Free fall and projectile motion calculator
Fall time, range, maximum height and impact speed without air resistance, with editable g (Earth, Moon, Mars) and the worked calculation.
Result
40.79 m
- Initial components
- vx = 14.14 m/s; vy = 14.14 m/s
- Calculation
- t = (vy + √(vy² + 2·g·h₀)) ÷ g = (14.14 + √(14.14² + 2 × 9.807 × 0)) ÷ 9.807 = 2.884 s
- Flight time
- 2.884 s
- Maximum height
- 10.2 m
- Impact speed
- 20 m/s, that is 72 km/h
Calculation, assumptions and sources
OverviewFree fall: time and impact speed from a height.
Free fall: time and impact speed from a height. Projectile motion: range, flight time, maximum height and impact speed, from the ground or from a height, angled up or down. Air resistance neglected; g editable, standard gravity by default.
Method & formulasfree fall: t = √(2h/g), v = g·t
Horizontally, v_x is constant; vertically, the acceleration is g downward, independent of the horizontal motion. The flight time is the positive root of y(t) = 0. The impact speed follows from energy conservation and depends only on v₀ and h₀, not on the launch angle.
- Range (R)
- horizontal distance between launch and return to the ground (zero altitude).
- Maximum height (H)
- height of the apex of the trajectory, where the vertical speed vanishes, measured from the ground.
- Standard gravity (9.80665 m/s²)
- exact conventional value, fixed by the 3rd CGPM (1901). Real gravity varies by about ±0.3% with latitude and altitude.
Validity domainAir resistance is neglected.
Air resistance is neglected. The approximation holds for a dense, compact object at moderate speed, not for a ping-pong ball, a shuttlecock or a fast projectile. The model also assumes uniform g (trajectories short compared with Earth’s radius) and a point mass without spin, hence no Magnus effect and no lift.
Maximum-range angleThe range is maximal at 45° when launching from the arrival altitude (h₀ = 0).
The range is maximal at 45° when launching from the arrival altitude (h₀ = 0). From a height, the optimal angle is less than 45° and depends on v₀ and h₀.