Further Mechanics 1: Momentum and impulseEdexcel A-Level Further Maths: Topic test

1. 1
   Railway truck $A$, of mass $8000$ kg, moves at $3$ m s$^{-1}$ along a straight horizontal track towards truck $B$, of mass $4000$ kg, which is at rest. The trucks collide and couple together. Resistance to motion may be ignored.

   (a)

   What is the speed of the coupled trucks after the collision?

   [1 mark]

   • A$3$ m s$^{-1}$
   • B$2$ m s$^{-1}$
   • C$4$ m s$^{-1}$
   • D$6$ m s$^{-1}$

   (b)

   What is the magnitude of the impulse exerted on truck $B$?

   [1 mark]

   • A$12000$ N s
   • B$24000$ N s
   • C$16000$ N s
   • D$8000$ N s

   (c)

   The coupling takes $0.5$ s to complete. Find the magnitude of the average force exerted on $B$ during the coupling.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A squash ball of mass $0.25$ kg is moving horizontally with velocity $(6\mathbf{i}-8\mathbf{j})$ m s$^{-1}$ when it is struck by a racket. Immediately after being struck it has velocity $(-2\mathbf{i}+4\mathbf{j})$ m s$^{-1}$. The ball is modelled as a particle, $\mathbf{i}$ and $\mathbf{j}$ are perpendicular horizontal unit vectors, and the weight of the ball may be ignored during the contact.

   (a)

   What is the impulse exerted on the ball by the racket?

   [1 mark]

   • A$(-2\mathbf{i}+3\mathbf{j})$ N s
   • B$(2\mathbf{i}-3\mathbf{j})$ N s
   • C$(\mathbf{i}-\mathbf{j})$ N s
   • D$(-8\mathbf{i}+12\mathbf{j})$ N s

   (b)

   What is the magnitude of this impulse?

   [1 mark]

   • A$13$ N s
   • B$5$ N s
   • C$3.61$ N s
   • D$14.4$ N s

   (c)

   The contact between the racket and the ball lasts $0.02$ s. Find the magnitude of the average force exerted on the ball by the racket.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   Two spheres, $X$ of mass $1.5$ kg and $Y$ of mass $0.5$ kg, are modelled as particles moving in opposite directions along a straight line on a smooth horizontal surface. $X$ has speed $8$ m s$^{-1}$ and $Y$ has speed $4$ m s$^{-1}$. They collide directly. Immediately after the collision, $X$ is moving at $5$ m s$^{-1}$ in its original direction of motion.

   (a)

   Find the speed of $Y$ immediately after the collision and state its direction of motion.

   [3 marks]

   (b)

   The collision lasts $0.01$ s. Find the magnitude of the impulse exerted on $Y$ and the magnitude of the average force on $Y$ during the collision.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two ice hockey pucks, $H$ of mass $0.4$ kg and $K$ of mass $0.2$ kg, move on smooth horizontal ice and are modelled as particles. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal and perpendicular. The pucks collide. Before the collision $H$ has velocity $(5\mathbf{i}+2\mathbf{j})$ m s$^{-1}$ and $K$ has velocity $(-4\mathbf{i}-\mathbf{j})$ m s$^{-1}$. Immediately after the collision $H$ has velocity $(\mathbf{i}+3\mathbf{j})$ m s$^{-1}$.

   (a)

   Find the velocity of $K$ immediately after the collision, and the impulse exerted on $H$ by $K$.

   [6 marks]

   (b)

   In a second collision on the same ice, $H$ moves along a straight line at $6$ m s$^{-1}$ and hits $K$, which is at rest. The two pucks then move on together. Find their common speed, the magnitude of the impulse on $K$, and the kinetic energy lost in the collision.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A golf ball of mass $0.046$ kg is at rest on a tee. It is struck by a club and leaves the club with speed $60$ m s$^{-1}$. The contact lasts $5\times10^{-4}$ s. The ball is modelled as a particle, and its weight may be ignored during the contact.

   (a)

   What is the magnitude of the impulse exerted on the ball by the club?

   [1 mark]

   • A$1.38$ N s
   • B$1300$ N s
   • C$2.76$ N s
   • D$166$ N s

   (b)

   What is the magnitude of the average force exerted on the ball during the contact?

   [1 mark]

   • A$5520$ N
   • B$1.38\times10^{-3}$ N
   • C$5.52$ N
   • D$55\,200$ N

   (c)

   The club has mass $0.3$ kg and is moving horizontally at $40$ m s$^{-1}$ immediately before it strikes the ball. Find the speed of the club immediately after the impact, assuming it continues in the same direction.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A particle $R$ of mass $1.5$ kg is at rest on a smooth horizontal plane. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal and perpendicular. $R$ receives an impulse of $(9\mathbf{i}+12\mathbf{j})$ N s.

   (a)

   What is the velocity of $R$ immediately after the impulse?

   [1 mark]

   • A$(9\mathbf{i}+12\mathbf{j})$ m s$^{-1}$
   • B$(13.5\mathbf{i}+18\mathbf{j})$ m s$^{-1}$
   • C$(-6\mathbf{i}-8\mathbf{j})$ m s$^{-1}$
   • D$(6\mathbf{i}+8\mathbf{j})$ m s$^{-1}$

   (b)

   What is the speed of $R$ immediately after the impulse?

   [1 mark]

   • A$14$ m s$^{-1}$
   • B$10$ m s$^{-1}$
   • C$100$ m s$^{-1}$
   • D$15$ m s$^{-1}$

   (c)

   $R$ then receives a second impulse, after which it moves with velocity $2\mathbf{i}$ m s$^{-1}$. Find the second impulse.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A bullet of mass $0.02$ kg is fired horizontally at $400$ m s$^{-1}$ into a block of mass $4.98$ kg which is at rest on a horizontal surface. The bullet embeds in the block and the two move off together. The surface may be assumed smooth during the impact.

   (a)

   Find the speed of the block, with the bullet embedded, immediately after the impact.

   [3 marks]

   (b)

   Find the magnitude of the impulse exerted on the block by the bullet. The block and bullet then move onto a rough part of the surface and come to rest after $0.8$ s. Assuming the resistance is constant, find its magnitude.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two spheres, $C$ of mass $m$ kg and $D$ of mass $2$ kg, are modelled as particles on a smooth horizontal surface. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are horizontal and perpendicular.

   (a)

   $C$ and $D$ move towards each other along a straight line with speeds $6$ m s$^{-1}$ and $4$ m s$^{-1}$ respectively, and collide directly. After the collision $C$ has speed $2$ m s$^{-1}$ in the opposite direction to its original motion and $D$ has speed $1$ m s$^{-1}$ in its original direction of motion. Find $m$, and find the magnitude and direction of the impulse exerted on $D$.

   [6 marks]

   (b)

   At a later time $D$ is moving with velocity $(3\mathbf{i}+4\mathbf{j})$ m s$^{-1}$ when it receives an impulse of magnitude $10$ N s in the direction of $(-3\mathbf{i}+4\mathbf{j})$. Find the velocity of $D$ after the impulse, and the angle through which its direction of motion is turned.

   [6 marks]

   Total for question 8: 12 marks

End of questions