Projectile Motion Calculator

What is a projectile motion calculator?

A projectile motion calculator finds how far and how high an object travels after it is launched, using its initial speed, launch angle and starting height. It returns the range, maximum height, time of flight and time to the peak, assuming no air resistance. Physics students, lab groups, teachers and engineers use it to solve trajectory problems and check homework fast.

What is the projectile motion formula?

Projectile motion is split into independent horizontal and vertical parts. The launch speed v₀ at angle θ becomes a horizontal component vₓ = v₀·cos(θ) and a vertical component v_y = v₀·sin(θ), and gravity only acts on the vertical part. From these the calculator computes:

• Range = vₓ · t — horizontal distance, in metres (m).
• Max height = h + v_y² / (2g) — highest point reached, in metres (m).
• Time of flight = (v_y + √(v_y² + 2gh)) / g — total time in the air, in seconds (s).

Every symbol has a fixed unit so the answers come out in metres and seconds:

• v₀ — initial launch speed, in metres per second (m/s).
• θ — launch angle above the horizontal, in degrees (converted to radians inside the formula).
• vₓ, v_y — horizontal and vertical speed components, in m/s.
• h — initial height above the landing level, in metres (m); default 0.
• g — gravitational acceleration, in m/s²; default 9.81 on Earth.
• t — time of flight, in seconds (s).

Two more quantities follow directly from these. The time to the peak is v_y / g (when the upward speed reaches zero), and for a level launch (h = 0) the time of flight simplifies to 2·v_y / g — exactly twice the time to peak, because the rise and fall are symmetric. All results are reported to five significant figures, assuming no air resistance.

Worked examples

Each example uses only the formulas above with g = 9.81 m/s², so you can reproduce every answer by typing the same inputs into the calculator.

Example 1 — classic 45° launch from the ground

A ball is thrown at v₀ = 20 m/s, θ = 45°, from ground level (h = 0).

vₓ = v_y = 20 · cos(45°) = 20 · 0.70711 = 14.142 m/s Max height = 14.142² / (2 × 9.81) = 199.99 / 19.62 = 10.19 m Time of flight = (14.142 + 14.142) / 9.81 = 28.284 / 9.81 = 2.883 s Range = 14.142 × 2.883 = 40.77 m Time to peak = 14.142 / 9.81 = 1.441 s

The ball lands 40.77 m away after 2.883 s, peaking at 10.19 m. A 45° angle gives the maximum range on level ground.

Example 2 — a 30° launch at higher speed

A projectile leaves at v₀ = 30 m/s, θ = 30°, h = 0.

vₓ = 30 · cos(30°) = 30 · 0.86603 = 25.981 m/s v_y = 30 · sin(30°) = 30 · 0.5 = 15.000 m/s Max height = 15² / (2 × 9.81) = 225 / 19.62 = 11.468 m Time of flight = (15 + 15) / 9.81 = 30 / 9.81 = 3.0581 s Range = 25.981 × 3.0581 = 79.452 m

The lower 30° angle trades height for a flatter, faster path, giving a long 79.452 m range.

Example 3 — horizontal launch from a height

An object is launched horizontally (θ = 0°) at v₀ = 10 m/s from a cliff h = 20 m high. Here v_y = 0, so it never rises above the launch point.

vₓ = 10 · cos(0°) = 10 m/s, v_y = 10 · sin(0°) = 0 m/s Max height = 20 + 0² / (2 × 9.81) = 20 m Time of flight = (0 + √(0 + 2 × 9.81 × 20)) / 9.81 = √392.4 / 9.81 = 2.0193 s Range = 10 × 2.0193 = 20.193 m

The object falls for 2.0193 s and lands 20.193 m from the base of the cliff. Time to peak is 0 s, since it starts at the top of its arc.

Range and height by launch angle

This table shows the range, maximum height and time of flight for a fixed v₀ = 20 m/s from level ground (h = 0, g = 9.81 m/s²), computed straight from the formulas above. Notice that range peaks at 45° and that complementary angles give equal ranges — 30° and 60° both reach 35.312 m.

Launch angle θ	Max height (m)	Time of flight (s)	Range (m)
15°	1.3657	1.0553	20.387
30°	5.0968	2.0387	35.312
45°	10.194	2.8832	40.775
60°	15.291	3.5312	35.312
75°	19.022	3.9385	20.387
90°	20.387	4.0775	0

At 90° the object goes straight up and lands back at the start, so its range is zero while its height is greatest.

Common uses

Projectile calculations appear wherever something is thrown, fired or dropped while moving forward:

• Physics homework and exams — solving for range, height, flight time or peak time in kinematics problems.
• Lab and classroom demos — predicting where a launched ball or marble will land on a track or floor.
• Sports analysis — estimating the carry of a thrown ball, a long jump, a golf shot or a basketball arc.
• Engineering estimates — sizing water jets, ballistic trajectories or the clearance needed for launched parts.
• Game and simulation design — checking the math behind jump arcs and projectile weapons.

Tips and common mistakes

• Use the right units. Speed in m/s, angle in degrees and height in metres. Mixing in km/h or feet without converting first gives wrong distances.
• Angle is measured from the horizontal. A 0° launch is flat (horizontal), and 90° is straight up — not the other way around.
• Don’t forget the height term. When h is greater than 0, the projectile is in the air longer, so range grows; setting h = 0 assumes it lands at launch level.
• 45° is best only on level ground. When launching from a height, the optimum angle for maximum range drops below 45°.
• Five significant figures is display precision, not real-world accuracy. The answer is only as good as the speed and angle you enter.
• A negative or zero result usually means a degenerate case. A 90° launch gives near-zero range, and θ = 0° gives zero rise above the start — both are correct, not errors.

Limitations and accuracy

This calculator models ideal projectile motion: it assumes no air resistance, no wind, a constant gravitational acceleration (g = 9.81 m/s² by default), and a flat landing surface at the chosen height. Under those assumptions the horizontal speed stays constant and the path is a perfect parabola, so the arithmetic is exact for your inputs. Real projectiles experience drag, lift and spin, which shorten the range and lower the peak — the effect is small for dense, slow objects but large for light or very fast ones such as a ping-pong ball or a bullet. Treat the tool as an education and estimation aid: it is ideal for textbook problems and quick trajectory checks, but match the assumptions to your real situation before relying on a figure.

For related calculations, try the kinetic energy calculator to find the launch energy, the speed converter to get v₀ into m/s first, or the density calculator for the object’s mass — and browse more in the chemistry and physics category.