Further Mechanics 1: Elastic collisions in one dimensionEdexcel A-Level Further Maths: Topic test

1. 1
   Two smooth spheres $P$ and $Q$ of equal radii, with masses $0.5$ kg and $1.5$ kg, move in the same direction along a straight line on a smooth horizontal surface. $P$ has speed $6$ m s$^{-1}$ and overtakes $Q$, which has speed $2$ m s$^{-1}$. The spheres collide directly and the coefficient of restitution between them is $0.5$.

   (a)

   What is the total momentum of the two spheres before the collision, in kg m s$^{-1}$?

   [1 mark]

   • A$3$
   • B$6$
   • C$0$
   • D$8$

   (b)

   What is the speed at which the spheres separate after the collision?

   [1 mark]

   • A$4$ m s$^{-1}$
   • B$1$ m s$^{-1}$
   • C$8$ m s$^{-1}$
   • D$2$ m s$^{-1}$

   (c)

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

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two smooth spheres $C$ and $D$ of equal radii, with masses $4$ kg and $2$ kg, are moving towards each other along the same straight line on a smooth horizontal surface. $C$ has speed $3$ m s$^{-1}$ and $D$ has speed $4$ m s$^{-1}$. They collide directly. Immediately after the collision $C$ is at rest.

   (a)

   Take the direction of motion of $C$ before the collision as positive, and let $v$ m s$^{-1}$ be the velocity of $D$ after the collision. Which equation follows from conservation of momentum?

   [1 mark]

   • A$4\times3-2\times4=2v$
   • B$4\times3+2\times4=2v$
   • C$4\times3-2\times4=4v$
   • D$4\times3-2\times4=0$

   (b)

   What is the coefficient of restitution between the spheres?

   [1 mark]

   • A$\frac23$
   • B$\frac72$
   • C$\frac27$
   • D$\frac12$

   (c)

   Find the kinetic energy lost in the collision.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A smooth sphere $A$ of mass $4$ kg and a smooth sphere $B$ of mass $1$ kg, with equal radii, lie on a smooth horizontal surface. $B$ is at rest between $A$ and a fixed smooth vertical wall, and the line of centres of the spheres is perpendicular to the wall. $A$ is projected towards $B$ with speed $10$ m s$^{-1}$. The coefficient of restitution between $A$ and $B$ is $0.5$ and between $B$ and the wall is $0.5$.

   (a)

   Find the speeds of $A$ and $B$ immediately after their first collision.

   [3 marks]

   (b)

   Show that $A$ and $B$ collide again, and find the speed of $A$ immediately after the second collision.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A small ball is released from rest at a height of $4.9$ m above a smooth horizontal floor. The ball falls vertically, bounces repeatedly, and the coefficient of restitution between the ball and the floor is $0.6$. The ball is modelled as a particle, air resistance is ignored and $g=9.8$ m s$^{-2}$.

   (a)

   Find the total time from release until the ball stops bouncing.

   [6 marks]

   (b)

   Find the total distance the ball travels before it stops bouncing.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Two smooth spheres $A$ and $B$ of equal radii have masses $2m$ and $m$ respectively. They lie on a smooth horizontal surface, with $B$ at rest. $A$ moves with speed $u$ along the line of centres and collides directly with $B$. The coefficient of restitution between the spheres is $\frac12$.

   (a)

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

   [1 mark]

   • A$\frac{u}{2}$
   • B$\frac{2u}{3}$
   • C$u$
   • D$\frac{3u}{2}$

   (b)

   What fraction of the initial kinetic energy of the system is lost in the collision?

   [1 mark]

   • A$\frac14$
   • B$\frac13$
   • C$\frac12$
   • D$\frac34$

   (c)

   Find, in terms of $u$, the speed of $A$ immediately after the collision.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A sphere $S$ of mass $4$ kg, moving with speed $6$ m s$^{-1}$, overtakes a sphere $T$ of mass $2$ kg moving in the same direction with speed $2$ m s$^{-1}$ along a straight line on a smooth horizontal surface. The spheres have equal radii and collide directly. The coefficient of restitution between them is $e$.

   (a)

   What is the total momentum of the spheres before the collision, in kg m s$^{-1}$?

   [1 mark]

   • A$20$
   • B$24$
   • C$14$
   • D$28$

   (b)

   Which range gives all possible values of the speed $v_T$ of $T$ after the collision, in m s$^{-1}$?

   [1 mark]

   • A$2\le v_T\le6$
   • B$\frac{14}{3}\le v_T\le\frac{22}{3}$
   • C$0\le v_T\le\frac{22}{3}$
   • D$\frac{14}{3}\le v_T\le14$

   (c)

   Given that the speed of $S$ after the collision is $4$ m s$^{-1}$, find the value of $e$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A smooth sphere $P$ of mass $1$ kg moves with speed $8$ m s$^{-1}$ along a line perpendicular to a fixed smooth vertical wall, towards the wall, on a smooth horizontal surface. The coefficient of restitution between $P$ and the wall is $0.5$. After rebounding, $P$ collides directly with a sphere $Q$ of mass $3$ kg and the same radius, which is moving towards the wall with speed $4$ m s$^{-1}$ along the same line. The coefficient of restitution between $P$ and $Q$ is $0.5$.

   (a)

   Find the speeds of $P$ and $Q$ immediately after their collision.

   [3 marks]

   (b)

   Show that $P$ and $Q$ collide again, and find the speed of $Q$ immediately after this second collision.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two smooth spheres $A$ and $B$ of equal radii have masses $m$ and $2m$. The sphere $B$ is at rest on a smooth horizontal surface, between $A$ and a fixed smooth vertical wall, with the line of centres perpendicular to the wall. $A$ is projected towards $B$ with speed $u$. The coefficient of restitution between the spheres is $e$ and between $B$ and the wall is $\frac12$.

   (a)

   Find, in terms of $u$ and $e$, the speeds of $A$ and $B$ immediately after their first collision, and show that the direction of motion of $A$ is reversed if and only if $e>\frac12$.

   [6 marks]

   (b)

   Given that $e=\frac14$, show that $A$ and $B$ collide a second time, find their speeds immediately after the second collision, and show that there are no further collisions.

   [6 marks]

   Total for question 8: 12 marks

End of questions