Projectile Motion Calculator
Calculate the range, maximum height, time of flight, and impact speed of a projectile from its launch speed, angle, and height.
How to use this calculator
- 1Enter the launch speed in metres per second.
- 2Enter the launch angle above horizontal, from 0° (flat) to 90° (straight up).
- 3Add a launch height if the projectile does not start at ground level.
- 4Change gravity to model the Moon, Mars, or another world.
How it works
Projectile motion (no air resistance)
vₓ = v·cos(θ) (constant) v_y = v·sin(θ) (changes under gravity) time of flight = (v_y + √(v_y² + 2·g·h₀)) ÷ g range = vₓ × time of flight max height = h₀ + v_y² ÷ (2g)
The key insight is that horizontal and vertical motion are completely independent. Gravity acts only downward, so the horizontal velocity never changes while the vertical velocity slows, stops at the peak, and reverses. Splitting the launch velocity into these two components lets you treat the flight as constant-speed motion sideways and free-fall up-and-down at the same time, then recombine them for the trajectory.
Worked example
Launching at 20 m/s and 45° from ground level on Earth (g = 9.81): the components are 14.14 m/s each way, the flight lasts 2.88 s, the peak height is 10.19 m, and the range is 40.77 m — the maximum range for that speed, which 45° always gives when launch and landing are level.
Projectile Motion Calculator: the complete guide
Two independent motions at once
The whole of projectile motion rests on one idea: horizontal and vertical motion do not affect each other. Gravity pulls straight down, so it changes only the vertical velocity, leaving the horizontal velocity constant throughout the flight. A projectile therefore drifts sideways at a steady rate while simultaneously rising and falling under gravity, exactly as if you were watching two separate motions.
This is why a bullet fired horizontally and a bullet dropped from the same height hit the ground at the same instant — their vertical motions are identical, and the forward speed of the fired bullet is irrelevant to how fast it falls. Splitting the launch velocity into horizontal and vertical components is the first step in every projectile problem for exactly this reason.
Why 45° gives the maximum range
For a launch and landing at the same height, 45° sends a projectile farthest. Steeper angles buy more air time but less forward speed; shallower angles give more forward speed but too little air time. Forty-five degrees is the balance point, and range falls off symmetrically on either side — 30° and 60° produce the same distance.
The symmetry breaks once the launch height changes. Throwing from a cliff or a raised platform, the optimal angle drops below 45°, because the extra falling distance already provides air time and a flatter, faster launch covers more ground. Shot-putters and long-jumpers, launching from above the landing point, optimise at angles well under 45° for this reason.
The limits of the ideal model
These equations assume a vacuum. In real air, drag removes energy continuously, so every figure here is an overestimate — and increasingly so at high speed, where drag grows with the square of velocity. A baseball hit at 45° in a vacuum would travel far beyond any real home run; air resistance cuts its range by roughly 40%.
The model is still genuinely useful. For dense, compact, relatively slow projectiles over short distances — a thrown ball, a jumping athlete, water from a hose — air resistance is a small correction and the vacuum equations predict the motion well. It is only for light objects, high speeds, or long flights that drag dominates and a numerical simulation becomes necessary.
Frequently asked questions
At what angle does a projectile travel farthest?
45° above horizontal, when the launch and landing heights are equal. It optimally balances air time against forward speed. If the projectile launches from above the landing point, the best angle is somewhat less than 45°.
Why do horizontal and vertical motion not affect each other?
Because gravity acts only in the vertical direction. It changes the vertical velocity but leaves the horizontal velocity untouched, so the two motions proceed independently. This is why a dropped object and a horizontally thrown object fall for the same time.
Does this account for air resistance?
No — it uses the ideal, drag-free model. That is accurate for dense, slow projectiles over short distances but overestimates range for fast or light objects. A real batted baseball, for instance, travels around 40% less far than these equations predict.
How do I model a throw on the Moon or Mars?
Change the gravity field. Use 1.62 m/s² for the Moon or 3.71 m/s² for Mars instead of Earth's 9.81. Lower gravity dramatically increases both the range and the flight time for the same launch, which is why the Apollo astronauts could throw so far.