Projectile motion calculator with trajectory, table and angle solver
This projectile motion calculator solves the classic physics problem: an object launched at speed v₀ and angle θ from a height h₀ under gravity g. It returns the range, maximum height, time of flight, time to the apex, impact speed and angle, the equation of the parabola, a trajectory table and a graph with the apex and landing point marked.
Switch to Solve for the launch angle to answer the reverse question — at what angle must I throw to hit a target 30 m away and 2 m up? — and get both the low (direct) and high (lob) solutions, or an “out of reach” verdict with the minimum speed required. Gravity presets for Earth, the Moon, Mars and Jupiter make it easy to compare worlds.
The model is the ideal one taught in school and first-year physics: constant gravity and no air resistance.
How to use the Projectile Motion Calculator
- Enter the launch speed in m/s (convert km/h or mph first with the unit converter), the launch angle in degrees above the horizontal and the launch height in metres (0 for ground level).
- Pick a gravity preset or choose “Custom…” and type your own g.
- Read the range, maximum height and flight time in the result panel, with the full breakdown and formulas in the details list.
- Scroll the trajectory table or press “Copy CSV” to paste it into a spreadsheet; the graph shows the same path.
- To find a launch angle, switch mode, enter the target distance and height, and read the two possible angles with their flight times and heights.
Projectile Motion Calculator features
Complete kinematics
Range, apex height and position, time of flight, impact velocity vector, trajectory equation and the optimum angle for maximum range — including the less-than-45° optimum when launching from a height.
Launch-height aware
Flight time uses the full quadratic solution t = (v_y + √(v_y² + 2gh₀))/g, so throws from a cliff, a table or a shoulder height are handled correctly.
Angle solver
Solves the quadratic in tan θ for any target distance and height, returning both trajectories, or the minimum speed needed when the target is out of reach.
Gravity presets
Earth 9.80665 m/s² (standard gravity), Moon 1.62, Mars 3.721, Jupiter 24.79, or any custom value.
Trajectory table, CSV and graph
Thirteen evenly spaced instants with x, y, vₓ, v_y and speed, copyable as CSV, plus an SVG plot with the launch point, apex and landing.
The projectile motion equations used
With x horizontal, y vertical (upwards positive), launch at (0, h₀) and velocity components vₓ = v₀ cos θ and v_y = v₀ sin θ, the motion is uniform horizontally and uniformly accelerated vertically:
| Quantity | Formula | Notes |
|---|---|---|
| Position | x = vₓ t · y = h₀ + v_y t − ½ g t² | Parabola: y = h₀ + x tan θ − g x² / (2 v₀² cos² θ) |
| Time of flight | t = (v_y + √(v_y² + 2 g h₀)) / g | Reduces to 2 v₀ sin θ / g when h₀ = 0 |
| Range | R = vₓ · t | Flat ground: R = v₀² sin 2θ / g, maximum at 45° |
| Maximum height | H = h₀ + v_y² / (2g) | Reached at t = v_y / g |
| Impact speed | √(vₓ² + v_y² + 2 g h₀) | Equals v₀ when h₀ = 0 (energy conservation) |
| Angle to hit (R, y) | tan θ = [v₀² ± √(v₀⁴ − g(gR² + 2(y − h₀)v₀²))] / (gR) | Two solutions, one, or none |
Worked example: 20 m/s at 45° on Earth
The components are vₓ = v_y = 20 cos 45° = 14.142 m/s. Time to the apex is 14.142 / 9.80665 = 1.442 s, so the total flight time on flat ground is 2.884 s. The range is 14.142 × 2.884 = 40.79 m, which matches v₀² sin 90° / g = 400 / 9.80665. The maximum height is 14.142² / (2 × 9.80665) = 10.20 m, and the ball lands at 20 m/s, 45° below the horizontal — the mirror image of the launch.
On the Moon (g = 1.62 m/s²) the same throw travels 400 / 1.62 = 246.9 m and stays airborne for 17.5 s. Launching from a 10 m platform at 30° instead gives a range of about 48.1 m and shows why the best angle drops below 45° when there is extra height to fall through.
Why real projectiles fall short
Air drag grows roughly with the square of speed and always opposes the motion, so real ranges are shorter and the optimum launch angle is lower than 45° (around 30–40° for a golf ball or football). Spin adds lift (the Magnus effect) and wind shifts the landing point. The calculator leaves these out because they depend on the object’s shape, mass and spin and have no closed-form solution. For a shot put or a thrown ball at modest speed the ideal model is within a few per cent; for an arrow or a bullet, treat the numbers as an upper bound.
- Angles are measured from the horizontal; negative angles (throwing downward) are allowed when h₀ > 0. The landing is taken at y = 0 — for a raised target use the angle-solver mode.
- Use the speed calculator for the launch speed from distance and time, or the quadratic equation solver to see the time-of-flight quadratic solved by hand.
Projectile Motion Calculator FAQ
Which launch angle gives the maximum range?
45° on flat ground without air resistance. From a raised launch point the optimum is lower — the calculator reports it (for example 40.6° for 20 m/s from 10 m up). With air resistance the real optimum is lower still.
Does the projectile motion calculator include air resistance?
No. It uses the ideal model with constant gravity and no drag, which is what textbook problems assume. The page explains how real trajectories differ.
How is the time of flight calculated from a height?
From y = h₀ + v_y t − ½ g t² = 0, solved as a quadratic: t = (v_y + √(v_y² + 2 g h₀)) / g. The negative root is discarded because it is before the launch.
Why are there two launch angles for one target?
A projectile can reach a point by a flat, fast path or by a high lob; both satisfy the same quadratic in tan θ. They merge at the limit of the range, beyond which no angle works.
What value of g should I use?
Standard gravity 9.80665 m/s² is the default; textbooks often use 9.8 or 9.81, which changes answers by under 0.1 %. Choose “Custom…” to match your course.