About the Projectile Motion Calculator
A projectile is anything thrown, kicked, fired or launched that then moves under gravity alone: a football, a javelin, a stream of water from a hose, a cannonball. Its path is a graceful curve called a parabola, and with a little physics you can predict exactly how far it will travel, how high it will go and how long it will stay in the air.
This projectile motion calculator takes the launch speed, launch angle and launch height and works out the range, the time of flight, the maximum height, and the speed and angle at which the projectile lands. It can use Earth's gravity or that of the Moon, Mars, Venus or Jupiter. Like most textbook calculations, it ignores air resistance, which gives the ideal result that real throws fall short of.
How to Use the Projectile Motion Calculator
Enter the launch speed and choose its unit.
Enter the launch angle above the horizontal: 0° for a horizontal throw, 45° for a diagonal one, 90° for straight up.
Enter the launch height above the landing level, or leave it at 0 for ground level.
Choose the gravity if the projectile is not on Earth.
The Formulas
horizontal velocity vx = v × cos θ (stays constant)
vertical velocity vy = v × sin θ (changes by −g every second)
time of flight t = (vy + √(vy² + 2 g h₀)) ÷ g
range R = vx × t
maximum height H = h₀ + vy² ÷ 2g
time to the top vy ÷ g
from ground level (h₀ = 0): R = v² sin 2θ ÷ g, t = 2 vy ÷ gStep-by-Step Examples
20 m/s at 45° from the ground.
vx = vy = 20 × 0.7071 = 14.142 m/s
t = 2 × 14.142 ÷ 9.80665 = 2.8842 s
R = 14.142 × 2.8842 = 40.789 m
H = 14.142² ÷ (2 × 9.80665) = 10.197 mThe same speed at 30° and 60° both travel about 35.324 m, but the 60° shot peaks at 15.296 m and the 30° shot at only 5.0986 m.
A horizontal throw from a height. A ball thrown horizontally at 15 m/s from 10 m up:
t = √(2 × 10 ÷ 9.80665) = 1.4281 s
R = 15 × 1.4281 = 21.421 mA long throw. 30 m/s at 40° from 2 m travels about 92.704 m.
Two Independent Motions
The key idea in projectile motion is that the horizontal and vertical motions are independent. Horizontally, with no air resistance, nothing pushes or pulls the projectile, so it moves at a steady speed. Vertically, gravity pulls it down, slowing it on the way up and speeding it up on the way down, exactly as if it had been thrown straight up. Combining a steady sideways motion with a uniformly accelerated vertical motion gives the parabola. A striking consequence is that a bullet fired horizontally and a bullet dropped from the same height at the same moment hit the ground together.
The Best Launch Angle
From level ground, the range formula R = v² sin 2θ ÷ g is largest when sin 2θ is 1, which happens at 45°. Angles the same distance either side of 45° — such as 30° and 60°, or 40° and 50° — give the same range, the higher one taking longer and flying higher. When the projectile is launched from above the landing point, as with a shot put released at shoulder height, the best angle is somewhat less than 45°. Air resistance lowers the optimum further, which is why long jumpers take off at around 20° and golf drives leave the tee at low angles with plenty of backspin.
Air Resistance and Real Projectiles
Real projectiles move through air, which pushes against them and saps their speed. The effect grows rapidly with speed and matters most for light objects with large surfaces. A table-tennis ball or shuttlecock falls far short of the ideal range; a heavy shot put comes much closer. Spin adds further effects: backspin gives a golf ball lift, while sidespin makes a football curve. Engineers and sports scientists use computer models that include drag and spin, but the ideal calculation remains the starting point and gives an upper limit on the range.
Projectiles on Other Worlds
Weaker gravity means longer, higher flights. On the Moon, where gravity is about a sixth of Earth's, the same throw goes about six times as far and six times as high, and with no air there is no drag at all. Astronauts on Apollo 14 famously hit golf balls on the Moon. On Jupiter, with roughly two and a half times Earth's surface gravity, a projectile would fall back much more quickly and cover far less ground.
Projectiles in Sport
Sport is full of projectiles. Basketball players arc their shots steeply so the ball drops into the hoop from above, giving a larger effective target. Footballers taking a free kick must clear the defensive wall yet bring the ball down before the crossbar, often using spin to bend its path. Javelin and discus throwers balance speed, angle and aerodynamic lift, and coaches use video analysis of release speed and angle to improve technique.
Understanding Your Result
The headline is the horizontal range.
The flight time and peak line gives the time in the air and the highest point reached.
The landing line gives the impact speed and the angle below horizontal.
The velocity components line shows the horizontal and vertical parts of the launch velocity.
When Should You Use This Calculator?
Use it for physics homework on projectiles and kinematics.
Use it to explore how angle and speed affect range in sport.
Use it to estimate where a launched object will land.
Common Mistakes
Using degrees in a radian calculation. The calculator handles angles in degrees.
Forgetting the launch height. Throws from above the ground travel further.
Expecting real throws to match. Air resistance shortens real ranges.
Assuming 45° is always best. It is only best from level ground without air.