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

1. 1
   Particle $A$ of mass $2$ kg moves with speed $5$ m s$^{-1}$ along a straight line on a smooth horizontal surface and collides directly with particle $B$ of mass $3$ kg, which is at rest. The coefficient of restitution between the particles is $0.5$.

   (a)

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

   [1 mark]

   • A$2.5$ m s$^{-1}$
   • B$3$ m s$^{-1}$
   • C$2$ m s$^{-1}$
   • D$4.5$ m s$^{-1}$

   (b)

   Find the kinetic energy lost in the collision.

   [1 mark]

   • A$11.25$ J
   • B$13.75$ J
   • C$25$ J
   • D$22.5$ J

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Particles $P$ and $Q$, of mass $3$ kg and $5$ kg, move towards each other along the same straight line on a smooth horizontal surface with speeds $4$ m s$^{-1}$ and $2$ m s$^{-1}$ respectively. They collide directly. After the collision the direction of motion of $P$ is reversed and its speed is $1$ m s$^{-1}$.

   (a)

   Find the velocity of $Q$ immediately after the collision.

   [1 mark]

   • A$0.2$ m s$^{-1}$ in the original direction of motion of $Q$
   • B$1$ m s$^{-1}$ in the original direction of motion of $Q$
   • C$1$ m s$^{-1}$ in the original direction of motion of $P$
   • D$5$ m s$^{-1}$ in the original direction of motion of $P$

   (b)

   Find the coefficient of restitution between $P$ and $Q$.

   [1 mark]

   • A$\frac12$
   • B$\frac16$
   • C$3$
   • D$\frac13$

   (c)

   Find the total kinetic energy lost in the collision.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two smooth spheres $A$ and $B$ of equal radii and masses $m$ and $3m$ lie on a smooth horizontal table. Sphere $A$ is projected with speed $u$ directly towards $B$, which is at rest. The coefficient of restitution between the spheres is $e$.

   (a)

   Show that the speed of $B$ immediately after the collision is $\frac{(1+e)u}{4}$.

   [3 marks]

   (b)

   Given that the collision loses $\frac{9}{16}$ of the initial kinetic energy of $A$, find the value of $e$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Particles $A$ and $B$, of mass $0.5$ kg and $0.3$ kg, move in the same direction along a straight line on a smooth horizontal surface. $A$ has speed $8$ m s$^{-1}$ and $B$ has speed $2$ m s$^{-1}$, so $A$ catches up with $B$ and they collide directly. The coefficient of restitution between $A$ and $B$ is $e$.

   (a)

   (i) Show that the speed of $B$ immediately after the collision is $\frac{23+15e}{4}$ m s$^{-1}$.
   (ii) Hence find the least and greatest possible speeds of $B$ after the collision, and state what happens to the particles when $B$ has the least speed.

   [6 marks]

   (b)

   In a second trial the speed of $B$ immediately after the collision is measured as $7$ m s$^{-1}$. Find the value of $e$ and the kinetic energy lost in the collision.

   [6 marks]

   Total for question 4: 12 marks

End of questions