Projectile motion calculator

Launch speed and angle in, and you get range, maximum height and time of flight. The working shows the split into horizontal and vertical components, which is the part that makes the whole topic click.

Launch speed (m/s)

Launch angle (degrees)

Launch height (m)

Gravity g (m/s²)

Launch height is 0 for something thrown from ground level. Gravity defaults to 9.8, which GCSE, AP, HSC, VCE and CBSE all use. UK A-level specifies 9.81 and OCR Gateway uses 10, so change it to match your data sheet. Air resistance is ignored, as it is in almost every school question.

The one idea the whole topic rests on

Horizontal and vertical motion are completely independent. Gravity pulls downward, so it only affects the vertical part. Nothing pushes or pulls horizontally once the object has left your hand, so the horizontal speed simply stays the same for the entire flight.

That means a projectile is really two separate problems happening at once. Vertically it is an object thrown straight up and falling back down. Horizontally it is an object moving at constant speed. The only thing connecting them is time: both parts of the motion take exactly as long as each other.

The classic demonstration: fire a bullet horizontally and drop an identical one from the same height at the same moment, and they hit the ground together. The fired one travels far further, but gravity pulls both down at exactly the same rate, because horizontal speed has no effect on vertical motion.

Working through it

Split the launch velocity. Speed u at angle θ becomes horizontal u cos θ and vertical u sin θ. This is just trigonometry on the velocity arrow.

Do the vertical part. Treat it as SUVAT with a = −g. At the highest point the vertical velocity is momentarily zero, which is what lets you find the time to the peak and the maximum height.

Find the total flight time. From level ground the flight is symmetrical, so it takes as long coming down as going up. From a height, the descent is longer and you solve a quadratic instead.

Do the horizontal part. Range is simply horizontal speed multiplied by total time, because that speed never changes.

Common questions

Why is 45° the best angle for maximum range?: Because it is the best compromise. A steeper angle gives more air time but less horizontal speed; a shallower one gives more horizontal speed but less time. At 45° from level ground the two balance exactly. If you launch from a height, the optimum drops slightly below 45°.
Why is vertical velocity zero at the top?: Because that is what "the top" means. The object stops rising and begins falling, so at that instant its vertical velocity passes through zero. It is still moving horizontally at full speed, so it has not stopped.
Do 30° and 60° really give the same range?: Yes, from level ground. Any pair of angles adding to 90° gives the same range, because sin(2θ) is the same for both. The 60° launch goes much higher and stays up longer, but lands at the same point.
What difference does air resistance make?: A real one, which is why these answers are idealised. Air resistance shortens the range, lowers the peak, and makes the descent steeper than the ascent so the path is no longer a symmetrical parabola. School questions almost always ignore it, and will say so.