Key takeaways
- Projectile range is R = v² × sin(2θ) ÷ g, using g = 9.81 m/s².
- The formula uses sin(2θ), so the angle is doubled before finding its sine.
- A 45° launch angle gives the maximum range for a given speed.
- Angles that add up to 90°, like 30° and 60°, give the same range.
In this guide
What is projectile range?
Projectile range is the horizontal distance a launched object travels before it lands, assuming it starts and ends at the same height and ignoring air resistance. It depends on how fast and at what angle the object is launched.
Why does projectile range matter?
Every arcing shot in sport, and every ballistics or engineering problem involving a launched object, comes down to this one relationship between speed, angle, and gravity.
A ball launched at 40° travels exactly as far as one launched at 50°, at the same speed. Move away from 45° in either direction, and you give up some range — 45° is the one angle that gets the most out of a given speed.
The formula
| Symbol | What it means | Example |
|---|---|---|
| R | The range, in metres (m) | 22.59 m |
| v | The launch speed, in metres per second (m/s) | 15 m/s |
| θ | The launch angle, measured from the ground, in degrees (°) | 40° |
How to calculate projectile range step by step
- 1Find the launch speed. Measure it in m/s.
- 2Find the launch angle. Measure it in degrees.
- 3Double the angle and find its sine. Use the sin, cos, tan calculator for
sin(2θ). - 4Square the speed, multiply by that sine value, then divide by 9.81.
Try the calculator below with your own speed and angle to see every step worked out.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using radians by mistake. The angle here is in degrees; mixing up degrees and radians, see the degrees to radians calculator, gives a completely wrong answer.
- Assuming a steeper launch always goes further. Range is greatest at a 45° launch angle; both flatter and steeper angles give a shorter range for the same speed.
- Ignoring that the object must land at the same height it started. This formula doesn't apply if the landing point is higher or lower than the launch point.
Tips and tricks
- Angles that add up to 90° always give the same range, so a 20° shot and a 70° shot at the same speed land in the same spot.
- This formula builds on the same falling-time ideas as free fall distance, just with a sideways launch added in.
- For a rough sense of scale, 22.59 m is roughly the length of a tennis court.
Where you'll use it in real life
- Sport: a soccer free kick, a basketball shot, or a javelin throw all follow this same relationship between angle and range.
- Ballistics: engineers and analysts use projectile motion to predict where a launched object will land.
- Water features: fountain designers use launch angle and speed to control how far water arcs before landing.
- Physics education: this formula is a classic example of combining trigonometry with motion under gravity.
Quick summary
To find projectile range, square the launch speed, multiply by the sine of double the launch angle, then divide by gravity (9.81 m/s²). A 45° launch angle always gives the maximum range for a given speed. Use the calculator above to try your own speed and angle.
Worked example
You kick a soccer ball at 15 m/s at a launch angle of 40° above the ground. Ignoring air resistance, how far does it travel before landing?
- 01R = v² sin(2θ) / g (g = 9.81 m/s²)
- 02R = 15² × sin(80°) / 9.81
- 03R = 225 × 0.9848 / 9.81
- 04R = 22.59 m
These numbers are pre-filled in the calculator above.
Frequently asked questions
How do I calculate the range of a projectile?
Square the launch speed, multiply by the sine of double the launch angle, then divide by gravity (9.81 m/s²): R = v²sin(2θ) ÷ g.
What launch angle gives the maximum range?
45° gives the longest range for a given launch speed, when the object lands at the same height it was launched from.
Why do 40° and 50° give the same range?
Because sin(2θ) is the same for angles that add up to 90°, so 40° and 50°, which sum to 90°, produce identical ranges at the same speed.
Does this formula account for air resistance?
No, it assumes ideal conditions with no air resistance, so real-world objects like a thrown ball will usually travel a bit less far than this formula predicts.
