What is the range formula for a projectile?

For a launch and landing at the same height, range is R = v₀² · sin(2θ) / g, where v₀ is launch speed, θ is launch angle, and g = 9.81 m/s². When the launch height h₀ differs from zero the formula becomes a quadratic in time: solve 0.5·g·t² − v_y₀·t − h₀ = 0 for t, then R = vₓ · t.

What angle gives maximum range on flat ground?

45° gives the maximum range on level ground, because sin(2 × 45°) = sin(90°) = 1, which is the maximum value of the sine function. Any angle above or below 45° produces a shorter range for the same launch speed.

How is maximum height calculated?

The peak occurs when the vertical velocity equals zero. From v_y₀ = v₀·sin θ and v_y = v_y₀ − g·t, the time to peak is t_peak = v_y₀ / g. Maximum height is H = h₀ + v_y₀² / (2·g). At 45° and 50 m/s from ground level that gives H ≈ 31.9 m.

Does air resistance affect the result?

This calculator assumes a vacuum (no air resistance), which is the standard first-year physics model. In reality drag reduces range and maximum height, especially for lightweight or high-speed objects. For accurate ballistics you would need drag coefficient, cross-sectional area, and air density.

Can I use kilometres per hour instead of metres per second?

The calculator expects SI units (m/s, metres, seconds, m/s²). To convert: divide km/h by 3.6 to get m/s. For example, 72 km/h = 72 ÷ 3.6 = 20 m/s. Results are then in metres and seconds.

What does the impact angle mean?

The impact angle is the angle below the horizontal at which the projectile hits the ground. It is calculated from the vertical and horizontal velocity components at the moment of impact using arctan(|v_y| / vₓ). A steeper impact angle (closer to 90°) means the object is falling more vertically on arrival.

Projectile Range Calculator

Projectile motion is one of the most fundamental topics in classical mechanics. When an object is launched with an initial speed at an angle to the horizontal — and the only force acting on it is gravity — its path traces a parabola. This calculator applies those exact kinematics equations to give you range, maximum height, total flight time, impact speed, and every intermediate quantity, all solved in your browser without any data leaving your device.

The tool covers five solve modes: compute the horizontal range from a given speed and angle; find the required launch speed to hit a target range; find the required angle; find the minimum launch height; or compute the total flight time. A collapsible “Show working” panel walks you through every step of the arithmetic.

How it works

The motion is split into two independent components:

Horizontal (constant velocity, no force): x(t) = vₓ · t, where vₓ = v₀ · cos θ
Vertical (constant downward acceleration g): y(t) = h₀ + v_y₀ · t − 0.5 · g · t²

where v_y₀ = v₀ · sin θ and g = 9.81 m/s².

Flight time is found by setting y(t) = 0 (landing level) and solving the resulting quadratic:

0.5 · g · t² − v_y₀ · t − h₀ = 0

The positive root is the total time of flight T. The range is then simply R = vₓ · T.

For a flat launch (h₀ = 0) this simplifies to the classic formula:

R = v₀² · sin(2θ) / g

Maximum height is reached when vertical velocity = 0, at t_peak = v_y₀ / g:

H_max = h₀ + v_y₀² / (2g)

Impact speed is the vector magnitude at landing: √(vₓ² + v_y²), where v_y = v_y₀ − g · T. The impact angle below horizontal follows from arctan(|v_y| / vₓ).

When solving for an unknown (launch speed, angle, or height) the calculator uses a binary-search or fine-grid scan to find the value that produces the target range to six significant figures.

Worked example

A ball is kicked at 50 m/s and 45° from ground level (h₀ = 0):

vₓ = 50 · cos 45° = 35.36 m/s
v_y₀ = 50 · sin 45° = 35.36 m/s
T = 2 · v_y₀ / g = 2 × 35.36 / 9.81 = 7.211 s
R = 35.36 × 7.211 = 254.9 m
H_max = 35.36² / (2 × 9.81) = 63.7 m
Impact speed = 50 m/s (energy conservation — same as launch, since h₀ = 0)

Now raise the launch point by 10 m (h₀ = 10): the quadratic gives T = 7.487 s, extending the range to 264.7 m — an extra 9.8 m purely from the extra height.

Launch speed	Angle	h₀	Range	Max height	Flight time
50 m/s	45°	0 m	254.9 m	63.7 m	7.211 s
50 m/s	30°	0 m	220.9 m	31.9 m	5.097 s
50 m/s	60°	0 m	220.9 m	95.6 m	8.830 s
50 m/s	45°	10 m	264.7 m	73.7 m	7.487 s
100 m/s	45°	0 m	1019.4 m	254.8 m	14.42 s

Note that 30° and 60° give the same range — they are complementary angles (they sum to 90°), a symmetry of the sin(2θ) formula.

Formula reference

All standard projectile-motion equations assume no air resistance and constant g = 9.81 m/s²:

Range (level ground): R = v₀² · sin(2θ) / g
Range (elevated, general): solve 0.5·g·T² − v_y₀·T − h₀ = 0, then R = vₓ·T
Max height: H = h₀ + v_y₀² / (2g)
Time to peak: t_peak = v_y₀ / g
Impact speed: v_impact = √(vₓ² + (v_y₀ − g·T)²)
Impact angle: arctan(|v_y₀ − g·T| / vₓ) below horizontal

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