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Calculators/Projectile Motion
Classical Mechanics

Projectile Motion Calculator

Calculate range, maximum height, time of flight and full trajectory for any projectile. Enter launch speed and angle — every result updates instantly with step-by-step working.

Inputs

Trajectoryg = 9.81 m/s²
Enter valid inputs to calculate.

Projectile motion equations

Horizontal position
x = v₀cosθ · t
Vertical position
y = h₀ + v₀sinθ · t − ½gt²
Range (flat ground)
R = v₀²sin(2θ) / g
Maximum height
H = h₀ + (v₀sinθ)² / 2g
Time of flight
t = (v₀sinθ + √(v₀²sin²θ + 2gh₀)) / g
Optimal angle for max range
θ = 45° (flat ground, no air drag)

How to use the projectile motion calculator

Enter your launch speed in metres per second, your launch angle in degrees, and optionally a launch height if the projectile starts above the landing surface. The calculator instantly shows range, maximum height, flight time and final speed, with a live trajectory diagram and full step-by-step solution.

The gravity preset lets you compare the same throw on Earth, Moon, Mars or Jupiter — a useful way to build intuition for how g affects range.

What is projectile motion?

Projectile motion describes the path of an object launched into the air that moves only under the influence of gravity (ignoring air resistance). The key insight is that horizontal and vertical motion are completely independent — horizontal velocity stays constant throughout the flight, while vertical velocity changes at exactly g = 9.81 m/s² on Earth.

This independence is what creates the parabolic trajectory — constant horizontal displacement combined with accelerating vertical displacement produces a parabola. For a full explanation, see our article on Projectile Motion and Why Projectile Motion Is a Parabola.

Frequently asked questions

What angle gives maximum range?↓
45° gives maximum range on flat ground with no air resistance. With air resistance, the optimal angle is slightly below 45° (around 38–42°). From an elevated position, the optimal angle is also less than 45°.
Does mass affect projectile range?↓
No — in the ideal model (no air resistance), mass cancels out of the equations. A feather and a cannonball launched at the same speed and angle follow the same trajectory in a vacuum.
How does air resistance change the trajectory?↓
Air resistance adds a drag force proportional to v² that opposes motion. It reduces range, lowers maximum height, and makes the trajectory asymmetric — the descent is steeper than the ascent.
What is the range formula?↓
For a flat surface: R = v₀²·sin(2θ)/g. This simplifies to R = v₀²/g at 45°, which gives the maximum range. For launch from a height, a more complex formula involving the quadratic formula is needed.

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