Mechanics: KinematicsEdexcel A-Level Maths: Topic test

1. 1
   A remote-controlled car moves along a straight line. Its displacement from its starting point $O$ is $s$ metres at time $t$ seconds. For $0\le t\le4$ the car moves away from $O$ at a constant speed and reaches $s=12$. For $4\le t\le10$ the car is at rest. For $10\le t\le16$ the car returns to $O$ at a constant speed.

   (a)

   What is the velocity of the car when $t=2$?

   [1 mark]

   • A$1.5$ m s$^{-1}$
   • B$6$ m s$^{-1}$
   • C$3$ m s$^{-1}$
   • D$-3$ m s$^{-1}$

   (b)

   What is the average speed of the car over the whole $16$ s?

   [1 mark]

   • A$0$ m s$^{-1}$
   • B$1.5$ m s$^{-1}$
   • C$0.75$ m s$^{-1}$
   • D$1.2$ m s$^{-1}$

   (c)

   Find the velocity of the car when $t=13$, and state what the sign of your answer shows about the motion.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A lorry decelerates uniformly on the approach to a roundabout, slowing from $24$ m s$^{-1}$ to $6$ m s$^{-1}$ in $9$ s along a straight road. Assume the lorry continues to decelerate at the same rate until it stops.

   (a)

   What is the acceleration of the lorry?

   [1 mark]

   • A$-2$ m s$^{-2}$
   • B$2$ m s$^{-2}$
   • C$-2.7$ m s$^{-2}$
   • D$-3.3$ m s$^{-2}$

   (b)

   How far does the lorry travel in the $9$ s?

   [1 mark]

   • A$54$ m
   • B$216$ m
   • C$270$ m
   • D$135$ m

   (c)

   Find the further distance the lorry travels, from the moment its speed is $6$ m s$^{-1}$, before it comes to rest.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A bead moves along a straight horizontal wire. At time $t$ seconds, $t\ge0$, its displacement from a fixed point $O$ on the wire is $x$ metres, where $x=2t^3-15t^2+24t+3$.

   (a)

   Find the values of $t$ at which the bead is instantaneously at rest.

   [3 marks]

   (b)

   Find the total distance travelled by the bead during the first $5$ seconds.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A ball is thrown from the top of a tower, $10$ m above horizontal ground, with velocity $(8\mathbf{i}+5\mathbf{j})$ m s$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors horizontally and vertically upwards. The ball is modelled as a particle moving freely under gravity, with $g=10$ m s$^{-2}$.

   (a)

   Show that the ball hits the ground $2$ s after it is thrown. Hence find the horizontal distance from the foot of the tower at which the ball lands, and the speed of the ball as it hits the ground.

   [6 marks]

   (b)

   Taking the foot of the tower as the origin, with $x$ horizontal and $y$ vertically upwards, show that the path of the ball has equation $y=10+\frac58x-\frac{5}{64}x^2$. Hence find the greatest height of the ball above the ground.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A tennis ball is hit from ground level with velocity $(12\mathbf{i}+16\mathbf{j})$ m s$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors horizontally and vertically upwards. The ball is modelled as a particle moving freely under gravity, with $g=9.8$ m s$^{-2}$, and it lands on horizontal ground.

   (a)

   How long does the ball take to reach its greatest height?

   [1 mark]

   • A$1.22$ s
   • B$3.27$ s
   • C$2.04$ s
   • D$1.63$ s

   (b)

   What is the greatest height of the ball above the ground?

   [1 mark]

   • A$13.1$ m
   • B$7.35$ m
   • C$20.4$ m
   • D$26.1$ m

   (c)

   Find the horizontal distance between the point where the ball is hit and the point where it lands.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A radio-controlled helicopter moves in a horizontal plane with constant acceleration $(-0.4\mathbf{i}+0.3\mathbf{j})$ m s$^{-2}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors directed east and north. At time $t=0$ its velocity is $(4\mathbf{i}-\mathbf{j})$ m s$^{-1}$.

   (a)

   What is the velocity of the helicopter when $t=5$?

   [1 mark]

   • A$(6\mathbf{i}-2.5\mathbf{j})$ m s$^{-1}$
   • B$(-2\mathbf{i}+1.5\mathbf{j})$ m s$^{-1}$
   • C$(2\mathbf{i}+0.5\mathbf{j})$ m s$^{-1}$
   • D$(2\mathbf{i}-2.5\mathbf{j})$ m s$^{-1}$

   (b)

   What is the displacement of the helicopter from its position at $t=0$ to its position at $t=10$?

   [1 mark]

   • A$(40\mathbf{i}-10\mathbf{j})$ m
   • B$(20\mathbf{i}+5\mathbf{j})$ m
   • C$(-20\mathbf{i}+15\mathbf{j})$ m
   • D$(60\mathbf{i}-25\mathbf{j})$ m

   (c)

   Find the time at which the helicopter is travelling due east.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A cyclist rides along a straight road. She starts from rest and accelerates uniformly to $8$ m s$^{-1}$ in $5$ s. She then rides at this constant speed for $T$ seconds and finally decelerates uniformly to rest in $4$ s. The total distance she travels is $148$ m.

   (a)

   Use the area under the velocity-time graph to find the value of $T$.

   [3 marks]

   (b)

   Find the average speed of the cyclist for the whole journey, and the distance she has travelled when $t=12$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   Two particles $P$ and $Q$ move along the same straight line through a fixed point $O$. At time $t=0$ both are at $O$. At time $t$ seconds, $t\ge0$, the acceleration of $P$ is $(12-6t)$ m s$^{-2}$ and its velocity at $t=0$ is $-9$ m s$^{-1}$. Particle $Q$ moves with constant acceleration and has velocity $6$ m s$^{-1}$ at $t=0$. Both particles are at $O$ again when $t=3$.

   (a)

   For $P$: (i) find its velocity at time $t$; (ii) find the times at which it is instantaneously at rest; (iii) find the greatest distance of $P$ from $O$ for $0\le t\le3$.

   [6 marks]

   (b)

   For $Q$: (i) show that its acceleration is $-4$ m s$^{-2}$; (ii) find its greatest distance from $O$ for $0\le t\le3$; (iii) find the time, in the interval $0<t<3$, when $P$ and $Q$ have the same velocity.

   [6 marks]

   Total for question 8: 12 marks

End of questions