What is projectile motion?

Projectile motion is the curved path an object follows when it is launched into the air and then left to gravity alone. Once the object is released nothing pushes it forward any more, so its horizontal motion stays constant while gravity steadily pulls it downward. The combination of steady horizontal drift and accelerating vertical fall produces the familiar parabolic arc — the shape of a thrown ball, a struck golf ball or a stream of water from a hose.

How do you calculate the range of a projectile?

Split the launch velocity into components, find how long the object stays airborne, then multiply. The horizontal component is v₀·cos θ and the vertical component is v₀·sin θ. For a launch from height h₀ the flight time is t = [v_y₀ + √(v_y₀² + 2gh₀)] ÷ g, and the range is simply R = v₀·cos θ × t. When the launch and landing are at the same level this reduces to the familiar R = v₀²·sin(2θ) ÷ g.

Why is 45 degrees the best launch angle — and when is it not?

From ground level, range depends on sin(2θ), which is largest when 2θ = 90°, so θ = 45°. This also explains the neat pairing of complementary angles: 30° and 60° give exactly the same distance, as do 20° and 70°. But the 45° rule only holds when the launch and landing heights match. Launch from a raised position and the extra fall time rewards a flatter, faster shot, so the optimum drops below 45° — the higher you start, the lower the best angle becomes.

How high does a projectile go?

The object rises until its vertical velocity reaches zero, which takes t = v_y₀ ÷ g. Substituting that back gives the maximum height H = h₀ + v_y₀² ÷ (2g). Notice the horizontal velocity plays no part at all — only the vertical component and gravity determine the peak. Doubling the vertical launch speed quadruples the height, because the term is squared.

Why do a dropped object and a horizontally fired one land at the same time?

Because the horizontal and vertical motions are completely independent. Gravity acts only downward, so it changes the vertical velocity in exactly the same way whether or not the object is also moving sideways. A bullet fired horizontally and one simply dropped from the same height both take √(2h/g) seconds to reach the ground — the fired one just lands a long way downrange. This independence is the single most useful idea in the whole topic.

What is the impact speed, and how does it compare to the launch speed?

For a launch and landing at the same height, the impact speed exactly equals the launch speed and the impact angle mirrors the launch angle — the path is symmetric. When the object starts higher up it lands faster, and energy conservation gives the clean result v = √(v₀² + 2gh₀). The horizontal component never changes throughout the flight; all of the extra speed comes from the vertical part.

How does gravity on the Moon or Mars change the flight?

Both flight time and range scale as 1/g, while maximum height also scales as 1/g. Since lunar gravity is about 1.62 m/s² against Earth's 9.81, the same throw on the Moon travels roughly six times as far and hangs in the air roughly six times as long. On Mars, at about 3.72 m/s², everything stretches by around 2.6×. Jupiter works the other way: at 24.79 m/s² the same shot covers less than half its Earth distance.

How much does air resistance change these results?

This calculator uses the idealised vacuum model, where gravity is the only force. That is the standard model taught in physics courses and it is close to reality for dense, compact, relatively slow objects — a shot put or a dropped stone. It becomes optimistic for anything light, large or fast: a real golf ball or football falls short of the vacuum prediction, the trajectory becomes asymmetric with a steeper descent than ascent, and the optimum launch angle drops well below 45°. Treat these numbers as a clean upper bound.

Does the mass of the object matter?

Not in this model. Gravity accelerates every object at the same rate regardless of mass, so a marble and a cannonball launched at identical speeds and angles trace identical paths — mass does not appear anywhere in the equations. Mass only starts to matter once air resistance enters, because a heavier object of the same size carries more momentum relative to the drag force and so is slowed less. That is why a feather and a hammer fall together on the Moon but not on Earth.

Projectile Motion Calculator

Work out exactly where a launched object will land, how high it will climb and how long it will stay in the air. Enter an initial speed, a launch angle and a launch height, and this calculator returns the horizontal range, the maximum height, the total flight time and the impact speed — then animate...

LAUNCH SETUP

INITIAL SPEED (m/s)

LAUNCH ANGLE45°

LAUNCH HEIGHT (m)

GRAVITY — 9.81 m/s²

REAL-WORLD PRESETS

HORIZONTAL RANGE

40.77 m

MAX HEIGHT

10.19 m

FLIGHT TIME

2.88 s

IMPACT

20.0 m/s

Horizontal vₓ

14.142 m/s

Vertical v_y₀

14.142 m/s

Time to apex

1.442 s

Impact angle

45.0°

Optimal angle

45.0°

Range at optimum

40.77 m

STEPS

Split the launch: vₓ = 20 × cos(45°) = 14.142 m/s, v_y₀ = 20 × sin(45°) = 14.142 m/s

Rise time: t_apex = v_y₀ ÷ g = 14.142 ÷ 9.81 = 1.442 s

Peak: H = h₀ + v_y₀² ÷ (2g) = 0 + 200.00 ÷ 19.62 = 10.194 m

Flight: t = [v_y₀ + √(v_y₀² + 2gh₀)] ÷ g = 2.883 s

Range: R = vₓ × t = 14.142 × 2.883 = 40.775 m

FLIGHT SIMULATION

TIME

0.00 s

0.0 m

HEIGHT

0.0 m

SPEED

20.0 m/s

▬ resultant v▬ horizontal vₓ▬ vertical v_y

RANGE vs LAUNCH ANGLE

From ground level the curve peaks at exactly 45°, and angles either side of it pair up (30° and 60° travel the same distance).

Vacuum model. Gravity is the only force acting — no air resistance, wind or spin. Real projectiles fall short of these numbers, and the gap widens with speed and surface area.

Live simulation · drawn to scale · updates as you type

Created with❤️byeaglecalculator.com

HOW TO USE

1
Enter the INITIAL SPEED — how fast the object is moving the instant it is released, in metres per second. This is the total speed along the launch direction, not the horizontal or vertical part; the calculator splits it into components for you.
2
Set the LAUNCH ANGLE with the slider or the quick chips, measured up from the horizontal. 0° is a flat, horizontal throw and 90° is straight up. Tap the green ★ chip to jump straight to the angle that maximises range for your setup.
3
Add a LAUNCH HEIGHT if the object does not start at ground level — a cliff, a table, a shooter's hand. Leave it at 0 for a ground-level launch. Then pick a planet if you want to model gravity somewhere other than Earth.
4
Read the black result box for range, maximum height, flight time and impact speed, and watch the simulation on the right. Press PLAY or REPLAY to run the flight, or drag the scrubber to step through it and read the position, height and speed at any instant.

THE FORMULAS

Horizontal velocityvₓ = v₀ · cos θ

Vertical velocityv_y₀ = v₀ · sin θ

Position at time tx = vₓt · y = h₀ + v_y₀t − ½gt²

Time to apext_apex = v_y₀ / g

Maximum heightH = h₀ + v_y₀² / (2g)

Time of flightt = [v_y₀ + √(v_y₀² + 2gh₀)] / g

RangeR = vₓ · t

Range (level ground)R = v₀² · sin(2θ) / g

Impact speedv = √(v₀² + 2gh₀)

Optimal angleθ = arctan( v₀ / √(v₀² + 2gh₀) )

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Last updated: July 29, 2026 · Formula verified · Eagle-eyed accuracy for every calculation.