๐ฏ Projectile Motion Calculator
Enter the initial velocity, launch angle, and initial height to calculate the time of flight, maximum height, and range of projectile motion (ignoring air resistance) using standard kinematics formulas. Standard gravity (9.80665 m/sยฒ) is used.
How to use
- Enter the initial velocity (m/s).
- Enter the launch angle (degrees, 0โ90ยฐ from horizontal).
- Optionally enter the initial height (m, height above the ground).
- The time of flight, maximum height, and range are calculated automatically.
How the calculation works
This tool calculates the path of an object launched at an angle, under ideal conditions with no air resistance. Gravity is standard gravity, 9.80665 m/sยฒ. With launch speed vโ, angle ฮธ and starting height hโ: Horizontal speed vx = vโ cos ฮธ (constant) Initial vertical speed vy = vโ sin ฮธ Maximum height H = hโ + vyยฒ รท (2g) Time of flight T = (vy + โ(vyยฒ + 2g hโ)) รท g Range R = vx ร T Time of flight is the time until the object returns to height 0 (the ground). With a starting height, it falls further than it rose, so both flight time and range increase.
Worked example
Speed 20 m/s, angle 45ยฐ, starting height 0 m Time of flight: about 2.884 s Maximum height: about 10.197 m Range: about 40.789 m The same speed from a height of 10 m 45ยฐ gives about 49.10 m and 40ยฐ gives about 49.78 m, so a lower angle goes further.
Things to be aware of
- Real balls and shells meet air resistance, so they travel less far than calculated and their best angle is lower than 45ยฐ.
- At 90ยฐ (straight up) the range is 0 and the maximum height is at its highest.
- Gravity varies slightly by location (about 9.78 m/sยฒ at the equator and 9.83 m/sยฒ at the poles), but standard gravity is fine for everyday calculations.
FAQ
Does this account for air resistance?
No, this tool calculates ideal projectile motion (as in a vacuum), ignoring air resistance. Real-world range will be shorter due to drag.
What value of gravity does this use?
It uses the internationally defined standard gravity, 9.80665 m/sยฒ.
Why does a 45-degree launch angle give the longest range?
With zero initial height, range is proportional to sin(2 ร angle), which is maximized at 45ยฐ (sin 90ยฐ = 1). With a nonzero initial height, the optimal angle is slightly less than 45ยฐ.