Projectile Motion Calculator
How far, how high and how long a projectile flies from its launch speed, angle and height, with the speed and angle it lands at, ignoring air resistance.
Result
Horizontal range: 208.4 ft
No air resistance: close for a heavy, slow object, but a real ball or shell falls well short of this.
- Time of flight
- 2.735 s
- Highest point above the landing level
- 30.09 ft
- Time to the highest point
- 1.368 s
- Landing speed
- 60 mph (88 ft/s)
- Landing angle below horizontal
- 30 °
- Horizontal velocity
- 51.96 mph (76.21 ft/s)
- Initial vertical velocity
- 30 mph (44 ft/s)
https://calcgallery.com/physics/projectile-motion-calculator/
Results come from the published equations applied to the values you enter. They are not electrical design, conductor sizing, laboratory safety or medical guidance. Weather indices describe a standard person in standard conditions and are not a forecast. Where codes apply, local rules vary and must be checked. Read the full disclaimer
How to use the Projectile Motion Calculator
- Enter the launch speed and the angle above the horizontal. Use a negative angle for something thrown downwards.
- Enter how far above the landing point it is launched, zero on level ground, or a negative height if it lands higher up.
- Read the range, flight time and highest point, and the speed and angle it lands at. Change gravity under more options to match a textbook.
The formulas
vx = v₀ cos θ, vy = v₀ sin θtime of flight T = (vy + √(vy² + 2 g h₀)) ÷ grange R = vx × Tpeak height h₀ + vy² ÷ 2g (reached at vy ÷ g)landing speed √(vx² + vy² + 2 g h₀)level ground (h₀ = 0): T = 2 vy ÷ g, R = v₀² sin 2θ ÷ gHeights are measured from the landing level. g is standard gravity, 9.80665 m/s², unless you change it.
Limits
The path is a parabola only without air resistance, with gravity constant and straight down, and over ground flat enough that the Earth’s curve does not matter. Spin, wind and drag are not modelled. The page does not say whether anything is safe to throw, fire or jump.
With your numbers
- vx = v₀ cos θ = 76.2102 ft/s, vy = v₀ sin θ = 44 ft/s
- T = (vy + √(vy² + 2g h₀)) ÷ g = 2.73512 s
- R = vx × T = 208.444 ft
Quick reference
No air resistance: close for a heavy, slow object, but a real ball or shell falls well short of this.
| Example | Horizontal range |
|---|---|
| 60 mph at 30° on level ground | 208.4 ft |
| 20 m/s at 45° on level ground | 40.79 m |
| 20 m/s at 30° from 10 m up | 48.06 m |
| Thrown level at 10 m/s from 5 m | 10.1 m |
| Firework shell, 70 m/s at 75° | 249.8 m |
| Tennis ball, 30 m/s at 45°, caught 10 m higher | 80.35 m |
| Straight up at 10 m/s | 0 m |
Where the numbers come from
The fixed figures this calculator relies on, and the published source each one was checked against. How the calculators are built and checked
1 source
SI Brochure, 9th ed. (BIPM, 2019), Appendix 1, 3rd CGPM (1901), Declaration on the unit of mass and on the definition of weight, p. 154: "The value adopted in the International Service of Weights and Measures for the standard acceleration due to gravity is 980.665 cm/s2", https://www.bipm.org/documents/20126/41483022/SI-Brochure-9-EN.pdf
Used for
- Standard acceleration of gravity, gₙ, a conventional value fixed by definition. Real gravity differs slightly from place to place, and on other worlds far more, so pages let you change it.
Common questions about the Projectile Motion Calculator
- How do I calculate the range of a projectile?
- On level ground the range is R = v₀² sin 2θ ÷ g. At 20 m/s and 45° that is 400 × 1 ÷ 9.80665 ≈ 40.8 m. From a height, the page first finds the flight time, T = (vy + √(vy² + 2gh₀)) ÷ g, and multiplies it by the horizontal velocity, v₀ cos θ.
- What angle gives the longest range?
- 45° on level ground, without air resistance, because sin 2θ is largest there. Angles the same distance either side, such as 30° and 60°, land at the same spot, the higher one taking longer. From a height, the best angle is a little below 45°.
- How high does a projectile go?
- The vertical velocity falls by g every second until it reaches zero at the top, after vy ÷ g seconds, having risen vy² ÷ 2g. OpenStax University Physics example 4.7 fires a shell at 70.0 m/s and 75°: with g = 9.8 it peaks at about 233 m after 6.90 s, and the page gives the same with gravity set to 9.8.
- Can it land higher than it was launched?
- Yes, if it is still rising fast enough. Enter the launch height as negative. OpenStax example 4.8 has a tennis ball hit at 30 m/s and 45° and caught 10 m higher; the page gives 3.79 s in the air and a speed of 26.5 m/s at 36.9° below the horizontal, the book’s figures. If the path never gets that high, the page says so.
- How much does air resistance change the answer?
- A great deal for anything light or fast. A baseball, a golf ball or an arrow falls well short of the drag-free range, and the path is no longer a symmetric parabola. The page is exact for the idealised case taught in physics, and a fair guide only for heavy, slow objects over short distances.