Projectile Motion Calculator
Find how far and how high a launched object travels, and how long it stays in the air.
How to use the Projectile Motion Calculator
- Enter the launch speed and pick its unit.
- Enter the launch angle and pick its unit.
- Enter the launch height and pick its unit.
- Choose the gravity.
- Press Calculate. The result updates automatically as you type, and you can switch the result unit at any time.
About the Projectile Motion Calculator
A projectile is any object that moves only under gravity after launch: a thrown ball, a kicked football, water from a hose or a jumping athlete. The motion splits neatly into two independent parts. Horizontally the speed stays constant; vertically gravity slows the rise and speeds the fall, tracing a parabola.
From level ground, 45° gives the longest range, and complementary angles such as 30° and 60° land at the same spot. Launching from a height shifts the best angle a little below 45°. In real life, air drag shortens range considerably for fast or light objects, and spin can add lift, so treat results as an ideal upper bound.
Formula
vx = v₀ cos θ, vy = v₀ sin θ
Time of flight t = (vy + √(vy² + 2gh₀)) ÷ g
Range R = vx × t
Max height H = h₀ + vy² ÷ (2g)
From ground level (h₀ = 0) these simplify to R = v₀² sin 2θ ÷ g and t = 2v₀ sin θ ÷ g. Example: 20 m/s at 45° gives R = 400 ÷ 9.807 ≈ 40.8 m and H ≈ 10.2 m.
Example calculation
Using the default values:
- Launch speed = 20 m/s
- Launch angle = 45 °
- Launch height = 0 m
- Gravity = Earth (9.807 m/s²)
the calculator returns:
- Horizontal range: 40.7886 m
- Maximum height: 10.1972 m
- Time of flight: 2.8842 s
- Time to maximum height: 1.4421 s
- Impact speed: 20 m/s
Frequently asked questions
What angle gives the maximum range?
45° when the launch and landing heights are equal and air resistance is ignored. From a raised position the optimum is slightly lower.
Does mass affect projectile motion?
Not without air resistance. A heavy and light ball launched at the same speed and angle follow the same path in a vacuum.
Why is the horizontal speed constant?
Gravity acts only vertically, so with no air drag there is no horizontal force to change the horizontal velocity.
Last updated: · Category: Science calculators