Projectile Range with Air Resistance Calculator
Estimate how far a projectile travels with quadratic air drag, and compare it to the ideal vacuum range, by numerical integration.
How it works
Vacuum range R₀ = v₀²·sin(2θ) ÷ g · With drag a = −g − (½ρC_dA ÷ m)·|v|·v (integrated numerically)
Without air the range has a tidy closed form, R₀ = v₀²·sin(2θ)/g, maximised at 45°. Air resistance has no simple formula because the drag force always opposes the velocity and changes direction throughout the flight, so this tool integrates the motion in small time steps. Drag both shortens the range and pulls the optimum launch angle below 45°, and the effect is largest for light, fast, bluff projectiles.
Worked example
A baseball (0.145 kg, area 0.0042 m², C_d = 0.3) launched at 40 m/s and 45° has a vacuum range of 40²·sin(90°)/9.81 = 1600/9.81 ≈ 163 m. Stepping the motion with quadratic drag in sea-level air brings the real range down to roughly 100 m — drag removes a large slice of the ideal distance.
Sources: Projectile motion — Wikipedia
Frequently asked questions
Why is there no exact formula for range with air resistance?
Quadratic drag couples the horizontal and vertical motion in a non-linear way, so the equations have no elementary closed-form solution. The standard approach — used here — is to integrate the motion numerically in small time steps.
Does 45° still give the maximum range?
Only in a vacuum. With air resistance the optimum launch angle drops below 45° — often to around 35–40° — because a lower, flatter trajectory spends less time fighting drag.
How accurate is the estimate?
It uses a constant drag coefficient and air density and a small fixed time step, which is good for a quick engineering estimate. Real flights add spin (the Magnus effect), altitude-varying density and Reynolds-number effects that this model omits.
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This calculator is for educational purposes. Double-check important results before acting on them.