Projectile Motion Calculator
Enter the launch speed, the angle and the height you launch from, and get the range, the maximum height, the time of flight and the speed and angle at impact — with every formula substituted and a to-scale trajectory plot. The model ignores air resistance, as textbook problems do. Runs in your browser; free, no sign-up.
Enter the launch speed, angle and height. The projectile moves under gravity alone — no air resistance — and lands on level ground at height zero.
Use a negative angle to throw downward (only useful from a height).
Leave blank or enter 0 for a launch from ground level.
Results
40.79 m
Range (horizontal distance)
2.884 s
Time of flight
10.2 m
Maximum height above the ground
20 m/s
Speed at impact
45°
Angle at impact, below horizontal
45°
Angle for the longest range
Longest possible range at this speed: 40.79 m
The working
- In SI units and degrees: v₀ = 20 m/s, θ = 45°, h₀ = 0 m, g = 9.80665 m/s².
- Split the launch velocity: vx = v₀ cos θ = 20 × cos 45° = 14.14 m/s, and vy = v₀ sin θ = 20 × sin 45° = 14.14 m/s.
- The flight ends when the height y = h₀ + vy·t − ½gt² reaches 0: t = (vy + √(vy² + 2gh₀)) / g = (14.142136 + √(14.142136² + 2 × 9.80665 × 0)) / 9.80665 = 2.884 s.
- Range: R = vx × t = 14.14 × 2.884 = 40.79 m.
- Peak: it is highest when vy has fallen to 0, at t = vy / g = 14.142136 / 9.80665 = 1.442 s, where H = h₀ + vy² / (2g) = 0 + 14.142136² / (2 × 9.80665) = 10.2 m.
- At impact vy = vy − g·t = 14.142136 − 9.80665 × 2.884 = -14.14 m/s, so the speed is √(vx² + vy²) = √(14.14² + -14.14²) = 20 m/s, at 45° below the horizontal.
Position during the flight
| Time (s) | Horizontal distance x (m) | Height y (m) |
|---|---|---|
| 0 | 0 | 0 |
| 0.5768 | 8.158 | 6.526 |
| 1.154 | 16.32 | 9.789 |
| 1.731 | 24.47 | 9.789 |
| 2.307 | 32.63 | 6.526 |
| 2.884 | 40.79 | 0 |
Trajectory
Both axes use the same scale, so the curve has its true shape. The path is a parabola.
Assumes no air resistance, a constant g, a point-like object and a flat landing level at height zero. Real projectiles — especially fast, light or tumbling ones — land shorter than this ideal model predicts.
How to Use the Projectile Motion Calculator
- 1. Enter the launch. Give the speed, the angle above the horizontal and the height of the launch point above the landing ground. Pick units for each — m/s, km/h, mph or ft/s; degrees or radians; metres or feet.
- 2. Choose the gravity. Earth uses the standard 9.80665 m/s². Moon, Mars or a custom value let you see how the same throw behaves elsewhere.
- 3. Read the results. Range, time of flight and maximum height come first, then the impact speed and angle and the launch angle that would have gone farthest.
- 4. Check the working and the plot. Each formula is shown with your numbers in it, and the trajectory is drawn with both axes at the same scale.
The Equations Behind It
Horizontal and vertical motion are independent. Horizontally the speed never changes; vertically gravity gives a constant downward acceleration g. Split the launch velocity into components, vx = v₀ cos θ and vy = v₀ sin θ, and the position at time t is x = vx·t and y = h₀ + vy·t − ½gt².
| Quantity | General formula | Launch from ground level (h₀ = 0) |
|---|---|---|
| Time of flight | t = (vy + √(vy² + 2gh₀)) / g | 2v₀ sin θ / g |
| Range | R = vx · t | v₀² sin 2θ / g |
| Maximum height | H = h₀ + vy² / 2g | v₀² sin² θ / 2g |
| Time to the peak | t = vy / g | v₀ sin θ / g |
| Speed at impact | √(v₀² + 2gh₀) | v₀ |
The impact speed follows from energy conservation. Reference: OpenStax University Physics, Volume 1, section 4.3.
Worked Examples
Level ground. A ball is kicked at 20 m/s at 45° from the ground (the calculator's default). The components are both 14.14 m/s. It stays up for 2.884 s, lands 40.79 m away, and reaches 10.2 m at its peak — after 1.442 s, exactly half the flight. It lands at 20 m/s, the launch speed, and 45° below the horizontal.
From a height. Throw a stone at 15 m/s at 30° from a 10 m cliff. Now vx = 12.99 m/s and vy = 7.5 m/s. It rises to 12.87 m, lands after 2.385 s and 30.98 m away, at 20.52 m/s and 50.73° below the horizontal. The longest range from this speed and height would use 36.16° and reach 31.39 m, because launching from a height favours flatter throws.
Another world. The 20 m/s, 45° kick on the Moon (1.62 m/s²) flies 17.46 s and 246.9 m. Range scales with 1/g, so it goes about six times as far.
What the Model Leaves Out
- Air resistance. Drag slows real objects, shortens the range and makes the fall steeper than the climb. For a slow, dense object over a short distance the error is small; for a fast, light or spinning ball it is not.
- A flat landing level. The calculator lands the projectile at height zero. To hit a target at a different height, use the Kinematics Calculator for the vertical motion and solve for the time.
- Constant g and a flat Earth. Over trajectories of a few kilometres the change in g and the curvature of the planet are negligible, but they matter for artillery and rockets.
- A point object. Spin, wind and the size of the projectile are ignored.
Frequently Asked Questions
- What launch angle gives the maximum range?
- From ground level it is 45°, because sin 2θ is largest there. From a raised launch point the best angle is below 45°: tan θ = v₀ / √(v₀² + 2gh₀). The calculator shows this angle and the range it would give.
- Why do two angles give the same range?
- On level ground, range depends on sin 2θ, and sin 2θ is the same for θ and 90° − θ. So 30° and 60° land at the same spot, although the steeper throw stays in the air longer and goes higher.
- How do I find the time of flight?
- Set the height equation y = h₀ + vy·t − ½gt² equal to zero and take the positive root: t = (vy + √(vy² + 2gh₀)) / g. From ground level this simplifies to 2v₀ sin θ / g.
- Does the mass of the projectile matter?
- Not in this model. Without air resistance every object follows the same path for the same launch, whatever its mass. In real air a heavier object of the same size is slowed less.
- Can I launch downward or horizontally?
- Yes. Use 0° for a horizontal launch from a cliff or table, or a negative angle for a downward throw. From ground level neither goes anywhere, so the calculator asks for a launch height.
- What gravity value does it use?
- 9.80665 m/s², the standard value fixed by definition, for Earth. The Moon is 1.62 m/s² and Mars 3.73 m/s² (NASA planetary fact sheets, mean surface values), or you can enter your own.
- Is the maximum height measured from the ground or the launch point?
- From the landing ground, so a launch from 10 m includes that 10 m. Subtract the launch height to get the rise above the launch point.