Range, maximum height and flight time

Three key quantities

For a launch from the ground (y0 = 0) with a speed v0 and an angle alpha, three results are worth knowing:

• Total flight time: T = 2v0sin(alpha) / g
• Maximum height: H = (v0sin(alpha))^2 / (2g)
• Range (total horizontal distance): L = v0^2 * sin(2*alpha) / g

Where do these formulas come from?

Maximum height is reached when vy(t) = 0, that is, when the vertical speed becomes zero (the peak of the parabola). The total flight time corresponds to the moment when y(t) becomes zero again. The range is obtained by plugging this flight time into x(t).

Concrete example

A projectile is launched at v0 = 20 m/s with alpha = 30 degrees, g = 9.8 m/s^2.

• T = 220sin(30) / 9.8 = 2200.5 / 9.8 = 20 / 9.8 = 2.04 s
• H = (200.5)^2 / (29.8) = 100 / 19.6 = 5.1 m
• L = 20^2 * sin(60) / 9.8 = 400*0.87 / 9.8 = 35.5 m

Summary table

Quantity	Formula	Condition for it
Flight time T	2v0sin(alpha)/g	y(t) = 0
Max height H	(v0sin(alpha))^2/(2g)	vy(t) = 0
Range L	v0^2sin(2alpha)/g	x(T)

Common pitfall

The range is maximal for alpha = 45 degrees (because sin(2*alpha) = sin(90) = 1, its maximum value). Many students wrongly think that a larger angle always gives a longer range: beyond 45 degrees, the range decreases again.