Mechanics 2 (M2): Kinematics of a particle moving in a straight line or planeEdexcel International A Level Further Maths: Topic test

1. 1
   A pebble is thrown vertically upwards with speed $19.6\text{ m s}^{-1}$ from a point on a bridge $24.5$ m above a river. The pebble is modelled as a particle moving freely under gravity and later falls into the river. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the greatest height of the pebble above the bridge.

   [1 mark]

   • A$39.2$ m
   • B$19.6$ m
   • C$78.4$ m
   • D$44.1$ m

   (b)

   Find the time taken for the pebble to reach the river.

   [1 mark]

   • A$5$ s
   • B$4$ s
   • C$1$ s
   • D$9$ s

   (c)

   Find the speed of the pebble as it reaches the river.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A particle moves on a straight line. At time $t$ seconds, $t\ge0$, its velocity is $v\text{ m s}^{-1}$, where $v=t^2-6t+5$.

   (a)

   Find the acceleration of the particle when $t=4$.

   [1 mark]

   • A$-3\text{ m s}^{-2}$
   • B$8\text{ m s}^{-2}$
   • C$14\text{ m s}^{-2}$
   • D$2\text{ m s}^{-2}$

   (b)

   Find the displacement of the particle between $t=0$ and $t=3$.

   [1 mark]

   • A$3$ m
   • B$-4$ m
   • C$-3$ m
   • D$15$ m

   (c)

   Find the times at which the particle is instantaneously at rest.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A ball is projected from a point $O$ on horizontal ground with speed $24.5\text{ m s}^{-1}$ at an angle $\alpha$ above the horizontal, where $\tan\alpha=\frac43$. The ball moves freely under gravity. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the time for which the ball is in the air before it returns to the ground.

   [3 marks]

   (b)

   Find the speed of the ball when it is $14.7$ m above the ground.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A ball moves freely under gravity. At time $t$ seconds its position vector relative to a fixed point $O$ on horizontal ground is $\mathbf{r}=12t\,\mathbf{i}+(19.6t-4.9t^2)\,\mathbf{j}$ metres, where $\mathbf{i}$ is horizontal and $\mathbf{j}$ is vertically upwards.

   (a)

   Find the greatest height of the ball above the ground, justifying your method using the velocity of the ball.

   [6 marks]

   (b)

   The ball returns to ground level at the point $A$. Find the distance $OA$ and the speed of the ball at $A$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A marble leaves the edge of a horizontal table of height $0.784$ m with horizontal speed $2\text{ m s}^{-1}$. It then moves freely under gravity until it hits the horizontal floor. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Find the time taken by the marble to reach the floor.

   [1 mark]

   • A$0.16$ s
   • B$0.08$ s
   • C$0.4$ s
   • D$0.8$ s

   (b)

   Find the horizontal distance from the edge of the table to the point where the marble hits the floor.

   [1 mark]

   • A$5$ m
   • B$0.32$ m
   • C$1.6$ m
   • D$0.8$ m

   (c)

   Find the vertical component of the velocity of the marble as it hits the floor.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A particle moves on a straight line and passes through the point $O$ when $t=0$ with velocity $12\text{ m s}^{-1}$. At time $t$ seconds its acceleration is $(12t-18)\text{ m s}^{-2}$.

   (a)

   Find the velocity of the particle when $t=3$.

   [1 mark]

   • A$12\text{ m s}^{-1}$
   • B$18\text{ m s}^{-1}$
   • C$0\text{ m s}^{-1}$
   • D$54\text{ m s}^{-1}$

   (b)

   Find the displacement of the particle from $O$ when $t=2$.

   [1 mark]

   • A$0$ m
   • B$4$ m
   • C$-20$ m
   • D$16$ m

   (c)

   Find the least velocity of the particle for $t\ge0$.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A particle $P$ moves in a horizontal plane. At time $t$ seconds its velocity is $\mathbf{v}=\left[(4t-3)\mathbf{i}+(6-t^2)\mathbf{j}\right]\text{ m s}^{-1}$. When $t=0$ the position vector of $P$ relative to a fixed origin $O$ is $(2\mathbf{i}-\mathbf{j})$ m.

   (a)

   Find the magnitude of the acceleration of $P$ when $t=2$.

   [3 marks]

   (b)

   Find the position vector of $P$ when $t=3$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A ball is hit from a point $O$ on horizontal ground with speed $20\text{ m s}^{-1}$ at an angle $\alpha$ above the horizontal, where $\tan\alpha=\frac34$. A vertical wall of height $5$ m stands on the ground at a horizontal distance of $24$ m from $O$, in the vertical plane of the motion. The ball is modelled as a particle moving freely under gravity. Take $g=9.8\text{ m s}^{-2}$.

   (a)

   Show that the ball passes over the wall and find the vertical distance by which it clears the top of the wall.

   [6 marks]

   (b)

   Find the speed of the ball, and the angle its direction of motion makes with the horizontal, at the instant it passes over the wall.

   [6 marks]

   Total for question 8: 12 marks

End of questions