What is projectile motion?
After launch, a projectile moves under gravity alone (neglecting air resistance). Its horizontal motion has constant velocity while its vertical motion has constant downward acceleration g ≈ 9.81 m/s²; the two are independent. Combining them gives a parabolic path — a result Galileo established well before Newton wrote the equations of motion.
Range, height and time of flight
For a launch speed v at angle θ over flat ground, the range is R = v² sin(2θ) / g (greatest at 45°), the maximum height is H = v² sin²θ / (2g), and the time of flight is t = 2v sinθ / g. Launching from a height extends the range and the flight time; the tool solves the full quadratic for you.
How to use the calculator
Enter the launch speed, angle and height. The tool resolves the velocity into components vₓ = v cosθ and v_y = v sinθ, integrates the kinematic equations and reports range, apex, flight time and impact speed, with the trajectory plotted.
Note: this is the idealized (vacuum) model with no air resistance, constant gravity and a flat surface. Real projectiles experience drag, so their range and peak height are smaller and the path is no longer a perfect parabola.
Frequently asked questions
What is projectile motion?
Projectile motion is the motion of an object launched into the air and moving under gravity alone. Its horizontal velocity is constant and its vertical motion accelerates downward at g, producing a parabolic path when air resistance is ignored.
What is the range formula for a projectile?
Over flat ground, range R = v^2 * sin(2*theta) / g, where v is the launch speed, theta the launch angle and g the gravitational acceleration. The tool also handles launches from a non-zero height.
At what angle is the range maximum?
On flat ground and without air resistance, the range is greatest at a launch angle of 45 degrees. Launching from a height lowers the optimal angle slightly below 45 degrees.
Does mass affect projectile motion?
No, in the ideal (no-drag) case: gravity gives every object the same acceleration, so range, height and flight time do not depend on mass. Mass matters only once air resistance is included.
References
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics (Wiley) — kinematics and projectile motion.
- R. A. Serway & J. W. Jewett, Physics for Scientists and Engineers (Cengage).
- The range, height and time formulas follow from the constant-acceleration kinematic equations (Galileo Galilei, 1638).