Momentum and impulse in vector formEdexcel International A Level Further Maths: Subtopic test

1. 1
   A ball of mass $0.4$ kg is moving with velocity $(6\mathbf{i}+8\mathbf{j})$ m s$^{-1}$ when it is hit by a bat. Immediately after being hit it moves with velocity $(-2\mathbf{i}+5\mathbf{j})$ m s$^{-1}$. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are perpendicular and lie in a horizontal plane.

   (a)

   Find the impulse exerted on the ball by the bat.

   [1 mark]

   • A$(3.2\mathbf{i}+1.2\mathbf{j})$ N s
   • B$(-3.2\mathbf{i}-1.2\mathbf{j})$ N s
   • C$(-8\mathbf{i}-3\mathbf{j})$ N s
   • D$(-0.8\mathbf{i}+2\mathbf{j})$ N s

   (b)

   Find the magnitude of the impulse.

   [1 mark]

   • A$3.42$ N s
   • B$8.54$ N s
   • C$4.40$ N s
   • D$11.7$ N s

   (c)

   Find the angle between the direction of the impulse and the vector $-\mathbf{i}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Two particles $A$ and $B$, of masses $2$ kg and $3$ kg, move on a smooth horizontal plane and collide. Before the collision $A$ has velocity $(4\mathbf{i}-\mathbf{j})$ m s$^{-1}$ and $B$ has velocity $(-2\mathbf{i}+3\mathbf{j})$ m s$^{-1}$. After the collision $A$ has velocity $(-\mathbf{i}+2\mathbf{j})$ m s$^{-1}$. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are perpendicular and lie in the plane.

   (a)

   Find the total momentum of the two particles before the collision.

   [1 mark]

   • A$(2\mathbf{i}+2\mathbf{j})$ kg m s$^{-1}$
   • B$(8\mathbf{i}-2\mathbf{j})$ kg m s$^{-1}$
   • C$(2\mathbf{i}-11\mathbf{j})$ kg m s$^{-1}$
   • D$(2\mathbf{i}+7\mathbf{j})$ kg m s$^{-1}$

   (b)

   Find the velocity of $B$ after the collision.

   [1 mark]

   • A$(-2\mathbf{i}+4\mathbf{j})$ m s$^{-1}$
   • B$(4\mathbf{i}+3\mathbf{j})$ m s$^{-1}$
   • C$\left(\frac43\mathbf{i}+\mathbf{j}\right)$ m s$^{-1}$
   • D$(-\mathbf{i}+2\mathbf{j})$ m s$^{-1}$

   (c)

   Find the speed of $B$ after the collision.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A tennis ball of mass $0.06$ kg is moving with velocity $(-20\mathbf{i}+5\mathbf{j})$ m s$^{-1}$ when it is struck by a racket. The racket exerts an impulse of $(3\mathbf{i}+1.5\mathbf{j})$ N s on the ball. The unit vectors $\mathbf{i}$ and $\mathbf{j}$ are perpendicular and lie in a horizontal plane. Ignore the weight of the ball.

   (a)

   Find the velocity of the ball immediately after it is struck.

   [3 marks]

   (b)

   Find the speed of the ball immediately after it is struck, and the angle through which its direction of motion is turned.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   Two smooth spheres $P$ and $Q$, of masses $0.5$ kg and $1.5$ kg, move on a smooth horizontal table. Just before they collide, $P$ has velocity $(6\mathbf{i}+2\mathbf{j})$ m s$^{-1}$ and $Q$ has velocity $(-2\mathbf{i}+4\mathbf{j})$ m s$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are perpendicular unit vectors.

   (a)

   In one collision, $P$ and $Q$ coalesce to form a single particle. Find the velocity of the combined particle, and the kinetic energy lost in the collision.

   [6 marks]

   (b)

   In a different collision, with the same velocities before, $P$ has velocity $(-3\mathbf{i}+2\mathbf{j})$ m s$^{-1}$ after the collision. Find the velocity of $Q$ after the collision, and the impulse exerted on $P$ by $Q$. State the impulse exerted on $Q$ by $P$.

   [6 marks]

   Total for question 4: 12 marks

End of questions