Projectile Motion Calculator
Find the time of flight, maximum height, range and landing speed of anything thrown or launched. Enter the speed, angle and starting height. Calvo shows a picture of the path and explains every step, one at a time.
Projectile motion
Air resistance is ignored (ideal projectile). Use a negative angle to throw downwards. Speeds in m/s, distances in metres.
The formulas
| Quantity | Formula (from the ground) | With starting height h₀ |
|---|---|---|
| Sideways speed | vx = v₀ cosθ | same |
| Up speed | vy = v₀ sinθ | same |
| Time to top | t = vy / g | same |
| Maximum height | H = vy² / 2g | H = h₀ + vy² / 2g |
| Time of flight | T = 2vy / g | T = (vy + √(vy² + 2gh₀)) / g |
| Range | R = v₀² sin 2θ / g | R = vx × T |
Worked example
A ball is kicked at 20 m/s at 45° from the ground (g = 9.81). Then vx = vy = 14.14 m/s. Time to the top = 14.14 / 9.81 = 1.44 s, so the total time is 2.88 s. Maximum height = 14.14² / (2 × 9.81) = 10.19 m. Range = 14.14 × 2.88 = 40.77 m.
Useful facts
On level ground the longest range is at 45°. Two angles that add up to 90° (like 30° and 60°) give the same range, but the higher angle stays in the air longer. On the Moon, with gravity about six times smaller, the same throw goes about six times farther.
FAQ
How do you find the range of a projectile?
From level ground use R = v0^2 x sin(2 theta) / g. If it starts from a height h0, find the time of flight T = (vy + sqrt(vy^2 + 2 g h0)) / g and then R = vx x T.
What angle gives the maximum range?
45 degrees, when the ball starts and lands at the same height and air resistance is ignored. Launching from a height makes the best angle a little smaller than 45 degrees.
What is the time of flight of a projectile?
From the ground, T = 2 v0 sin(theta) / g. It is the time between launch and landing.
What is the maximum height of a projectile?
H = h0 + (v0 sin(theta))^2 / (2 g). The ball is highest when its upward speed becomes zero.
Does mass change the path of a projectile?
No. Without air resistance every object follows the same path, whatever its mass, because gravity speeds up all objects equally.
Can I use it for objects thrown downwards or off a cliff?
Yes. Use a negative angle to throw downwards, or enter a starting height for a cliff or table. Angle 0 means thrown straight sideways.