Projectile motion
Model projectile motion using horizontal and vertical vector components; derive and apply equations for time of flight, range, and maximum height; interpret results in real-world contexts.
Worked examples
Time of flight, range, and maximum height
Straightforward
Problem
A ball is projected at 20 m/s at 30° above the horizontal. Take m/s². Find (a) the time of flight, (b) the range, (c) the maximum height.
1
Find the time of flight using .
2
Find the range using .
3
Find the maximum height using .
Answer
(a) The time of flight is s. (b) The range is m. (c) The maximum height is m.
Velocity at a given time
Moderate
Problem
A ball is projected at 25 m/s with and . Take m/s². Find the velocity vector and speed at s.
1
Find the horizontal velocity, which is constant.
2
Find the vertical velocity at s using .
The negative value means the ball is moving downward at s.
3
Find the speed using .
Answer
The velocity vector at s is m/s, and the speed is m/s.
Working backwards from range
Challenging
Problem
A ball is kicked at 45° and lands 90 m away on level ground. Take m/s². Find the initial speed.
1
Rearrange the range formula to make the subject.
2
Substitute the known values.
3
Take the square root to find .
Answer
The initial speed is m/s. (Note: the maximum range is achieved at 45° because is maximised when , giving .)
Practise
Q1·Straightforward
A ball is projected at 20 m/s at an angle of 30° above the horizontal. Taking m/s², find the time of flight in seconds (the time until it returns to the same height).
Explanation
The ball returns to launch height when the vertical displacement is zero again:
Q2·Straightforward
A projectile is launched at 30 m/s at 45° above the horizontal. Taking m/s², find the horizontal range in metres.
Explanation
Launch at 45° gives maximum range for a given speed — is maximised when , i.e., .
Q3·Straightforward
A projectile is launched at 20 m/s at 60° above the horizontal. Taking m/s², find the maximum height reached in metres.
Explanation
Q4·Straightforward
A projectile is launched at 25 m/s at an angle where and . Find the initial horizontal velocity component in m/s.
Explanation
The horizontal velocity remains constant throughout the flight (no air resistance). The vertical initial velocity is m/s.
Q5·Moderate
A ball is projected at 25 m/s with , . Taking m/s², find the vertical velocity in m/s at s.
Explanation
At s, the ball is still moving upward (positive vertical velocity). It will reach maximum height when , at s.
Q6·Moderate
A ball is projected at 25 m/s with , (same ball as Q5). Taking m/s², find the height of the ball in metres at s.
Explanation
Q7·Moderate
A projectile is launched at 20 m/s at 60° above the horizontal. Taking m/s², find the time to reach maximum height. Give your answer to 2 decimal places.
Explanation
At maximum height, :
This is exactly half the total time of flight.
This is exactly half the total time of flight.
Q8·Moderate
A projectile is launched at 25 m/s with , . Taking m/s², find the maximum height in metres.
Explanation
Q9·Moderate
A projectile is launched at 30 m/s at 30° above the horizontal. Taking m/s², find the horizontal range in metres to 2 decimal places.
Explanation
**Note:** and give the same range, since , i.e., .
Q10·Challenging
A ball is projected at 20 m/s at 30° above the horizontal. Taking m/s², find the speed of the ball in m/s at s. Give your answer to 2 decimal places.
Explanation
Horizontal velocity (constant):
Vertical velocity at s:
The negative sign means the ball is descending.
Speed:
Vertical velocity at s:
The negative sign means the ball is descending.
Speed:
Q11·Challenging
A ball is kicked at 45° and lands 80 m away on the same level. Taking m/s², find the initial speed to 2 decimal places.
Explanation
Q12·Challenging
A projectile is launched at 30 m/s with , . Taking m/s², find the horizontal range in metres.
Explanation
**Check the time of flight:** s. Range: m ✓
Open Math
Projectile motion
Vectors · ME-12-03
Name:
Date:
Q1Straightforward
A ball is projected at 20 m/s at an angle of 30° above the horizontal. Taking m/s², find the time of flight in seconds (the time until it returns to the same height).
Q2Straightforward
A projectile is launched at 30 m/s at 45° above the horizontal. Taking m/s², find the horizontal range in metres.
Q3Straightforward
A projectile is launched at 20 m/s at 60° above the horizontal. Taking m/s², find the maximum height reached in metres.
Q4Straightforward
A projectile is launched at 25 m/s at an angle where and . Find the initial horizontal velocity component in m/s.
Q5Moderate
A ball is projected at 25 m/s with , . Taking m/s², find the vertical velocity in m/s at s.
Q6Moderate
A ball is projected at 25 m/s with , (same ball as Q5). Taking m/s², find the height of the ball in metres at s.
Q7Moderate
A projectile is launched at 20 m/s at 60° above the horizontal. Taking m/s², find the time to reach maximum height. Give your answer to 2 decimal places.
Q8Moderate
A projectile is launched at 25 m/s with , . Taking m/s², find the maximum height in metres.
Q9Moderate
A projectile is launched at 30 m/s at 30° above the horizontal. Taking m/s², find the horizontal range in metres to 2 decimal places.
Q10Challenging
A ball is projected at 20 m/s at 30° above the horizontal. Taking m/s², find the speed of the ball in m/s at s. Give your answer to 2 decimal places.
Q11Challenging
A ball is kicked at 45° and lands 80 m away on the same level. Taking m/s², find the initial speed to 2 decimal places.
Q12Challenging
A projectile is launched at 30 m/s with , . Taking m/s², find the horizontal range in metres.
Worked solutions and answers at openmath.au/year-12/extension-1/introduction-to-vectors/projectile-motion