Mechanics (A2): projectiles and two-dimensional motionAQA A-Level Maths: Topic test

1. 1
   A drone moves in a horizontal plane. At time $t=0$ its position vector relative to a fixed point $O$ is $(2\mathbf{i}+5\mathbf{j})$ m and its velocity is $(3\mathbf{i}-4\mathbf{j})$ m s$^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north. The drone has constant acceleration $(-0.5\mathbf{i}+2\mathbf{j})$ m s$^{-2}$.

   (a)

   What is the velocity of the drone when $t=4$?

   [1 mark]

   • A$(-2\mathbf{i}+8\mathbf{j})$ m s$^{-1}$
   • B$(\mathbf{i}+4\mathbf{j})$ m s$^{-1}$
   • C$(5\mathbf{i}+4\mathbf{j})$ m s$^{-1}$
   • D$(\mathbf{i}-12\mathbf{j})$ m s$^{-1}$

   (b)

   What is the position vector of the drone when $t=4$?

   [1 mark]

   • A$(6\mathbf{i}+21\mathbf{j})$ m
   • B$8\mathbf{i}$ m
   • C$(14\mathbf{i}-11\mathbf{j})$ m
   • D$(10\mathbf{i}+5\mathbf{j})$ m

   (c)

   Find the speed of the drone when $t=4$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A basketball player releases a ball from a point $2.1$ m above level horizontal ground. The ball is released with speed $10$ m s$^{-1}$ at an angle $\alpha$ above the horizontal, where $\sin\alpha=0.8$. Model the ball as a particle moving freely under gravity, with $g=9.8$ m s$^{-2}$.

   (a)

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

   [1 mark]

   • A$5.37$ m
   • B$3.27$ m
   • C$8.63$ m
   • D$7.20$ m

   (b)

   How long after release does the ball hit the ground?

   [1 mark]

   • A$1.63$ s
   • B$0.816$ s
   • C$1.86$ s
   • D$1.30$ s

   (c)

   The ball hits the ground $1.86$ s after release. Find the horizontal distance travelled by the ball.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A remote-control boat moves on a lake. At time $t$ seconds, $t\ge0$, its position vector relative to a fixed point $O$ is $\mathbf{r}=\left[(t^3-3t)\mathbf{i}+(2t^2-8t)\mathbf{j}\right]$ m, where $\mathbf{i}$ and $\mathbf{j}$ are unit vectors due east and due north.

   (a)

   Show that the boat is never at rest.

   [3 marks]

   (b)

   Find the position vector of the boat and its speed at the instant when it is travelling due east.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A football is kicked from a point $O$ on level ground with initial velocity $(24\mathbf{i}+18\mathbf{j})$ m s$^{-1}$, where $\mathbf{i}$ is horizontal and $\mathbf{j}$ is vertically upwards. Model the football as a particle moving freely under gravity, with $g=9.8$ m s$^{-2}$.

   (a)

   Find (i) the time of flight, (ii) the horizontal range, (iii) the greatest height reached above the ground.

   [6 marks]

   (b)

   A tree $12$ m tall stands on the ground $60$ m from $O$, in the plane of the football's flight. (i) Show that the football passes over the tree, finding its height above the ground as it passes. (ii) Find the speed of the football at this instant. (iii) Explain why the real football may not clear the tree, even though the model predicts that it does.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A particle $P$ moves in a plane. At time $t$ seconds, $t\ge0$, its velocity is $\mathbf{v}=\left[(6t-2)\mathbf{i}+(3-t^2)\mathbf{j}\right]$ m s$^{-1}$. When $t=0$, $P$ is at the origin $O$.

   (a)

   What is the acceleration of $P$ when $t=3$?

   [1 mark]

   • A$(16\mathbf{i}-6\mathbf{j})$ m s$^{-2}$
   • B$(6\mathbf{i}+6\mathbf{j})$ m s$^{-2}$
   • C$(6\mathbf{i}-6\mathbf{j})$ m s$^{-2}$
   • D$(6\mathbf{i}-2t\mathbf{j})$ m s$^{-2}$

   (b)

   What is the position vector of $P$ when $t=2$?

   [1 mark]

   • A$\left(8\mathbf{i}+\frac{10}{3}\mathbf{j}\right)$ m
   • B$(10\mathbf{i}-\mathbf{j})$ m
   • C$\left(8\mathbf{i}+\frac{26}{3}\mathbf{j}\right)$ m
   • D$\left(12\mathbf{i}+\frac{10}{3}\mathbf{j}\right)$ m

   (c)

   Find the value of $t$ at which $P$ is moving parallel to $\mathbf{j}$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A marble rolls off the horizontal edge of a bench of height $0.9$ m with horizontal speed $2.4$ m s$^{-1}$ and then moves freely under gravity until it hits the level floor. Model the marble as a particle, with $g=9.8$ m s$^{-2}$.

   (a)

   How long does the marble take to reach the floor after leaving the bench?

   [1 mark]

   • A$0.303$ s
   • B$0.0918$ s
   • C$0.857$ s
   • D$0.429$ s

   (b)

   How far from the foot of the bench, horizontally, does the marble land?

   [1 mark]

   • A$0.727$ m
   • B$1.03$ m
   • C$2.06$ m
   • D$2.40$ m

   (c)

   Find the speed of the marble at the instant it hits the floor.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A ball is thrown from a point $O$. At time $t$ seconds after it is thrown, its position vector relative to $O$ is $\mathbf{r}=\left[6t\,\mathbf{i}+(15t-4.9t^2)\mathbf{j}\right]$ m, where $\mathbf{i}$ is horizontal and $\mathbf{j}$ is vertically upwards.

   (a)

   Find the velocity of the ball at time $t$, and the time at which the ball is at its greatest height.

   [3 marks]

   (b)

   Find the speed of the ball when it returns to the height of $O$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A ball $P$ is projected from a point $O$ on level ground with velocity $(12\mathbf{i}+24.5\mathbf{j})$ m s$^{-1}$, where $\mathbf{i}$ is horizontal and $\mathbf{j}$ is vertically upwards. At the same instant a second ball $Q$ is released from rest at the point with position vector $(36\mathbf{i}+73.5\mathbf{j})$ m relative to $O$. Both balls are modelled as particles moving freely under gravity, with $g=9.8$ m s$^{-2}$.

   (a)

   Show that $P$ and $Q$ collide, and find the time of the collision and the position vector of the point of collision.

   [6 marks]

   (b)

   (i) Find the velocity and the speed of $P$ at the instant of collision. (ii) Find the velocity of $Q$ at this instant. (iii) State two modelling assumptions made about the balls.

   [6 marks]

   Total for question 8: 12 marks

End of questions