Projectile Motion
Launch speed, angle and height in — range, peak, flight time and the full trajectory out. Includes real air resistance, so you can see how far the textbook parabola is from the truth.
Instant answers
Works offline once loaded
Nothing you type is sent anywhere
Drawn trajectory
Optional air drag
Other worlds
Launch
metres per second
0° is horizontal, 90° is straight up
metres above the landing surface
0.47 for a sphere, 0.8 for a cube, 0.04 for a streamlined shape
Range
—
No air resistance
With air resistance
Apex
Time of flight—
Peak height—
Time to peak—
Impact speed—
Impact angle—
Horizontal speed—
Best angle and target practice
Angle for maximum range—
That maximum range—
Complementary angle—
metres away, at the launch surface
Two solutions normally exist — a flat shot and a lob
Position second by second
| Time | Distance | Height | Speed | Direction |
|---|
Range against launch angle
Every angle from 1° to 89° at your launch speed. The curve is symmetric about 45° only when you launch and land at the same height.
Instructions
How to use this calculator
Step by step
- Enter the launch speed, the angle above horizontal, and how high above the landing surface the projectile starts. Zero height means launching and landing on the same level.
- Pick your units. Speed can be given separately in km/h or mph, since that is how most people describe a thrown or hit ball.
- Leave air resistance on "none" for the textbook problem. Every physics course starts here, and the answers match what you will be marked on.
- Switch air resistance on and pick a real object to see how much the vacuum answer overstates things. The trajectory turns from a parabola into a lopsided curve with a steeper descent.
- Read the trajectory chart: the blue curve is the vacuum path and the amber one, when shown, is the real path with drag.
- The best-angle panel tells you the angle that maximises range for your speed and height. It is 45° only when you launch and land at the same level with no air.
- Type a target distance to get the launch angles that reach it. There are usually two — a flat, fast shot and a high lob — and both land in the same place.
Good to know
- Horizontal and vertical motion are independent. Gravity does not care how fast you are moving sideways, which is why a bullet fired horizontally and one dropped from the same height hit the ground together.
- Range in a vacuum is v²sin(2θ)/g, so 30° and 60° give exactly the same distance. Launching from a height breaks that symmetry and pushes the best angle below 45°.
- Drag scales with the square of speed and with cross-sectional area, but the deceleration divides by mass. A dense small object like a shot put barely notices the air; a table-tennis ball is dominated by it.
- Doubling the launch speed quadruples the vacuum range, because range depends on v². With drag it does much less than quadruple.
- Time of flight depends only on the vertical component of velocity and the launch height. Two shots with the same vertical speed hang in the air the same length of time regardless of how far they travel.
- This page ignores spin. A golf ball or a football with backspin generates lift and can travel considerably further than the drag-only number here, which is why real golf drives beat the physics.
The maths behind it
- Velocity components vₓ = v·cos θ, v_y = v·sin θ Splitting the launch into independent horizontal and vertical motion.
- Height over time y = h + v_y·t − ½g·t² The vertical equation; horizontal is simply x = vₓ·t in a vacuum.
- Time of flight t = (v_y + √(v_y² + 2gh)) / g The positive root of y = 0. Reduces to 2v_y/g when h = 0.
- Range R = vₓ · t With h = 0 this simplifies to v²·sin(2θ)/g.
- Peak height y_max = h + v_y² / 2g Reached at t = v_y/g, exactly halfway through the flight when h = 0.
- Best launch angle θ = arcsin(1 / √(2 + 2gh/v²)) Maximises range from a raised launch point; equals 45° when h = 0.
- Quadratic drag a = −g ĵ − (ρ·C_d·A / 2m)·|v|·v Integrated numerically with fourth-order Runge-Kutta; ρ = 1.225 kg/m³.
Why is 45° not always the best angle?
45° is optimal only when the projectile lands at the same height it was launched from and there is no air. Launch from a hill or a raised hand and the extra fall time rewards a flatter, faster shot, so the best angle drops below 45°. Add air resistance and it drops further — for a baseball at typical speeds the optimum is around 40°.
How much difference does air resistance really make?
It depends entirely on the object. A shot put loses only a few percent of its range; a baseball loses roughly a third; a table-tennis ball loses most of it. The controlling quantity is the ratio of drag force to weight, which favours dense, small, smooth objects.
Why are there two angles that hit the same target?
Because range rises from zero at 0°, peaks near 45°, and falls back to zero at 90°. Any distance short of the maximum is therefore reached twice: once on the way up with a flat, fast trajectory, and once on the way down with a high, slow lob. Mortars use the high one; rifles use the low one.
Does mass affect the trajectory?
In a vacuum, not at all — that is Galileo's result, and the mass cancels out of every equation on this page. With air resistance mass matters a great deal, because the drag force is independent of mass but the resulting deceleration is divided by it. A heavy ball pushes through the air; a light one is stopped by it.
What about spin, wind and altitude?
None of them are modelled here. Spin produces a sideways Magnus force that can add serious distance to a golf ball or curve a football. Wind adds directly to the relative airspeed. Altitude thins the air and reduces drag, which is why balls fly further in Denver. Treat the drag numbers here as the still-air, no-spin baseline.
Can I use this for a physics assignment?
Yes — with air resistance set to none, the results are the exact closed-form solutions your textbook uses, and the formulas panel shows every one of them. The second-by-second table is useful for checking your own working line by line.
Why does the drag calculation take a moment on some settings?
Because there is no closed-form answer. The page integrates the equations of motion in two-millisecond steps with fourth-order Runge-Kutta, and the best-angle and target-solving panels each run that integration dozens of times. It is still fast enough to feel instant, but that is what is happening behind the number.