Projectile motionEdexcel A-Level Maths: Subtopic test

1. 1
   A ball is kicked from a point on horizontal ground with speed $20$ m s$^{-1}$ at $30^\circ$ above the horizontal. Model the ball as a particle moving freely under gravity, with no air resistance. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the time taken for the ball to reach its greatest height.

   [1 mark]

   • A$2.04$ s
   • B$1.77$ s
   • C$1.02$ s
   • D$0.51$ s

   (b)

   Find the greatest height of the ball above the ground.

   [1 mark]

   • A$10.2$ m
   • B$15.3$ m
   • C$20.4$ m
   • D$5.10$ m

   (c)

   Find the horizontal range of the ball, that is, the distance from the kick to where it first lands.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A stone is thrown horizontally with speed $12$ m s$^{-1}$ from the top of a vertical cliff, $45$ m above the level sea. Model the stone as a particle moving freely under gravity. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the time taken for the stone to reach the sea.

   [1 mark]

   • A$3.03$ s
   • B$4.59$ s
   • C$9.18$ s
   • D$1.52$ s

   (b)

   Find the horizontal distance from the foot of the cliff to the point where the stone enters the sea.

   [1 mark]

   • A$55.1$ m
   • B$36.4$ m
   • C$110$ m
   • D$29.7$ m

   (c)

   Find the speed of the stone as it enters the sea.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A golf ball is hit from a point on horizontal ground with speed $25$ m s$^{-1}$ at an angle $\alpha$ above the horizontal, where $\tan\alpha=\frac34$. Model the ball as a particle moving freely under gravity, in a vertical plane, with no air resistance. Take $g=9.8$ m s$^{-2}$.

   (a)

   Find the time of flight of the ball and its horizontal range.

   [3 marks]

   (b)

   A thin vertical pole of height $8$ m stands on the ground, $40$ m horizontally from where the ball was hit, in the plane of the ball's motion. Show that the ball passes over the pole.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A particle is projected from a point $O$ on horizontal ground with speed $U$ at an angle $\alpha$ above the horizontal. The particle moves freely under gravity in a vertical plane, with constant gravitational acceleration $g$ and no air resistance.

   (a)

   Show that the time of flight is $T=\frac{2U\sin\alpha}{g}$ and that the horizontal range is $R=\frac{U^2\sin2\alpha}{g}$.

   [6 marks]

   (b)

   (i) Show that the equation of the path of the particle is $y=x\tan\alpha-\frac{gx^2}{2U^2\cos^2\alpha}$, where $x$ and $y$ are the horizontal and vertical distances from $O$.
   (ii) Given that $U=14$ m s$^{-1}$, $\tan\alpha=\frac34$ and $g=9.8$ m s$^{-2}$, find the two horizontal distances from $O$ at which the particle is $2$ m above the ground.

   [6 marks]

   Total for question 4: 12 marks

End of questions