Enter launch speed and angle — instantly get range, max height, flight time and full trajectory.
Range (Horizontal Distance)
Max Height
—
m
Flight Time
—
s
Vx (horizontal)
—
m/s
Vy (initial vertical)
—
m/s
Step-by-Step Solution
What is Projectile Motion?
Projectile motion is the curved path an object follows when launched into the air and subject only to gravity. Whether it is a football kicked at an angle, a cannonball fired from a cliff, or water from a fountain nozzle, all projectiles follow the same parabolic trajectory governed by two independent components: constant horizontal velocity and uniformly accelerated vertical motion.
R = v₀²sin(2θ) / g
Range for level ground | H = v₀²sin²θ / 2g | T = 2v₀sinθ / g
Variables Explained
Symbol
Name
SI Unit
Description
v₀
Initial Speed
m/s
The launch speed of the projectile
θ
Launch Angle
degrees
Angle above the horizontal at launch
g
Gravity
m/s²
Gravitational acceleration (9.81 on Earth)
R
Range
m
Total horizontal distance travelled
H
Max Height
m
Highest point above launch height
T
Flight Time
s
Total time in the air
Key Physics Principles
The horizontal and vertical components of projectile motion are completely independent. Horizontally, no force acts (ignoring air resistance), so the object moves at constant velocity: x = v₀cosθ × t. Vertically, gravity accelerates the object downward: y = v₀sinθ × t − ½gt².
💡 Maximum range is always achieved at 45° on level ground. Complementary angles (e.g. 30° and 60°) give the same range but different heights and flight times.
Real-World Applications
⚽
Sports Science
Optimising the angle of a penalty kick, javelin throw, or basketball shot for maximum distance or height.
🚀
Artillery & Ballistics
Calculating gun barrel angles to hit targets at known distances — the original use of projectile equations.
🌊
Water Fountains
Engineers design fountain nozzle angles to produce specific water arc shapes and landing distances.
🎮
Game Physics
Every physics engine in video games uses projectile equations for realistic movement of thrown or fired objects.
Frequently Asked Questions
What angle gives maximum range on flat ground?
45° always gives the maximum range when the projectile lands at the same height it was launched. This is because sin(2×45°) = sin(90°) = 1, which is the maximum value of the sine function.
Does mass affect projectile range?
No — in the absence of air resistance, mass has no effect on range, height, or flight time. Galileo proved this by dropping objects of different masses from the Leaning Tower of Pisa. Both hit the ground simultaneously.
How do I use this calculator for cliff launches?
Enter the cliff height as the initial height (h₀). The calculator accounts for the additional falling distance and gives the correct range and flight time for launches above the landing level.
What about air resistance?
This calculator assumes a vacuum (no air resistance). In reality, air drag reduces range significantly, especially for fast or light objects. The equations here are the standard physics model used for A-level and university physics.
