Projectile motion calculator
Range, flight time and apex height of a body thrown at an angle.
Fill in the fields and the result will appear here automatically.
Calculate an ideal launch from 0° to 90° until it meets horizontal ground y=0. Enter initial height above that plane. Horizontal velocity is constant and vertical acceleration −9.80665 m/s². An exactly vertical launch has zero range; small nonzero angles are not erased by an arbitrary threshold.
How it works
Formula and logic
The speed splits into components: vy = v·sin α, vx = v·cos α. Flight time = (vy + √(vy² + 2 gh)) ÷ g, and range = vx × time. Gravity is taken as 9.80665 m/s².
Example
A throw of 20 m/s at 45° from the ground carries 40.789 m in 2.884 s.
Fields and units
- Initial speed — m/s
- Angle to the horizon — °
- Launch height — m
How to use
- — Enter the initial speed in metres per second.
- — Set the angle to the horizon between 0 and 90 degrees.
- — Give the launch height; leave it at zero for a throw from the ground.
- — Compare the time to apex with the total time: launching from height makes them unequal halves.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax: independent motion components without drag
- Limitation
- Constant standard g, flat ground, a point body and no air drag. Actual local gravity, wind, rotation, terrain and aerodynamics are not modelled. The maximum-range result at 45° applies to equal launch and landing heights.
FAQ
Why is 45° the best angle?
Only from ground level. Raise the launch point and the optimum drops below 45°: the body falls for longer, so the horizontal component is worth more than the vertical one.
Is air resistance included?
No. Its influence depends on mass, area, shape, speed and medium; a “low speed” alone does not guarantee small error. Significant drag makes this parabolic trajectory unsuitable for actual flight.
Why is the range exactly zero at 90°?
Because there is no horizontal component. In binary arithmetic cos 90° comes out as 6·10⁻¹⁷, and without snapping that to zero the range would read as a trillionth of a millimetre instead of an honest nought.
Which value of g is used?
Constant standard g, flat ground, a point body and no air drag. Actual local gravity, wind, rotation, terrain and aerodynamics are not modelled. The maximum-range result at 45° applies to equal launch and landing heights.