Physics · Mechanics
Projectile motion calculator.
Enter a launch speed and angle (and a launch height if you like) to get the range, peak height, time in the air and impact speed. No air resistance.
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Projectile motion formulas
With no air resistance, a projectile moves at a constant horizontal speed while gravity changes its vertical speed at g. Split the launch speed into vx = v0 cos θ and vy = v0 sin θ. For launch and landing at the same level, the range is R = v0² sin(2θ) ÷ g and the peak height is h = vy² ÷ 2g (OpenStax). With a launch height h0 the calculator solves 0 = h0 + vyt − ½gt² for the landing time and multiplies by vx.
Worked example. 20 m/s at 30° from ground level on Earth (g = 9.80665 m/s²): vx = 17.32 m/s, vy = 10 m/s, time of flight 2.039 s, range 35.32 m, peak height 5.099 m. Launch the same ball from a 2 m platform at 15 m/s and 40° and it stays up 2.156 s and lands 24.77 m away at 16.26 m/s.
Other worlds. Range scales as 1 ÷ g. The same 20 m/s, 30° launch on the Moon (g = 1.62 m/s²) flies 213.8 m.
This is an idealised textbook model. Real balls, arrows and shells lose speed to air drag, so measured ranges are shorter. It is for study, not for ballistics, sports-safety or engineering design.
Frequently asked questions
What is the formula for projectile range?
R = v0² × sin(2θ) ÷ g for launch and landing at the same height. 20 m/s at 30° gives 35.32 m on Earth.
Which launch angle gives the longest range?
On level ground with no air resistance the factor sin(2θ) is largest at 45°, so 45° gives the longest range. With a raised launch point the best angle is a little lower.
How do I find the time of flight?
For level ground t = 2 v0 sin θ ÷ g. In the example that is 2.039 s. With a launch height the calculator uses the full quadratic solution.
Does mass change the range?
No. Without air resistance mass cancels out, so a heavy and a light object launched the same way follow the same path.
Why does the answer differ from a real throw?
Air resistance, spin and wind are ignored here. They shorten real trajectories, especially for light or fast objects.
Sources
Checked
- OpenStax University Physics Volume 1, 4.3 Projectile Motion
- NIST CODATA: standard acceleration of gravity (9.806 65 m/s2, exact)
- NASA NSSDC Planetary Fact Sheet (surface gravity row)
- NASA NSSDC Moon Fact Sheet
- NASA NSSDC Mars Fact Sheet
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Scope
Idealised textbook model with no air drag. Not for ballistics, sports-safety or engineering design. Results are arithmetic from the formulas shown, for study and general interest, and not professional or engineering advice.