Horizontal and vertical components, time of flight, range, greatest height, the equation of the trajectory, projectiles launched from a height.
12 original questions, 91 marks. Type your answers below for an instant mark and a score (algebra in any equivalent form, e.g. 2sqrt(3) or 2√3), or print the worksheet, mark it with the answers sheet, and use the solutions to see every step written out by hand.
A ball is thrown horizontally with speed 12 m s−1 from a point O at the top of a vertical wall. The point O is 19.6 m above horizontal ground. The ball is modelled as a particle moving freely under gravity.
Find the time taken for the ball to reach the ground.
Find the horizontal distance from the foot of the wall to the point where the ball lands.
Find the speed of the ball just before it hits the ground.
5 marks
Show answer
a) 2 sb) 24 mc) 23.0 m s-1
2
A particle is projected from a point O on horizontal ground with speed 20 m s−1 at an angle of 30° above the horizontal. The particle moves freely under gravity.
Find
the greatest height of the particle above the ground,
the time of flight,
the horizontal distance from O to the point where the particle lands.
6 marks
Show answer
a) 5.10 mb) 2.04 sc) 35.3 m
3
A small stone is projected from a point on horizontal ground with speed 25 m s−1 at an angle α above the horizontal, where tan α = 34
The stone is modelled as a particle moving freely under gravity.
Find the speed of the stone 2 seconds after it is projected.
Find the direction of motion of the stone at this time.
6 marks
Show answer
a) 20.5 m s-1b) 13.0° below the horizontal
4
A stone is thrown from a point O at the top of a vertical cliff. The point O is 30 m above the sea. The stone is thrown with speed 15 m s−1 at an angle of 40° above the horizontal, as shown in the diagram. The stone hits the sea at the point S.
The stone is modelled as a particle moving freely under gravity.
Find the time taken for the stone to travel from O to S.
Find the distance from the foot of the cliff to S.
6 marks
Show answer
a) 3.65 sb) 41.9 m
5
A golf ball is hit from a point A on horizontal ground. It leaves the ground with speed U m s−1 at an angle of 35° above the horizontal and first lands at the point B on the ground, where AB = 80 m.
The ball is modelled as a particle moving freely under gravity.
Show that U2 sin 35° cos 35° = 392
Find the value of U.
Find the time the ball takes to travel from A to B.
7 marks
Show answer
a) U2 sin 35° cos 35° = 392b) U = 28.9c) 3.38 s
6
A ball is thrown from a point O on a balcony, 25 m above horizontal ground. The ball is thrown with speed 18 m s−1 at an angle of 30° below the horizontal, as shown in the diagram.
The ball is modelled as a particle moving freely under gravity. Find
the vertical component of the velocity of the ball just before it hits the ground,
the time taken for the ball to reach the ground,
the speed and direction of motion of the ball just before it hits the ground.
8 marks
Show answer
a) 23.9 m s-1 downwardsb) 1.52 sc) 28.5 m s-1 at 56.9° below the horizontal
7
A particle is projected from a point O on horizontal ground with speed 14 m s−1 at an angle α above the horizontal, where tan α = 43
At time t seconds the particle is x metres horizontally and y metres vertically from O. The particle moves freely under gravity.
Show that y = 43x − 572x2
Use your answer to part (a) to find the horizontal distance from O to the point where the particle lands.
A vertical wall of height 5 m stands 15 m horizontally from O, in the plane of the motion. Determine whether the particle passes over the wall.
9 marks
Show answer
a) y = 4x/3 - 5x2/72b) 19.2 mc) no: at x = 15, y = 4.375 m < 5 m
8
A girl throws a ball from a point A, 1.5 m above horizontal ground. The ball leaves A with speed U m s−1 at an angle α above the horizontal. The highest point reached by the ball is 4 m above the ground, and at this moment the ball has travelled 12 m horizontally from A.
The ball is modelled as a particle moving freely under gravity.
Show that U sin α = 7
Find the value of U and the size of the angle α.
8 marks
Show answer
a) U sin α = 7b) U = 18.2, α = 22.6°
9
A firework is launched from a point on horizontal ground with speed 24.5 m s−1 at an angle of 60° above the horizontal. The firework is modelled as a particle moving freely under gravity.
Find the length of time for which the firework is at least 15 m above the ground.
Write down the least speed of the firework during its flight.
Find the speed of the firework when it is 15 m above the ground.
8 marks
Show answer
a) 2.55 sb) 12.25 m s-1c) 17.5 m s-1
10
A ball is projected from a point O on horizontal ground with speed 14 m s−1 at an angle α above the horizontal. The ball passes through the point P, which is 10 m horizontally and 4 m vertically from O, as shown in the diagram.
The ball is modelled as a particle moving freely under gravity.
Show that the time taken to reach P is 57 cos α seconds.
Show that 5 tan2 α − 20 tan α + 13 = 0
Find the two possible values of α.
For the smaller value of α, find the time taken for the ball to reach P.
10 marks
Show answer
a) t = 5/(7 cos α)b) 5 tan2 α - 20 tan α + 13 = 0c) α = 72.6° or 39.2°d) 0.922 s
11
A footballer takes a free kick from a point K on horizontal ground, 25 m from the goal line. The ball leaves the ground with speed 22 m s−1 at an angle of 25° above the horizontal, in a vertical plane perpendicular to the goal line. A line of defenders stands 9.15 m from K; they jump to a height of 2.2 m. The crossbar of the goal is 2.44 m above the ground.
The ball is modelled as a particle moving freely under gravity.
Show that the ball passes over the defenders.
Determine whether the ball passes under or over the crossbar, and by how much.
State one limitation of the model that could affect your answer to part (b).
8 marks
Show answer
a) at 9.15 m the ball is 3.23 m high > 2.2 mb) over the crossbar by 1.51 m (height 3.95 m)c) air resistance (or spin, or the size of the ball) is ignored
12
A small stone is projected from a point O at the top of a vertical cliff. The point O is 45 m vertically above the point N on horizontal ground. The stone is projected with speed U m s−1 at an angle α above the horizontal, where tan α = 34. The stone hits the ground at the point B, where NB = 60 m, as shown in the diagram.
The stone is modelled as a particle moving freely under gravity.
Show that U = 17.5
Find the length of time for which the stone is higher than O.
Find the speed of the stone just before it hits the ground.
10 marks
Show answer
a) U = 17.5b) 15/7 s = 2.14 sc) 34.5 m s-1
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Every question here is original and written by Robinson Tuition. It follows the Edexcel AS specification and suits AQA and OCR AS maths too; this is not an exam-board paper.