Free fall and projectile motion
free fall: t = √(2h/g), v = g·t
Two classic mechanics calculations: free fall (time and impact speed from a height) and projectile motion (range, flight time, maximum height, impact speed, from the ground or from a height, angled up or down). The model (no air resistance) is displayed, not hidden, and gravity g is an editable field whose default is the exact conventional standard gravity.
40.79 m
- Initial components
- vₓ = 14.14 m/s; v_y = 14.14 m/s
- Calculation
- t = (v_y + √(v_y² + 2·g·h₀)) ÷ g = (14.14 + √(14.14² + 2 × 9.807 × 0)) ÷ 9.807 = 2.884 s
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulasfree fall: t = √(2h/g), v = g·t
free fall: t = √(2h/g), v = g·t
components: vₓ = v₀·cos θ, v_y = v₀·sin θ
flight time: t = (v_y + √(v_y² + 2·g·h₀)) / g
range: R = vₓ · t
maximum height: H = h₀ + v_y² / 2g
impact speed: |v| = √(v₀² + 2·g·h₀)
Everything follows from two facts: horizontally, nothing accelerates (vₓ is constant); vertically, everything falls the same way (acceleration g downward, whatever the horizontal motion). The flight time comes from solving y(t) = 0, a quadratic whose positive root is kept. The impact speed comes from energy conservation: it does not depend on the launch angle, only on v₀ and the initial height.
- Range (R)
- · the horizontal distance covered between launch and return to zero altitude (the ground).
- Maximum height (H)
- · the highest point of the trajectory, reached when the vertical speed vanishes, measured here from the ground.
- Standard gravity (9.80665 m/s²)
- · an exact conventional value fixed by the 3rd CGPM (1901), a definition. Not a measurement. Real gravity varies by about ±0.3% with latitude and altitude.
Validity domainThe model neglects air resistance: it is reliable for a dense, compact object at moderate speed (a steel ball dropped from a few metres), and less and less beyond.
The model neglects air resistance: it is reliable for a dense, compact object at moderate speed (a steel ball dropped from a few metres), and less and less beyond. A ping-pong ball, a shuttlecock or a fast projectile are far outside it. It also assumes uniform g (trajectories short compared to Earth’s radius) and a point mass without spin: no Magnus effect, no lift.
Common pitfall: 45° is only optimal from the groundThe maximum-range angle is 45° only when launching from the arrival altitude (h₀ = 0).
The maximum-range angle is 45° only when launching from the arrival altitude (h₀ = 0). From a height, the optimal angle is less than 45°, and it depends on v₀ and h₀. Another trap: the impact speed does not depend on the angle (energy conservation); two launches at 20° and 70° from the ground land at exactly the same speed, just not in the same place.