Direct impact and Newton's law of restitutionEdexcel A-Level Further Maths: Subtopic test

1. 1
   Two smooth spheres $A$ and $B$ of equal radii, with masses $2$ kg and $3$ kg, move towards each other along the same straight line on a smooth horizontal surface. $A$ has speed $5$ m s$^{-1}$ and $B$ has speed $2$ m s$^{-1}$. They collide directly. The coefficient of restitution between the spheres is $0.5$.

   (a)

   Taking the initial direction of motion of $A$ as positive, find the total momentum of the two spheres before the collision.

   [1 mark]

   • A$16$ kg m s$^{-1}$
   • B$10$ kg m s$^{-1}$
   • C$4$ kg m s$^{-1}$
   • D$-4$ kg m s$^{-1}$

   (b)

   Find the speed of separation of the spheres after the collision.

   [1 mark]

   • A$3.5$ m s$^{-1}$
   • B$7$ m s$^{-1}$
   • C$2.5$ m s$^{-1}$
   • D$1.75$ m s$^{-1}$

   (c)

   Hence find the velocity of $B$ after the collision, stating its direction.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A smooth ball of mass $0.3$ kg strikes a fixed smooth vertical wall at right angles with speed $8$ m s$^{-1}$. The coefficient of restitution between the ball and the wall is $0.5$.

   (a)

   Find the speed of the ball immediately after impact.

   [1 mark]

   • A$16$ m s$^{-1}$
   • B$8$ m s$^{-1}$
   • C$7.5$ m s$^{-1}$
   • D$4$ m s$^{-1}$

   (b)

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

   [1 mark]

   • A$1.2$ N s
   • B$3.6$ N s
   • C$2.4$ N s
   • D$7.2$ N s

   (c)

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

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two smooth spheres $A$ and $B$ of equal radii have masses $m$ and $3m$. They lie on a smooth horizontal surface. $A$ moves with speed $6$ m s$^{-1}$ and collides directly with $B$, which is at rest. The coefficient of restitution between the spheres is $\frac12$.

   (a)

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

   [3 marks]

   (b)

   Find the fraction of the initial kinetic energy of the system that is lost in the collision.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two smooth spheres $A$ and $B$ of equal radii, with masses $3$ kg and $1$ kg, move along the same straight line on a smooth horizontal surface. The coefficient of restitution between the spheres is $e$.

   (a)

   Initially $A$ moves at $4$ m s$^{-1}$ and $B$ moves at $1$ m s$^{-1}$ in the same direction, with $A$ behind $B$. After they collide directly, $B$ moves at $4$ m s$^{-1}$.
   (i) Find the speed of $A$ after the collision.
   (ii) Show that $e=\frac13$.
   (iii) Find the loss of kinetic energy in the collision.

   [6 marks]

   (b)

   In a second experiment $A$ moves at $4$ m s$^{-1}$ and $B$ moves at $9$ m s$^{-1}$ towards $A$, and they collide directly.
   (i) Show that the velocity of $A$ after the collision, in its original direction of motion, is $\frac{3-13e}{4}$ m s$^{-1}$.
   (ii) Hence find the range of values of $e$ for which the direction of motion of $A$ is reversed.

   [6 marks]

   Total for question 4: 12 marks

End of questions