Projectile Motion

Throw, kick or launch something: where does it land, how high does it go and how long is it in the air? Fill in any two of speed, angle, range, height and flight time — from any launch height, on any planet.

MechanicsKinematicsGravity

Inputs

Input examples

Click Use on exactly two of the five trajectory values. Launch height and gravity are always used. Air resistance is ignored.

Launch speed v₀

Launch angle θ

Range R — horizontal distance to landing

Maximum height H — above the launch point

Flight time T

Launch height h₀

Gravity g

What is Projectile Motion?

Once launched, a projectile moves sideways at a constant speed while gravity pulls it down at a constant acceleration. The two motions are independent: a bullet fired horizontally and one dropped from the same height hit the ground at the same moment.

Together they trace a parabola. On level ground the longest range comes at 45°, and complementary angles such as 30° and 60° land in the same spot — the higher one just takes longer.

45°max range (level)

30° & 60°same range

v_xstays constant

at the topv_y = 0

Trajectory

to scale

trajectory

Level-Ground Formulas

h₀ = 0

Quantity	Formula

Worked Techniques

Split into Components

Treat the horizontal and vertical motion separately. Only the vertical part feels gravity; time is what links them.

Example:

Launching from a Height

Solve y(t) = 0 with the quadratic formula and take the positive root. From a height, the best angle for range is a little below 45°.

Example: shot put from 2 m: optimal angle about 42°.

Two Angles for One Target

If the target is within reach there are two launch angles: a flat, fast one and a high lob. This calculator shows both.

Example:

Sources & Verification

References

D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics, 10th ed., Wiley, 2014 — Ch. 4 (Motion in Two and Three Dimensions).