About the Projectile Motion Calculator
This calculator works out the range, maximum height, and total flight time of an object launched at an angle, using the standard equations of projectile motion. It's built for the classic physics scenario, a ball, arrow, or any object launched into the air with some initial speed and angle, on level ground with no air resistance.
How It Works
You enter the initial launch velocity in meters per second and the launch angle in degrees, from 0 to 90. The calculator converts the angle to radians and applies the standard kinematic equations for projectile motion under constant gravitational acceleration to find how far the object travels horizontally, how high it rises, and how long it stays in the air before landing back at launch height.
Formula & Methodology
For an initial velocity of 25 m/s at a 45 degree angle: sin(2 x 45 degrees) = sin(90 degrees) = 1, so range = 625 x 1 / 9.80665, about 63.74 m. Maximum height uses sin^2(45 degrees) = 0.5, giving 625 x 0.5 / (2 x 9.80665), about 15.93 m. Flight time is 2 x 25 x sin(45 degrees) / 9.80665, about 3.61 seconds.
Examples
45 degree launch (maximum range angle)
A projectile launched at 25 m/s at a 45 degree angle travels a range of about 63.74 m, reaches a maximum height of 15.93 m, and stays airborne for roughly 3.61 seconds.
Lower angle, higher speed launch
A projectile launched at 40 m/s at a 30 degree angle covers a range of about 141.32 m, rises to a maximum height of 20.40 m, and has a flight time of about 4.08 seconds.
Advantages
- Computes all three key trajectory outputs, range, maximum height, and flight time, from just two inputs rather than requiring three separate formula lookups.
- Uses the standard g = 9.80665 m/s^2 constant, matching the value used consistently across the site's other physics calculators for comparable results.
- Makes it simple to compare how changing the launch angle affects range while holding speed constant, since all three outputs update from the same two entries.
Common Mistakes
- Assuming a steeper launch angle always means a longer range, when the maximum range actually occurs at 45 degrees and falls off symmetrically on either side of it.
- Forgetting that this model excludes air resistance, so it overstates the range and flight time for anything with significant drag, like a lightweight ball or a paper projectile.
- Mixing up maximum height, the vertical peak reached mid-flight, with range, the horizontal landing distance, which are governed by different terms in the equations.
Edge Cases to Watch For
- A negative initial velocity is explicitly rejected with an error message, since a negative launch speed has no physical meaning in this equation set.
- The angle input is bounded between 0 and 90 degrees in the form, matching the physically sensible range for a launch above flat, level ground.
- The model completely ignores air resistance, so predicted range and height will be optimistic compared to real-world trajectories for anything other than a dense, compact, low-drag object.
Common Use Cases
- Physics students verifying projectile motion homework or lab results against a known-good calculation.
- Hobbyists working out approximate trajectories for model rockets, water balloons, or similar launched objects.
- Teachers building example problems that illustrate how launch angle and velocity separately affect range, height, and flight time.