocity of the ball immediately after the impact.

   [1 mark]

   • A$(8\mathbf{i}+4.5\mathbf{j})$ m s⁻¹
   • B$(8\mathbf{i}+6\mathbf{j})$ m s⁻¹
   • C$(-8\mathbf{i}+4.5\mathbf{j})$ m s⁻¹
   • D$(6\mathbf{i}+4.5\mathbf{j})$ m s⁻¹

   (b)

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

   [1 mark]

   • A$1.35$ N s
   • B$1.8$ N s
   • C$3.15$ N s
   • D$2.4$ N s

   (c)

   Find the angle through which the direction of motion of the ball is deflected by the impact.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A smooth sphere of mass $0.5$ kg hits a fixed smooth vertical wall with speed $20$ m s⁻¹. Before the impact its direction of motion makes an angle $\alpha$ with the wall, where $\tan\alpha=\frac43$. The coefficient of restitution between the sphere and the wall is $\frac34$.

   (a)

   Find the speed of the sphere immediately after the impact.

   [3 marks]

   (b)

   Find the magnitude of the impulse exerted by the wall on the sphere, and the percentage of the sphere's kinetic energy lost in the impact.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two smooth spheres $A$ and $B$, of masses $2$ kg and $3$ kg, move on a smooth horizontal plane. Relative to unit vectors $\mathbf{i}$ and $\mathbf{j}$, the velocity of $A$ is $(8\mathbf{i}+6\mathbf{j})$ m s⁻¹ and the velocity of $B$ is $(-4\mathbf{i}-3\mathbf{j})$ m s⁻¹. The spheres collide directly. The coefficient of restitution between the spheres is $\frac12$.

   (a)

   (i) Show that the velocities of $A$ and $B$ are parallel, and find the speed of each sphere before the collision.
   (ii) Find the velocity of each sphere after the collision, giving your answers in the form $p\mathbf{i}+q\mathbf{j}$.

   [6 marks]

   (b)

   (i) Find the magnitude of the impulse exerted by $A$ on $B$ in the collision.
   (ii) Find the kinetic energy lost in the collision.

   [6 marks]

   Total for question 4: 12 marks

End of questions