Oblique impactEdexcel A-Level Further Maths: Subtopic test

1. 1
   A smooth particle of mass $0.4$ kg moves on a smooth horizontal surface and strikes a fixed smooth vertical wall with speed $10$ m s$^{-1}$. Its direction of motion makes an angle of $60^\circ$ with the wall. The coefficient of restitution between the particle and the wall is $0.5$.

   (a)

   Find the component of the velocity of the particle parallel to the wall immediately after impact.

   [1 mark]

   • A$5$ m s$^{-1}$
   • B$2.5$ m s$^{-1}$
   • C$5\sqrt3$ m s$^{-1}$
   • D$10$ m s$^{-1}$

   (b)

   Find the component of the velocity of the particle perpendicular to the wall immediately after impact.

   [1 mark]

   • A$2.5$ m s$^{-1}$
   • B$5\sqrt3$ m s$^{-1}$
   • C$\frac{5\sqrt3}{2}$ m s$^{-1}$
   • D$5$ m s$^{-1}$

   (c)

   Find the kinetic energy lost by the particle in the impact.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A smooth ball of mass $0.2$ kg hits a smooth horizontal floor. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertically upwards respectively. Immediately before the impact the velocity of the ball is $(6\mathbf{i}-8\mathbf{j})$ m s$^{-1}$. The coefficient of restitution between the ball and the floor is $0.75$.

   (a)

   Find the velocity of the ball immediately after impact.

   [1 mark]

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

   (b)

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

   [1 mark]

   • A$1.2$ N s
   • B$1.6$ N s
   • C$2.0$ N s
   • D$2.8$ N s

   (c)

   Find the speed of the ball immediately after impact, and the angle between its direction of motion and the floor.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two smooth spheres $A$ and $B$ of equal radii and equal mass $m$ lie on a smooth horizontal surface, with $B$ at rest. $A$ moves with speed $10$ m s$^{-1}$ and strikes $B$. At the instant of impact the direction of motion of $A$ makes an angle $\alpha$ with the line of centres, where $\cos\alpha=0.8$. The coefficient of restitution between the spheres is $0.5$.

   (a)

   Find the speed of $B$ immediately after the impact.

   [3 marks]

   (b)

   Find the speed of $A$ immediately after the impact, and the angle between the direction of motion of $A$ after the impact and the line of centres.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two smooth spheres $A$ and $B$ of equal radii have masses $2$ kg and $1$ kg. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ lie in the horizontal plane, and at the instant of collision the line of centres is parallel to $\mathbf{i}$. Before the collision $A$ has velocity $(5\mathbf{i}+3\mathbf{j})$ m s$^{-1}$ and $B$ has velocity $(-\mathbf{i}-2\mathbf{j})$ m s$^{-1}$. The coefficient of restitution between the spheres is $\frac12$.

   (a)

   Find the velocity of $A$ and the velocity of $B$ after the collision.

   [6 marks]

   (b)

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

   [6 marks]

   Total for question 4: 12 marks

End of questions