What angle gives the maximum projectile range?

45° when launch and landing share a height. From an elevated launch the optimum is always lower — atan(v/√(v²+2gh)) — because trading some height for horizontal speed pays when gravity already grants you falling time.

How is time of flight calculated with a launch height?

By solving y(t) = h + v·sinθ·t − ½gt² = 0 and keeping the positive root: t = (v·sinθ + √((v·sinθ)² + 2gh))/g. With h = 0 it collapses to the familiar 2·v·sinθ/g.

Why is impact speed higher than launch speed from a height?

Energy conservation: the projectile converts the extra potential energy of the launch height into kinetic energy on the way down. From equal heights, impact speed equals launch speed exactly — a useful sanity check.

How different is a throw on the Moon?

Gravity is 1.62 m/s² against Earth's 9.81 — about one sixth — so the same launch flies six times farther and six times longer. Set gravity in the calculator and the whole solution rescales.

Does the calculator account for air resistance?

No — it solves the idealised vacuum problem, matching what physics courses ask for. Drag turns the equations nonlinear with no closed-form answer; for dense, slow objects over short ranges the vacuum result is close, for light or fast ones it is an upper bound.

Projectile Motion Calculator

Solve launch problems: range, maximum height, flight time and impact speed from launch speed, angle and height — with a to-scale trajectory plot.

The Projectile Motion Calculator runs entirely in your browser. Inputs are processed on your device and never uploaded.

Open the Speed Calculator

About Projectile Motion Calculator

Projectile Motion Calculator solves the textbook launch problem in full. Velocity splits into components — horizontal v·cos θ that never changes, vertical v·sin θ that gravity claws back — and the flight time comes from solving the quadratic for when the projectile returns to the ground, launch height included. From there follow range, apex height, time to apex, and the speed and angle of impact. The famous result that 45° maximises range holds only for launches from ground level; from any elevation the true optimum is lower, and the calculator reports it exactly. Gravity is editable, so the same throw can be compared on Earth, the Moon at 1.62 m/s² or Mars at 3.71. Air resistance is ignored, as in the classroom setup — which is also why real golf balls and bullets behave differently.

Features

Range, apex, flight time and time to apex
Impact speed and impact angle below horizontal
Launch height support with the correct quadratic solution
True optimal angle for range from elevated launches
Editable gravity for Moon and Mars comparisons
Speed input in m/s, km/h, mph or ft/s
To-scale trajectory diagram
Everything computed on your device

How to use the Projectile Motion Calculator

Enter launch speed and pick its unit
Set the launch angle and, if any, the launch height
Adjust gravity for off-Earth questions
Read range, apex, times and impact values off the result

Example

Input

20 m/s at 45° from ground level

Output

Range 40.8 m · apex 10.2 m · flight 2.88 s · impact 20 m/s at 45°

The symmetric textbook case — impact mirrors the launch exactly.

Common errors & troubleshooting

45° did not give the longest throw from a cliff. — The 45° rule assumes launch and landing at the same height. From an elevation the optimum drops — the calculator's “best angle for range” line gives the exact value for your inputs.
Real-world objects land far short of the calculated range. — The model ignores air resistance, which is enormous for light or fast objects. A golf ball's real range involves drag and lift; the vacuum solution here is the physics-class baseline.
A 90° launch shows zero range. — Straight up means no horizontal velocity component, so the projectile lands where it started. The apex and flight time are still computed — it is the vertical-throw special case.
Negative launch heights are rejected. — The model measures height above the landing plane, so it must be zero or positive. For a target above the launch point, swap the frame: compute from the target's plane instead.

Frequently asked questions

What angle gives the maximum projectile range?: 45° when launch and landing share a height. From an elevated launch the optimum is always lower — atan(v/√(v²+2gh)) — because trading some height for horizontal speed pays when gravity already grants you falling time.
How is time of flight calculated with a launch height?: By solving y(t) = h + v·sinθ·t − ½gt² = 0 and keeping the positive root: t = (v·sinθ + √((v·sinθ)² + 2gh))/g. With h = 0 it collapses to the familiar 2·v·sinθ/g.
Why is impact speed higher than launch speed from a height?: Energy conservation: the projectile converts the extra potential energy of the launch height into kinetic energy on the way down. From equal heights, impact speed equals launch speed exactly — a useful sanity check.
How different is a throw on the Moon?: Gravity is 1.62 m/s² against Earth's 9.81 — about one sixth — so the same launch flies six times farther and six times longer. Set gravity in the calculator and the whole solution rescales.
Does the calculator account for air resistance?: No — it solves the idealised vacuum problem, matching what physics courses ask for. Drag turns the equations nonlinear with no closed-form answer; for dense, slow objects over short ranges the vacuum result is close, for light or fast ones it is an upper bound.

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