Projectile Motion Calculator
Solve projectile motion for range, maximum height, flight time and impact velocity, with a trajectory plot and the drag caveat that the standard equations ignore.
Last reviewed by the Radiatus Cloud team
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The equations assume no air
Standard projectile motion treats the only force after launch as gravity, which makes the path a parabola and gives closed-form answers for range, height and time. In air this is wrong, and how wrong depends on speed and size: a thrown ball at walking pace is close to the prediction, a cricket ball is noticeably short of it, and a bullet travels a small fraction of the vacuum range. The equations remain the right starting point and their answers are upper bounds rather than predictions.
Forty-five degrees is optimal only from level ground
The familiar result that 45 degrees gives maximum range assumes launch and landing at the same height. Throwing from a height, which is what a person does, shifts the optimum below 45, and the difference grows as the launch height increases relative to the range. Shot putters release at around 37 degrees for exactly this reason, not because of air resistance, and the same correction applies to anything launched from above the landing point.
The horizontal and vertical motions are independent
This is the insight that makes the problem tractable: gravity acts only vertically, so horizontal velocity is constant throughout and the vertical motion is the same as a ball thrown straight up. A ball dropped and a ball fired horizontally from the same height hit the ground at the same moment, which is counterintuitive and is the direct consequence.
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Frequently Asked Questions
Why is 45 degrees not always optimal?
Because that result assumes launch and landing at the same height. Launching from above the landing point shifts the optimum below 45, which is why shot putters release at around 37 degrees.
How much does air resistance change the answer?
It depends on speed and size. A thrown ball at walking pace is close to the prediction, a fast-moving ball falls noticeably short, and a bullet travels a small fraction of the vacuum range. Treat these answers as upper bounds.
Why do a dropped ball and a fired ball land together?
Because gravity acts only vertically, so the horizontal motion does not affect the vertical. The fired ball travels further and falls at exactly the same rate, so both reach the ground at the same moment.
Does mass affect the trajectory?
Not in a vacuum: the equations contain no mass term, so a heavy and a light projectile follow identical paths. In air mass matters considerably, because a heavier object of the same size is less affected by drag.
What launch angles give the same range?
Any two angles adding to 90 degrees give the same range from level ground, so 30 and 60 land in the same place. The steeper one takes longer and reaches higher, which matters when something has to be cleared.
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How to Use
Enter launch speed and angle to solve the trajectory.
Disclaimer: This tool is provided "as is" without warranty of any kind. Results are for educational and utility purposes.