Projectile Motion: formula
Enter the launch speed in meters per second, the launch angle in degrees above horizontal and the gravitational acceleration. The range is the speed squared times the sine of twice the angle divided by gravity, the maximum height is the speed squared times the sine squared of the angle over twice gravity and the flight time is twice the speed times the sine of the angle over gravity. These hold for level ground and no air resistance.
Projectile Motion: worked example
| Launch speed (m/s) | 20 |
|---|---|
| Launch angle (degrees) | 45 |
| Gravity (m/s²) | 9.8 |
| Range (m) | 40.8 |
| Maximum height (m) | 10.2 |
| Flight time (s) | 2.9 |
Air resistance shortens real trajectories a lot, especially for light objects and high speeds, so a real ball flies less than half as far as the ideal figure at high speed. Launching from a height or landing at another level changes the range. Wind and spin matter.
Gravity varies with the planet.
Projectile Motion: changing Launch speed (m/s)
| Launch speed (m/s) | Range (m) | Maximum height (m) | Flight time (s) |
|---|---|---|---|
| 16 | 26.1 | 6.5 | 2.3 |
| 18 | 33 | 8.3 | 2.6 |
| 20 (example) | 40.8 | 10.2 | 2.9 |
| 22 | 49.3 | 12.3 | 3.2 |
| 24 | 58.7 | 14.7 | 3.5 |
Projectile Motion: compare input ranges
| Input | Tested from | To | Range (m): low input | Range (m): high input | Swing |
|---|---|---|---|---|---|
| Launch speed (m/s) | 18 | 22 | 33 | 49.3 | 16.3 (40%) |
| Gravity (m/s²) | 9 | 11 | 44.4 | 36.4 | 8.1 (20%) |
| Launch angle (degrees) | 41 | 50 | 40.4 | 40.2 | 0.223 (0.5%) |
Projectile Motion: individual input comparisons
| Input | Value | Range (m) | Maximum height (m) | Flight time (s) |
|---|---|---|---|---|
| Launch speed (m/s) | 16 | 26.1 | 6.5 | 2.3 |
| 18 | 33 | 8.3 | 2.6 | |
| 20 | 40.8 | 10.2 | 2.9 | |
| 22 | 49.3 | 12.3 | 3.2 | |
| 24 | 58.7 | 14.7 | 3.5 | |
| Launch angle (degrees) | 36 | 38.8 | 7 | 2.4 |
| 41 | 40.4 | 8.8 | 2.7 | |
| 45 | 40.8 | 10.2 | 2.9 | |
| 50 | 40.2 | 12 | 3.1 | |
| 54 | 38.8 | 13.3 | 3.3 | |
| Gravity (m/s²) | 8 | 50 | 12.5 | 3.5 |
| 9 | 44.4 | 11.1 | 3.1 | |
| 10 | 40 | 10 | 2.8 | |
| 11 | 36.4 | 9.1 | 2.6 | |
| 12 | 33.3 | 8.3 | 2.4 |
Projectile Motion: combined input changes
| Launch speed (m/s) / Launch angle (degrees) | 36 | 45 | 54 |
|---|---|---|---|
| 16 | 24.8 | 26.1 | 24.8 |
| 18 | 31.4 | 33 | 31.4 |
| 20 | 38.8 | 40.8 | 38.8 |
| 22 | 46.9 | 49.3 | 46.9 |
| 24 | 55.8 | 58.7 | 55.8 |
Projectile Motion: which assumptions move the result?
Projectile Motion: values and definitions
| Quantity and definition | Value used |
|---|---|
| Launch speed (m/s) | 20 |
| Launch angle (degrees) | 45 |
| Gravity (m/s²) | 9.8 |
| Range (m) | 40.8 |
| Maximum height (m) | 10.2 |
| Flight time (s) | 2.9 |
Projectile Motion: half and double Launch speed (m/s)
| Launch speed (m/s) | Range (m) |
|---|---|
| 10 | 10.2 |
| 20 | 40.8 |
| 40 | 163.1 |
Common error: Entering the speed in miles per hour or feet per second. The formulas use meters and seconds, and mixed units give a range that is far off.
Physics & Mechanics calculators: Free Fall Calculator, Centripetal Force Calculator, Pendulum Period Calculator, Horsepower Calculator, Speed, Distance and Time Calculator, Kinetic Energy Calculator.
A launched object rises and falls under gravity while it drifts forward at a steady pace. At 20 meters per second and 45 degrees, it flies about 40.8 meters, peaks near 10.2 meters and spends about 2.9 seconds in the air. On level ground with no air resistance, 45 degrees gives the longest range, and angles equally far from it above and below give the same distance.
Remember the model leaves out air resistance and assumes level ground. A real throw lands shorter, and the best angle is a little lower than 45.
See the falling case on the free fall calculator, the speed relations on the uniform acceleration calculator and the energy on the kinetic and potential energy calculators. The force and centripetal force pages cover related mechanics, and the right triangle calculator handles the components.
How does launch speed affect range?
Range grows with the square of the speed, so doubling the launch speed quadruples the distance.
Why do real projectiles fall short of the formula?
Air resistance slows them, so the formula, which ignores it, overestimates range, especially at high speeds.
What launch angle gives the longest range?
45 degrees on level ground without air resistance, and 30 and 60 degrees give the same range as each other.