Projectile Motion Formulas – Practice
R = v² sin(2θ)/g — Level-ground range (also T and H formulas)
On level ground, time of flight is T = 2v sinθ/g, range is R = v² sin(2θ)/g, and max height is H = (v sinθ)²/(2g). Horizontal velocity stays constant.
Guide — how to use this formula
- 1. Use the given sinθ / cosθ / sin(2θ) values as stated.
- 2. T = 2v sinθ/g; R = (v cosθ)T or v² sin(2θ)/g; H = (v sinθ)²/(2g).
- 3. At 45°, R_max = v²/g on level ground.
- 4. Complementary angles share the same range.
Derivation
Split the launch into horizontal and vertical components:
- Time of flight (return to launch height):
T = \frac{2v\sin\theta}{g} - Range:
R = v\cos\theta\cdot T = \frac{v^{2}\sin 2\theta}{g} - Max height:
H = \frac{(v\sin\theta)^{2}}{2g}
Problems here give sinθ or cosθ (or sin(2θ)) so you can focus on the structure.
Examples
Example: v = 20, sinθ = 0.5, g = 10
T = 2 s; H = 5 m.
Example: 45° range
v = 30, g = 10 → R = 90 m.