Projectile Motion

Launch angle, speed, gravity and air drag, with the trajectory traced live.

0/3challenges

Target practice

t = 0.00 s

VacuumWith drag (baseball)

Launch speedm/s

Launch angle°

Launch heightm

Gravity

Air drag

Projectile

145.0 g, 7.3999999999999995 cm, C_d 0.35, k = 6.36e-3 1/m

Range: 65.5m
Max height: 20.2m
Flight time: 3.98s
Impact speed: 22.3m/s

Drag costs 30% of the vacuum range (93.2 m) and lands at 74% of the launch speed.

The model

Without air, horizontal speed never changes and gravity pulls the vertical speed down at g. The path is a parabola with range R = v₀ cos θ · t and flight time t = (v₀ sin θ + √((v₀ sin θ)² + 2gh₀)) / g.

With air, drag grows with the square of speed: a = -k|v|v, where k = ρ C_d A / 2m. There is no closed form, so the path is integrated numerically with fourth-order Runge-Kutta in 2 ms steps.

Try this

In a vacuum 45° always goes furthest. Turn drag on with a table tennis ball and the best angle drops well below 45°.

Raise the launch height and the best angle drops too, even without air.

Assumes flat ground, still air at sea-level density (1.225 kg/m³), a non-spinning sphere with constant drag coefficient, and uniform gravity. Educational model, not an engineering design or validation tool.

Achievements0/3

Bullseye +30Land within a metre of the target flag.
Moon shot +30Throw something more than 1 km on the Moon.
Against the air +30Reach 100 m high with air drag on, on Earth.

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