Projectile motion formulas, explained.
Projectile motion separates into constant horizontal velocity and vertical acceleration. Gravity changes the vertical component of velocity; it does not change the horizontal component in this ideal model.
The relationship to remember
Key formulas
- Horizontal velocity
- vₓ = v₀ cos θ
- Constant for the whole flight
- Initial vertical velocity
- vᵧ = v₀ sin θ
- Gravity reduces it by g each second
- Position at time t
- x = v₀ cos θ · t, y = v₀ sin θ · t − ½gt²
- The path is a parabola
- Time of flight
- T = 2v₀ sin θ / g
- Level launch and landing
- Maximum height
- H = v₀² sin²θ / (2g)
- Reached halfway through the flight
- Range
- R = v₀² sin 2θ / g
- Largest at θ = 45°
- Horizontal launch from height h
- t = √(2h/g), x = v₀√(2h/g)
- Launched horizontally, θ = 0
Worked example
Problem. A ball is kicked at 20 m/s and 30° above level ground. Using g = 9.81 m/s² and ignoring air resistance, find its time of flight, maximum height and range.
- Split the launch velocity: vₓ = 20 cos 30° = 17.32 m/s and vᵧ = 20 sin 30° = 10.00 m/s.
- Time of flight: T = 2vᵧ/g = 2 × 10.00 / 9.81 = 2.04 s.
- Maximum height: H = vᵧ²/(2g) = 100 / 19.62 = 5.10 m.
- Range: R = vₓT = 17.321 × 2.0387 = 35.31 m. Check: v₀² sin 60° / g = 400 × 0.8660 / 9.81 = 35.31 m.
Answer. The ball is in the air for about 2.04 s, rises 5.10 m and lands 35.3 m away.
A quick check
At 20 m/s and 45° under 9.81 m/s² gravity, flight lasts about 2.88 s, the range is 40.77 m, and the peak is 10.19 m.
Inside the default experiment
- Horizontal velocity = 20 cos(45°) = 14.14 m/s.
- Vertical launch velocity = 20 sin(45°) = 14.14 m/s.
- Flight time = 2vᵧ/g = 2.88 s. Range = vₓ × time = 40.77 m.
When this model applies
Point particle launched and landing at the same height; uniform gravity and no air resistance.
Make a prediction. Then test it.
Compare 30° and 60° at the same speed. What stays the same?
Try it: Projectile motion simulator ↗Frequently asked questions
Wondering about the why? Start here.
Why is 45° the best launch angle here?
For a level landing, range is v₀² sin(2θ)/g. The sine term reaches 1 when 2θ is 90°, so θ is 45°.
Does a heavier ball travel farther?
Mass does not appear in this ideal model. Air resistance can make real trajectories depend on mass, size, and shape.