Three launch pads, one panel.
Pick a mode below — everything recalculates live as you type, every result explains itself, and every arc is drawn to scale.
How fast it leaves the hand, ramp or barrel — 20 m/s is a strong throw.
From the horizontal. 45° flies farthest from ground level; use a minus angle to throw downhill from a height.
Height of the launch point above the landing ground — 0 m for level ground, ~1.6 m for a hand-held throw.
Earth by default — the Moon gives roughly six times the flight time.
Any positive value in m/s² — e.g. 8.87 for Venus.
Horizontal distance from the launch point to the target.
Height of the target above the landing ground — 3.05 m is a basketball hoop.
Where the projectile starts — shoulder height ≈ 1.6–2 m.
The fixed speed you can launch at — the solver finds every angle that hits.
The fixed angle you launch at — the solver finds the speed that reaches.
Heavier cuts through air better — drag pushes on area, mass pushes back.
0.47 for a smooth sphere; dimpled golf balls run near 0.25.
The silhouette facing the airflow — πr² for a ball, ≈ 0.0043 m² for a baseball.
1.225 kg/m³ at sea level, about 1.0 up a 2 km mountain. Set 0 for vacuum.
From three numbers to a flying curve.
No manual needed — pick, type, read. The graph does the rest.
Pick a mode
Trajectory analyses a full flight, Hit a target solves for the angle or speed that lands where you want, and Air drag compares the real world with the textbook.
Enter the launch
Speed, angle and height are enough — gravity presets run from the Moon to Jupiter, and everything recalculates as you type.
Read the results
Range, flight time, maximum height and impact speed all arrive with the exact formula that produced them — ready to copy for homework.
Watch it fly
Every arc is drawn to scale. Animate the flight to see the velocity vector swing, or overlay air drag against the vacuum curve.
The physics under the hood.
Ideal projectile motion is two simple motions stitched together: constant horizontal velocity and constant downward acceleration. Six equations do all the work.
The two motions
Horizontal velocity never changes; vertical velocity loses g metres per second, every second. The parabola is just these two stitched together.
Time of flight
Solves h₀ + v₀·sinθ·t − ½gt² = 0 — the moment the projectile returns to ground level.
Maximum height
The apex is where vertical velocity hits zero — plain kinematics in disguise.
Range on level ground
Famously maximised at 45° — but only when launch and landing heights match and the air is ignored.
Impact speed
Energy conservation: launch kinetic energy plus everything the height releases, all turned back into speed.
The path itself
Height as a function of distance — the parabola the graph draws, and the equation the target solver inverts.
Projectile questions, answered.
The things students ask most about throwing things through the air — each answer backed by the formulas above.
What is projectile motion?+
It's the motion of anything launched into the air that then moves only under gravity — a thrown ball, an arrow, a stone dropped off a cliff. It splits into two independent parts: a horizontal motion at constant velocity, and a vertical motion that accelerates downward at g. Stitch them together and you get the familiar parabolic arc.
How do I calculate the range of a projectile?+
On level ground, R = v₀²·sin(2θ)/g — launch speed, angle and gravity are all you need. From a raised launch point, first find the flight time t = (v₀sinθ + √(v₀²sin²θ + 2gh₀))/g, then multiply by the horizontal speed: R = v₀cosθ · t. The Trajectory mode does both instantly and draws the arc.
How do I find the maximum height?+
A projectile rises until its vertical velocity reaches zero, so the maximum height is H = h₀ + (v₀·sinθ)²/2g. Example: a 20 m/s launch at 30° from ground level peaks at (20 × 0.5)² / (2 × 9.81) ≈ 5.1 m.
What launch angle gives the maximum range?+
45° — but only when launch and landing heights are equal and air resistance is ignored. Launch from a cliff and the best angle drops below 45°, because the extra height already buys falling time, so a flatter, faster arc wins. Add real air and it drops further — most sports balls fly farthest at roughly 20–35°.
Why do two different angles hit the same target?+
Because the trajectory equation is quadratic in tanθ: for most reachable targets there are two roots — a flat, fast low arc and a slow, looping high arc — and both parabolas pass exactly through the target. Push the target to the edge of range and the two angles merge into one maximum-range shot. The Hit-a-target mode solves both.
How does air resistance change the trajectory?+
Drag continuously steals speed, so the path stops being a symmetric parabola: the descent is steeper than the climb, the range shrinks (often by 30–50% for balls), the apex drops, and the optimal launch angle moves lower. The Air-drag mode integrates the true drag force F = ½·ρ·Cd·A·v² with a 4th-order Runge–Kutta solver and overlays the result on the vacuum curve.
Does the projectile's mass matter?+
In a vacuum, no — gravity accelerates every mass equally, so a feather and a cannonball fired identically land together. With air resistance, yes: drag depends on size and shape, not mass, so light-and-large objects slow down much more. Doubling the mass halves the drag's deceleration — one reason heavy ammunition carries farther.
Is this calculator free, and is my data sent anywhere?+
It is completely free and runs entirely in your browser — nothing you type is uploaded, logged or sent to a server. No account, no sign-up, no limits.