Projectile Motion Calculator
Where the ball lands, ignoring air.
Projectile Motion Calculator
vy = v sin(angle); flight = (vy + sqrt(vy^2 + 2gh)) / g; range = v cos(angle) x flight.
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What This Calculator Does
The Projectile Motion Calculator finds flight time, horizontal range, maximum height, and the horizontal and vertical velocity components for a launch over level ground or from a raised height. It is the standard no-drag model used in introductory physics.
How It Works
The launch speed is split into vertical and horizontal components using sine and cosine. Vertical motion is accelerated by gravity at 9.81 m/s squared, so flight time is the positive root of the quadratic for the time when height returns to zero. Range is horizontal velocity times flight time. Maximum height adds the rise from the vertical component to the launch height. Gravity is assumed constant and air resistance is ignored.
Example
A ball launched at 20 m/s at 45 degrees from ground level has a vertical component of 14.14 m/s. Flight time is 2 x 14.14 / 9.81 = 2.88 s. Range is 14.14 x 2.88 = 40.77 m, and maximum height is 14.14 squared / 19.62 = 10.19 m.
Formula
vy = v sin(angle)t = (vy + sqrt(vy^2 + 2gh)) / gR = v cos(angle) x th max = h + vy^2 / (2g)Reading the Results
Flight time
Time from launch until the projectile returns to the ground plane at height zero. A raised launch increases the time.
Horizontal range
The horizontal distance at ground level, assuming no air resistance. Real range is shorter for most objects.
Maximum height
The peak height above the ground, including any launch height. It occurs when the vertical velocity reaches zero.
Sources and References
OpenStax College Physics - Projectile Motion
Kinematic equations for two-dimensional motion
Textbook derivation of range, time of flight, and maximum height without drag.
NASA - Equations of Motion
Constant acceleration and trajectory
Reference for the kinematic relationships used in the model.
References identify the method or data behind this calculator. Where a standard or tariff changes, the current edition applies, and the assumptions stated above take precedence over a remembered figure.
Frequently Asked Questions
What launch angle gives the maximum range?
At equal launch and landing heights, 45 degrees maximises range in a vacuum. With air resistance the optimum is usually lower, and from a raised launch point a lower angle can travel farther because the projectile stays in the air longer.
Why is the calculated range longer than what I observe?
The model ignores air resistance, which removes energy throughout the flight. A light or large projectile is affected more. Spin, wind, and an uneven landing surface also change the result, so treat the number as an upper bound.
Does launch height change the optimal angle?
Yes. When launching from a height, the projectile has extra fall time, and the angle that maximises range drops below 45 degrees. The calculator shows the range for the angle you enter; it does not search for the optimum.
Can I use this for a vertical launch?
Yes. At 90 degrees the horizontal component is zero, so the range is zero and the projectile rises and falls on the same line. The flight time and maximum height still apply. At negative angles the model treats the launch as downward.
Is gravity always 9.81 m/s squared?
It is a standard average at the Earth surface. The local value varies slightly with latitude and altitude, and other planets have very different gravity. For a school calculation, 9.81 is appropriate; for a precision trajectory, use the local value.