Projectile Motion Calculator
Range, maximum height, and time of flight for a launched projectile, with the trajectory plotted.
How this is calculated
Initial velocity splits into horizontal and vertical components using the launch angle. Time of flight is found by solving the vertical motion equation for when height returns to zero (accounting for any launch height), and range is the horizontal component times that time. Maximum height uses the vertical component alone, since gravity decelerates it to zero at the peak.
Real-world ranges typically fall noticeably short of this idealized calculation because of air resistance (drag), which this model does not include. Drag force grows with the square of velocity and depends on the object's drag coefficient — a streamlined object like a javelin loses much less range to drag than a less aerodynamic one like a shuttlecock, even launched at the identical speed and angle. Lighter, larger-surface-area objects are affected most, since drag has more relative effect on them than on dense, compact ones. For the force behind that deceleration, see the Force / Mass / Acceleration calculator.
Frequently asked questions
- Does this account for air resistance?
- No — this uses the idealized physics model (constant gravitational acceleration, no drag), the standard starting point for projectile motion problems. Air resistance would reduce both the range and height in practice, more significantly for lighter or less aerodynamic objects.
- What launch angle gives the maximum range?
- 45 degrees, when launching and landing at the same height. Launching from an elevated height shifts the optimal angle slightly below 45 degrees, since the projectile has extra time in the air to travel horizontally.
- What value of gravity does this use?
- g = 9.81 m/s², the standard value for Earth's surface gravity.
- Can I use this in my own app?
- Yes — every calculator on Stupidly Clever has a matching REST API and MCP tool that runs the same underlying logic.
- What is projectile motion?
- Projectile motion describes the curved path an object follows when launched into the air and subject only to gravity — the classic example being a ball thrown at an angle. Its path is a parabola, resulting from constant horizontal velocity combined with constantly accelerating vertical motion due to gravity.
- What assumptions does the calculator make?
- It assumes idealized physics: constant gravitational acceleration (9.81 m/s²), no air resistance, a point-mass projectile (no size or spin effects), and flat, level ground unless you specify an initial height. These are the standard simplifying assumptions used in introductory projectile motion problems.
- Why is the range the same for 30° and 60° launch angles?
- Range depends on sin(2×angle), and sin(2×30°) = sin(60°) = sin(120°) = sin(2×60°) — any pair of angles that add up to 90° produces the same range (assuming launch and landing at the same height), since they trade off horizontal and vertical velocity components symmetrically around the 45° optimum.