Projectile Motion Calculator
Enter the launch speed, angle and starting height to get the time of flight, range, maximum height and impact speed, then watch the trajectory animate on a scaled chart. It runs privately in your browser.
Step-by-step solution
Position over time
Solve the angle to hit a distance
Uses the speed, height and gravity above. The target is at landing level.
How to use the Projectile Motion Calculator
- Enter the initial speed and choose its unit (m/s, km/h, mph, ft/s or knots).
- Set the launch angle with the box or slider, and the initial height above the landing level.
- Pick the gravity: Earth is preset, or choose the Moon, Mars, another planet or a custom value.
- Read the results and press Play animation to watch the flight; tick Keep previous paths to compare shots.
- Solve for an angle: enter a target distance and press Solve angle, then Use to apply the answer.
Projectile motion equations
A projectile moves at constant horizontal speed while gravity accelerates it downward. Splitting the launch velocity into components, vₓ = v₀ cos θ and v₀ᵧ = v₀ sin θ, the position after time t is x = vₓ t and y = h₀ + v₀ᵧ t − ½ g t². Setting y = 0 gives the time of flight t = (v₀ᵧ + √(v₀ᵧ² + 2 g h₀)) / g; the range is R = vₓ t and the highest point is H = h₀ + v₀ᵧ² / (2g), reached at t = v₀ᵧ / g.
From level ground the formulas simplify to t = 2 v₀ sin θ / g, R = v₀² sin 2θ / g and H = v₀² sin² θ / (2g).
Range and height by launch angle
For a launch speed of 20 m/s from the ground on Earth (g = 9.80665 m/s²), ignoring air resistance:
| Angle | Time of flight | Range | Max height |
|---|---|---|---|
| 15° | 1.06 s | 20.4 m | 1.37 m |
| 30° | 2.04 s | 35.3 m | 5.10 m |
| 45° | 2.88 s | 40.8 m | 10.2 m |
| 60° | 3.53 s | 35.3 m | 15.3 m |
| 75° | 3.94 s | 20.4 m | 19.0 m |
Complementary angles (15° and 75°, 30° and 60°) land at the same spot, and 45° goes furthest. On the Moon (g = 1.62 m/s²) the same 45° kick travels about 247 m, six times further.
Tips and limits
- Launch height matters: a shot put released 2.1 m above the ground at 13.5 m/s reaches its longest throw at about 42°, not 45°.
- Air resistance is ignored. For fast or light objects (golf balls, shuttlecocks, bullets) real ranges are much shorter, and the best angle is usually lower.
- Units: enter speed in m/s, km/h, mph, ft/s or knots; switch results between metric and imperial at any time.
- For straight-line motion with constant acceleration, use the speed distance time calculator; to plot y = h₀ + x tan θ − g x² / (2 v₀² cos² θ) yourself, try the graphing calculator.
Frequently asked questions
What are the projectile motion formulas?
With launch speed v₀, angle θ, height h₀ and gravity g: vₓ = v₀ cos θ, v₀ᵧ = v₀ sin θ, time of flight t = (v₀ᵧ + √(v₀ᵧ² + 2gh₀)) / g, range R = vₓ·t and maximum height H = h₀ + v₀ᵧ² / (2g). From the ground (h₀ = 0) these simplify to t = 2v₀ sin θ / g and R = v₀² sin 2θ / g.
What angle gives the maximum range?
From level ground and without air resistance, 45° gives the maximum range, R = v₀² / g. When you launch from a height the best angle is a little below 45°, and the calculator's angle solver reports the exact optimum.
Why are there two angles that hit the same distance?
From level ground, complementary angles such as 30° and 60° give the same range. The low angle is a fast, flat shot; the high angle is a slow lob with a longer flight time and greater height.
How do I calculate a horizontal launch from a height?
Set the angle to 0° and enter the height. The time to fall is t = √(2h / g), independent of the horizontal speed, and the range is v₀ × t. A ball thrown at 15 m/s off a 30 m cliff lands after 2.47 s, 37.1 m away.
Does the calculator include air resistance?
No. It uses ideal projectile motion in a vacuum, which is accurate for dense, slow objects over short distances. Real balls, arrows and bullets fly shorter and land more steeply because of drag.
How fast does the projectile hit the ground?
The impact speed is √(vₓ² + vᵧ²) where vᵧ = v₀ sin θ − g t. Energy conservation gives the same result: v = √(v₀² + 2gh₀). Launched from the ground, it lands at the same speed and angle it was launched with.