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Mechanics · Projectiles

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Projectile motion formulae

The standard results for a projectile launched at speed u and angle θ over level ground — time of flight 2u sinθ/g, greatest height u²sin²θ/(2g), range u²sin2θ/g — and using them in reverse to find u or θ.

Mechanics · Projectiles

Projectile motion formulae

The standard results for a projectile launched at speed u and angle θ over level ground — time of flight 2u sinθ/g, greatest height u²sin²θ/(2g), range u²sin2θ/g — and using them in reverse to find u or θ.

Why it works

The three landmark results for a projectile launched from and landing at the same level, with speed uu at angle θ\theta, all come straight from the suvat equations — worth knowing as formulae once you've seen where they come from:

time of flight T=2usin⁡θg,greatest height H=u2sin⁡2θ2g,range R=u2sin⁡2θg.\text{time of flight } T = \frac{2u\sin\theta}{g}, \quad \text{greatest height } H = \frac{u^2\sin^2\theta}{2g}, \quad \text{range } R = \frac{u^2\sin 2\theta}{g}.

The range is worth a closer look. Range =ucos⁡θ×T=ucos⁡θ⋅2usin⁡θg=u2(2sin⁡θcos⁡θ)g=u2sin⁡2θg= u\cos\theta\times T = u\cos\theta\cdot\dfrac{2u\sin\theta}{g} = \dfrac{u^2(2\sin\theta\cos\theta)}{g} = \dfrac{u^2\sin 2\theta}{g}, using the double-angle identity. Since sin⁡2θ\sin 2\theta is largest when 2θ=90∘2\theta = 90^\circ, the maximum range is at θ=45∘\theta = 45^\circ, and equals u2g\dfrac{u^2}{g}.

These formulae are quickest when the launch and landing heights match. If they don't (a cliff, a different target height), don't force a formula — go back to the independent horizontal and vertical equations.