Further Mechanics 1: Elastic collisions in two dimensionsEdexcel A-Level Further Maths: Topic test

1. 1
   A smooth ball of mass $0.8$ kg moves on a smooth horizontal surface with speed $10$ m s$^{-1}$ and strikes a fixed smooth vertical wall. Immediately before the impact the path of the ball makes an angle $\alpha$ with the wall, where $\tan\alpha=\frac34$. The coefficient of restitution between the ball and the wall is $0.5$.

   (a)

   What is the component of the velocity of the ball parallel to the wall immediately after the impact?

   [1 mark]

   • A$8$ m s$^{-1}$
   • B$6$ m s$^{-1}$
   • C$4$ m s$^{-1}$
   • D$10$ m s$^{-1}$

   (b)

   What is the component of the velocity of the ball perpendicular to the wall immediately after the impact?

   [1 mark]

   • A$6$ m s$^{-1}$
   • B$4$ m s$^{-1}$
   • C$3$ m s$^{-1}$
   • D$12$ m s$^{-1}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two smooth spheres $A$ and $B$ of equal radii have masses $2$ kg and $3$ kg. Sphere $B$ is at rest on a smooth horizontal surface. Sphere $A$ moves with speed $10$ m s$^{-1}$ and collides obliquely with $B$. At impact the line of centres makes an angle $\theta$ with the direction of motion of $A$, where $\cos\theta=0.8$. The coefficient of restitution between the spheres is $0.5$.

   (a)

   Which component of the velocity of $A$ is unchanged by the impact?

   [1 mark]

   • Athe component along the line of centres, $8$ m s$^{-1}$
   • Bthe component perpendicular to the line of centres, $8$ m s$^{-1}$
   • Cthe whole velocity, $10$ m s$^{-1}$
   • Dthe component perpendicular to the line of centres, $6$ m s$^{-1}$

   (b)

   What is the speed of $B$ immediately after the impact?

   [1 mark]

   • A$3.2$ m s$^{-1}$
   • B$4.8$ m s$^{-1}$
   • C$6.4$ m s$^{-1}$
   • D$0.8$ m s$^{-1}$

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two fixed smooth vertical walls $W_1$ and $W_2$ meet at right angles on a smooth horizontal surface. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal, with $\mathbf{i}$ perpendicular to $W_1$ and $\mathbf{j}$ perpendicular to $W_2$. A smooth particle $P$ of mass $0.25$ kg moves with velocity $(-9\mathbf{i}-12\mathbf{j})$ m s$^{-1}$ and strikes $W_1$ and then $W_2$. The coefficient of restitution between $P$ and $W_1$ is $\frac23$ and between $P$ and $W_2$ is $\frac12$.

   (a)

   Find the velocity of $P$ after it has struck both walls.

   [3 marks]

   (b)

   Find the total kinetic energy lost by $P$ in the two impacts.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two smooth spheres $A$ and $B$ of equal radii have masses $3$ kg and $1$ kg and move on a smooth horizontal surface. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal. At the instant of impact the line of centres of the spheres is parallel to $\mathbf{i}$, the velocity of $A$ is $(6\mathbf{i}+5\mathbf{j})$ m s$^{-1}$ and the velocity of $B$ is $(-2\mathbf{i}+3\mathbf{j})$ m s$^{-1}$. The coefficient of restitution between the spheres is $\frac12$. After the collision, $B$ strikes a fixed smooth vertical wall that is parallel to $\mathbf{i}$. The coefficient of restitution between $B$ and the wall is $\frac13$.

   (a)

   Find the velocities of $A$ and $B$ immediately after their collision, and the kinetic energy lost in the collision.

   [6 marks]

   (b)

   Find the speed of $B$ immediately after it rebounds from the wall, and the angle between its new path and the wall.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Two identical smooth spheres $A$ and $B$, each of mass $m$ and with equal radii, lie on a smooth horizontal surface with $B$ at rest. $A$ moves with speed $10$ m s$^{-1}$ and collides obliquely with $B$. At impact the line of centres makes an angle $\alpha$ with the direction of motion of $A$, where $\cos\alpha=0.8$. The collision is perfectly elastic, so $e=1$.

   (a)

   What is the speed of $B$ immediately after the collision?

   [1 mark]

   • A$10$ m s$^{-1}$
   • B$8$ m s$^{-1}$
   • C$6$ m s$^{-1}$
   • D$4$ m s$^{-1}$

   (b)

   What is the angle between the directions of motion of the two spheres after the collision?

   [1 mark]

   • A$53.1^\circ$
   • B$36.9^\circ$
   • C$45^\circ$
   • D$90^\circ$

   (c)

   Find the speed of $A$ immediately after the collision.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A smooth ball of mass $0.3$ kg moves on a smooth horizontal floor with velocity $(5\mathbf{i}+12\mathbf{j})$ m s$^{-1}$ towards a corner formed by two fixed smooth vertical walls at right angles. The unit vector $\mathbf{i}$ is perpendicular to the first wall $W_1$ and $\mathbf{j}$ is perpendicular to the second wall $W_2$. The ball strikes $W_1$ and then $W_2$. The coefficient of restitution between the ball and each wall is $0.8$.

   (a)

   What is the angle between the direction of motion of the ball before the first impact and its direction after the second impact?

   [1 mark]

   • A$180^\circ$
   • B$90^\circ$
   • C$0^\circ$
   • D$120^\circ$

   (b)

   What is the speed of the ball after both impacts?

   [1 mark]

   • A$13$ m s$^{-1}$
   • B$8.32$ m s$^{-1}$
   • C$10.4$ m s$^{-1}$
   • D$16.25$ m s$^{-1}$

   (c)

   Find the total kinetic energy lost by the ball in the two impacts.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   Two smooth spheres $A$ and $B$ of equal radii have masses $1$ kg and $2$ kg. Sphere $B$ is at rest on a smooth horizontal surface. Sphere $A$ moves with speed $13$ m s$^{-1}$ and collides obliquely with $B$. At impact the line of centres makes an angle $\theta$ with the direction of motion of $A$, where $\cos\theta=\frac{12}{13}$. After the collision $A$ moves in a direction perpendicular to the line of centres.

   (a)

   Show that the coefficient of restitution between the spheres is $\frac12$.

   [3 marks]

   (b)

   Find the kinetic energy lost in the collision.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A small smooth ball is projected horizontally with speed $6$ m s$^{-1}$ from a point $4.9$ m above a smooth horizontal floor. The coefficient of restitution between the ball and the floor is $0.5$. The ball is modelled as a particle, air resistance is ignored and $g=9.8$ m s$^{-2}$.

   (a)

   Find the speed and direction of motion of the ball immediately after it first strikes the floor.

   [6 marks]

   (b)

   Find the total horizontal distance travelled by the ball from projection until it stops bouncing.

   [6 marks]

   Total for question 8: 12 marks

End of questions