Math
Projectile Motion Calculator
Solve projectile motion under gravity from initial velocity, angle, and launch height. Get range, max height, time of flight, and trajectory samples.
Launch parameters
Trajectory
- Range
- 40.79 m
- Max height
- 10.2 m
- Time of flight
- 2.88 s
- Time to apex
- 1.44 s
- Initial velocity
- 20 m/s
- Launch angle
- 45 deg
- Final speed
- 20 m/s
- Final angle
- 45 deg below horizontal
- Gravity
- 9.81 m/s^2
Frequently Asked Questions about the Projectile Motion Calculator
What equations does the calculator use?
It applies the standard kinematic equations under constant gravity g (default 9.80665 m/s^2): horizontal velocity vx = v0 cos(theta), vertical velocity vy(t) = v0 sin(theta) - g t, height y(t) = h0 + v0 sin(theta) t - 0.5 g t^2. Air resistance is ignored.
Why does 45 degrees give the maximum range?
When launch height h0 is zero, the range simplifies to v0^2 sin(2 theta) / g. The sine of 2 theta peaks at 1 when 2 theta = 90 degrees, i.e. theta = 45 degrees. With a non-zero launch height, the optimal angle drops slightly below 45 degrees.
How does launch height affect flight time and range?
A positive launch height extends both. Flight time goes from 2 v0 sin(theta) / g to (v0 sin(theta) + sqrt((v0 sin(theta))^2 + 2 g h0)) / g, and the longer flight gives the horizontal velocity more time to carry the projectile, so range goes up too.
Why doesn't the calculator include air resistance?
Air resistance is highly dependent on shape, surface roughness, spin, and air density. Modeling it accurately requires a differential equation with a drag coefficient and changes the answer by 10 to 50 percent for slow projectiles and much more for fast ones. This calculator uses the vacuum (no-drag) equations, which are the standard physics-class model.
Can I use this for projectiles on the Moon or Mars?
Yes. Set gravity to 1.62 m/s^2 for the Moon, 3.71 for Mars, or 24.79 for Jupiter. Lower gravity stretches the range and flight time dramatically: the same 20 m/s launch at 45 degrees reaches about 41 m on Earth and 247 m on the Moon.