Projectile Motion Range
Projectile Motion: Range, Height and Flight Time
A projectile launched over flat ground follows a parabolic path set entirely by its launch speed and angle (ignoring air resistance). This worksheet computes the horizontal range, the peak height and the total time of flight.
The range is R = v₀²·sin(2θ)/g, the peak height H = v₀²·sin²θ/(2g) and the flight time t = 2v₀·sinθ/g.
Inputs
Launch speed and angle (θ entered in radians; 0.7854 rad ≈ 45°, the range-maximising angle).
v_0 := 30 m/s=30 m/s
theta := 0.7854=0.79
g := 9.81 m/s^2=9.81 m/s^2
Range, height and flight time
R := v_0^2·sin(2·theta)/g=91.74 m
h_max := v_0^2·sin(theta)^2/(2·g)=22.94 m
t_flight := 2·v_0·sin(theta)/g=4.32 s
Results
The range is maximised at a 45° launch angle on level ground, where sin(2θ) reaches 1. The height and flight time peak at steeper angles. These closed-form results are the classic no-drag solution; real trajectories fall shorter once aerodynamic drag is included.
Assumptions & limitations
- No air resistance; constant gravitational acceleration.
- Launch and landing at the same elevation (level ground).
- Point mass; no spin or lift effects.