ProjectilesAQA A-Level Maths: Subtopic test

1. 1
   A ball is projected from a point $O$ on level ground with speed $20\ \text{m s}^{-1}$ at $30^\circ$ above the horizontal. Model the ball as a particle moving freely under gravity, with $g=9.8\ \text{m s}^{-2}$.

   (a)

   Find the vertical component of the initial velocity.

   [1 mark]

   • A$17.3\ \text{m s}^{-1}$
   • B$10\ \text{m s}^{-1}$
   • C$20\ \text{m s}^{-1}$
   • D$11.5\ \text{m s}^{-1}$

   (b)

   Find the time the ball is in the air before it lands.

   [1 mark]

   • A$1.02$ s
   • B$3.53$ s
   • C$4.08$ s
   • D$2.04$ s

   (c)

   Calculate the horizontal distance the ball travels before landing.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A stone is thrown horizontally at $15\ \text{m s}^{-1}$ from a window $19.6$ m above level ground. Model the stone as a particle moving freely under gravity, with $g=9.8\ \text{m s}^{-2}$.

   (a)

   Find the time the stone takes to reach the ground.

   [1 mark]

   • A$2.0$ s
   • B$1.4$ s
   • C$4.0$ s
   • D$3.0$ s

   (b)

   Find the horizontal distance from the foot of the wall to the point where the stone lands.

   [1 mark]

   • A$15$ m
   • B$19.6$ m
   • C$30$ m
   • D$49.4$ m

   (c)

   Find the speed of the stone as it hits the ground.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A particle is projected from a point $O$ on horizontal ground with initial velocity $(6\mathbf{i}+14.7\mathbf{j})\ \text{m s}^{-1}$, where $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertically upward unit vectors. The particle moves freely under gravity, with $g=9.8\ \text{m s}^{-2}$.

   (a)

   Find the velocity of the particle after $2$ s, and state whether it is moving upwards or downwards at that time.

   [3 marks]

   (b)

   Find the speed of the particle when it is $5$ m above the ground.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A golf ball is struck from a point $O$ on level ground with speed $25\ \text{m s}^{-1}$ at an angle $\alpha$ above the horizontal, where $\tan\alpha=\frac34$. A vertical wall $10$ m high stands on the ground $30$ m horizontally from $O$, in the plane of the ball's path. Model the ball as a particle moving freely under gravity, with $g=9.8\ \text{m s}^{-2}$.

   (a)

   Determine whether the ball passes over the wall and, if it does, by what vertical distance.

   [6 marks]

   (b)

   (i) Show that the path of the ball satisfies $y=0.75x-0.01225x^2$, where $x$ and $y$ are the horizontal and vertical distances from $O$ in metres.
   (ii) Hence find the range of values of $x$ for which the ball is more than $10$ m above the ground.

   [6 marks]

   Total for question 4: 12 marks

End of questions