Projectile Motion Calculator

Range, maximum height, time of flight and landing speed of a launch, with the launch height included. No air resistance, which is how the problem is set in class.

Range

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Range—

Maximum height—

Time of flight—

Time to the top—

Speed on landing—

Velocity components—

Angle for maximum range—

How this was worked out

The formula

x = v₀ · cos θ · t · y = h + v₀ · sin θ · t − ½ · g · t²

What it means

Projectile motion is two independent motions happening at once: a horizontal one at constant speed, because nothing slows it down, and a vertical one under gravity. You separate them by splitting the initial velocity into its two components, and from there each is solved on its own. The only thing they share is the time, which is the same for both, and that is why the range comes from multiplying the horizontal speed by the total time of flight.

How to work it out by hand

Split the initial velocity: horizontal is v₀·cos θ and vertical v₀·sin θ
Solve for when the height returns to zero: v₀·sin θ·t − ½·g·t² + h = 0
Multiply that time by the horizontal speed to get the range
Maximum height comes when the vertical speed reaches zero, at t = v₀·sin θ / g

What is worth knowing

The famous 45° for maximum range only holds when you launch and land at the same height. From a cliff, or from the hand of someone standing up, the best angle is smaller, which is why it is worked out separately. None of this includes air resistance, which on a tennis ball or a bullet cuts the range by a long way: the frictionless model is the one you solve by hand and the one exercises ask for, not the one that describes a real throw.

Frequently asked questions

Why is the best angle not always 45 degrees?

Because 45° assumes you launch and land at the same height. Launching from above, a slightly smaller angle goes further, and it is worked out here.

Does it include air resistance?

No. This is the frictionless model, the one you solve by hand. With air the real range is considerably shorter and the path stops being a parabola.

Can I throw downwards?

Yes, with a negative angle. There is then no upward phase and the time to the top is zero.

Why does it land at the same speed it left?

Only when you launch and land at the same height: the energy gained coming down is what was lost going up. From a height it lands faster.

Does it work on the Moon?

The physics is the same; the gravity is what changes. At a sixth of the gravity the same throw goes about six times further.