🎯 Projectile Motion Calculator
Find range, max height, and time of flight from initial velocity and launch angle.
How to Use This Calculator
Enter the initial speed (in m/s) and the launch angle (0° to 90°). The calculator gives you the horizontal range, maximum height, total time of flight, and both velocity components. The default value for g is 9.81 m/s², but you can change it to model other planets or run idealized problems with g = 10 m/s².
Enter the initial speed, v₀. This is the total speed at the moment of launch, not just the horizontal part.
Enter the launch angle. A 45° angle gives the maximum range on flat ground. For maximum height, use angles close to 90°.
Leave g at 9.81 for Earth. Change it to 1.62 for the Moon or 3.72 for Mars to compare trajectories.
Read off the five output values: range, max height, time of flight, and the horizontal and vertical speed components.
Projectile Motion Equations
The key insight is that horizontal and vertical motions are independent. The horizontal speed stays constant throughout (no air resistance), while the vertical component decelerates at g going up and accelerates coming back down. These two motions combine to produce the curved parabolic path.
Worked Examples
Where This Comes Up in Real Life
Any time something is launched and falls back down under gravity, you are looking at projectile motion. A footballer kicking a penalty or a basketball player shooting a free throw both rely on instinctively solving the same equations. At 30° a ball travels the same horizontal distance as at 60°, but the 60° shot takes much longer in the air. You can verify this with the calculator: both angles give the same range for a given speed.
In engineering, ballistics and artillery use these equations as a starting point, with corrections added later for air resistance and spin. Athletes like javelin throwers aim at angles close to 35-40° in practice because air drag shifts the optimal angle below the theoretical 45°. The same equations also describe the water stream from a garden hose or a ski jumper leaving the ramp.
Frequently Asked Questions
What is projectile motion?
Projectile motion is motion under gravity with no air resistance. Horizontal velocity is constant; vertical velocity changes due to g=9.81 m/s².
At what angle is range maximised?
45° gives maximum range (on flat ground). For any complementary angles (e.g. 30° and 60°), range is equal.
How do I calculate maximum height?
H = v₀²sin²(θ)/(2g). Maximum height at t = v₀sin(θ)/g.
What is the range formula?
R = v₀²sin(2θ)/g. Using sin(2θ)=2sinθcosθ.
What if launched from a height?
This calculator assumes launch from ground level. For height h₀, time is found from: h₀ + v₀sinθ·t − ½gt² = 0.