Projectile Motion Calculator

What the projectile motion calculator does

You have thrown something into the air, or you are about to, and you want to know where it lands, how high it climbs, and how long it stays up. A ball, an arrow, a jet of water, a stunt bike off a ramp. This calculator answers those questions for any launch you describe.

You give it the launch speed, the angle, the height you launch from, and the strength of gravity. Depending on what you pick, it returns the range, the time of flight, the maximum height, or the full picture at a moment you choose: where the object is and how fast it is moving at that instant. Because gravity is a field you can change, you can also drop it to 1.62 for the Moon or 3.72 for Mars and watch the same throw carry much further.

How to use it

1. Choose what to find. Range, time of flight, maximum height, or the flight parameters at a given time.
2. Enter the launch speed and angle. The speed in metres per second, and the angle in degrees measured up from the horizontal.
3. Enter the initial height. How far above the landing surface the object starts. Use 0 for a throw from ground level.
4. Set gravity. It starts at 9.81 m/s² for Earth. Change it for another planet if you like.
5. Press Calculate for the result, or Reset to clear it. If you picked the given-time option, a time box appears for the instant you want.

The physics: two motions at once

The whole subject rests on one idea, and it is a surprisingly old one. Galileo noticed that the up-and-down part of a thrown object's motion and the side-to-side part do not interfere with each other. They run at the same time, but each behaves as if the other were not happening.

The reason is that gravity only pulls downward. Sideways, once the object has left your hand, nothing pushes or pulls it, so its horizontal speed never changes. Vertically, gravity is always tugging, so the object slows as it rises, stops for an instant at the top, then speeds up as it falls, exactly like a ball thrown straight up.

So the calculator splits the launch velocity into those two parts. A launch at speed v₀ and angle θ becomes a steady horizontal speed and a starting upward speed:

v_x = v₀ cos θ   and   v_y = v₀ sin θ

The horizontal one carries the object forward at a constant rate. The vertical one rises and falls under gravity. Put the steady forward drift and the rise-and-fall together and you trace out the gentle arc that every thrown thing follows: a parabola.

The equations it uses

Everything below comes from two facts: horizontal speed stays at v_x, and vertical motion is free fall starting at v_y. The symbols are v₀ for launch speed, θ for launch angle, h₀ for the starting height, g for gravity, and t for time since launch.

Time of flight, the moment the object reaches the ground, found by asking when its height returns to zero:

T = ( v_y + √( v_y² + 2 g h₀ ) ) ÷ g

Range, the horizontal distance covered in that time:

R = v_x × T

When the launch and landing are at the same level, that whole thing collapses into the tidy form you may have seen, R = v₀² sin(2θ) ÷ g.

Maximum height, the starting height plus how far the upward motion climbs before gravity stops it:

H = h₀ + v_y² ÷ ( 2 g )

At any time t, the calculator can also give the position and velocity directly. The object has drifted x = v_x t sideways and reached a height y = h₀ + v_y t − ½ g t². Its vertical speed by then is v_y − g t, which turns negative once it is falling, and its actual speed through the air is the two parts combined, √( v_x² + (v_y − g t)² ).

Units, gravity, and precision

The calculator works in SI units throughout: speed in metres per second, height and distance in metres, time in seconds, and gravity in metres per second squared. The launch angle is the one exception, taken in degrees because that is how people read an angle. Behind the scenes it converts that angle to radians, which is the unit the sine and cosine actually need, by multiplying by π and dividing by 180.

Gravity starts at 9.81 m/s², the everyday value near Earth's surface. The formally defined figure, standard gravity, is 9.80665 m/s², and the true local value drifts a little with where you are, from about 9.78 near the equator to about 9.83 near the poles, and it eases off with altitude. For most throws the difference is too small to notice, which is why 9.81 is the sensible default, and the field is open so you can enter a more exact value or another planet's gravity when it matters.

Results are shown to a few decimal places. That is display precision, and it is worth being clear that it is not the same as real-world accuracy. The honest limit on these answers is not the rounding but the model itself, which treats the throw as clean and drag-free. The next sections are where that matters.

A worked example: 20 m/s at 45 degrees

Take a ball thrown from ground level at 20 m/s, angled 45 degrees up, with gravity at 9.81 m/s².

First split the launch. At 45 degrees the sine and cosine are equal, so the horizontal and vertical speeds come out the same: v_x = 20 cos 45° = 14.14 m/s, and v_y = 20 sin 45° = 14.14 m/s.

Since it starts and lands at the same level, the time of flight is T = 2 × 14.14 ÷ 9.81 = 2.88 s. The range is the horizontal speed across that time, R = 14.14 × 2.88 = 40.77 m. And the peak height is H = 14.14² ÷ (2 × 9.81) = 10.19 m.

So the ball is in the air for just under three seconds, lands about 41 metres away, and rises a little over 10 metres at the top of its arc.

Why 45 degrees gives the farthest throw

For a throw that lands at the same height it left, 45 degrees sends it the furthest, and there is a clean reason. The level-ground range depends on the angle only through sin(2θ), and a sine is at its largest when its input is 90 degrees. That happens when 2θ is 90, so θ is 45.

A nice consequence falls out of the same fact: any two angles that add up to 90 degrees give the same range. A 30 degree throw and a 60 degree throw land in the same spot, one on a flat, fast path and the other on a tall, slow arc. Push the angle all the way to 90 and the throw goes straight up, so the range drops to nothing, since there is no horizontal speed left to carry it forward.

Launch from a height and the best angle slips a little below 45 degrees. The extra time gained by falling further rewards a flatter, faster path, so a throw off a cliff or a rooftop carries furthest somewhat under 45.

What the model assumes

This calculator uses the clean, drag-free model of projectile motion, the same one taught in every first physics course. It is worth knowing what that model takes for granted, because it tells you when to trust the number as-is and when reality will shade it.

It assumes no air resistance. For a dense, compact object at a sensible speed, a thrown rock or a shot put, that is a close match to reality. For something fast and light, like a bullet or a golf ball, air drag is a real force that shortens the range and lowers the arc, and it pulls the best angle below 45 degrees. It assumes gravity is constant and the same everywhere along the path, which holds easily over any ordinary throw. It treats the object as a single point, so spin and lift are not in the picture. And it measures the landing at the height you set, so a throw onto a slope or a platform needs that height entered as where it actually comes down.

One result of the model is worth holding onto: mass does not appear in any of the equations. In the absence of drag, a heavy ball and a light one launched the same way follow the very same arc, which is Galileo's old observation in modern dress. When you do need to account for air, the usual approach is to take this ideal result first and then trim it with a correction from experience.

Questions people ask

What launch angle gives the maximum range?

For a launch and landing at the same height, 45 degrees. Launching from a height drops the best angle a little below 45, because the added fall time favours a flatter path.

Does a heavier object travel less far?

Not in this model. Without air resistance, mass cancels out of the motion, so two objects launched at the same speed and angle follow the same trajectory regardless of weight.

Why does my range change so much when I add a launch height?

Because starting higher keeps the object in the air longer, and the horizontal speed keeps carrying it the whole time. Even a few metres of height can add a surprising amount of range, which is why the simple level-ground formula does not fit an elevated launch.

Does it account for air resistance?

No. It uses the ideal, drag-free equations, which are a good match for dense, slow objects. For fast or light ones, real range and height will be lower than the calculator shows.

Why does a 90 degree launch give zero range?

At 90 degrees the whole launch speed is vertical, so the horizontal speed is zero. The object goes straight up and straight back down, covering no horizontal distance.

Can I use it for the Moon or other planets?

Yes. Change the gravity field to the value for that world, such as 1.62 m/s² for the Moon or 3.72 for Mars, and the same launch will reach much further under the weaker pull.

References

A quick note on where the physics comes from. The treatment of projectile motion as two independent one-dimensional motions, and the equations for range, time of flight, and maximum height, is standard classical mechanics, set out clearly in OpenStax's University Physics and in Georgia State University's HyperPhysics. The value of standard gravity, 9.80665 m/s², and the SI units used throughout follow the US National Institute of Standards and Technology.

1. OpenStax, University Physics Volume 1, Section 4.3, Projectile Motion. https://openstax.org/books/university-physics-volume-1/pages/4-3-projectile-motion
2. HyperPhysics, Georgia State University, Trajectory and Projectile Motion. http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html
3. National Institute of Standards and Technology (NIST), Special Publication 811, Guide for the Use of the International System of Units (SI) (standard gravity and units). https://www.nist.gov/pml/special-publication-811