Projectile Range Calculator

Projectile Range Calculator finds how far a projectile lands from its launch speed, angle and height, on level ground or on a slope. It also gives the best angle and maximum range.

Projectile Range Calculator Settings

The range of a projectile is the horizontal distance it covers between launch and landing. The Projectile Range Calculator finds it from four different starting points, set under Range From. The usual one is Speed, Angle and Height.

Velocity Parts, No Angle takes the horizontal and vertical launch velocities. Trajectory Equation takes the two coefficients of a path written as y = ax − bx². Launch Up or Down a Slope measures the range along a hillside. All four ignore air resistance.

The default launch is 20 m/s at 35° from 1.5 m above the ground, which lands 40.36 m away after 2.46 s. The cards show that the same speed could reach 42.26 m at the best angle of 43.98°, and that a launch at 52.87° would land on the same spot.

Projectile Range Formula

Range is always the horizontal speed times the time of flight. On level ground, where the launch and landing heights are equal, this collapses to one line. OpenStax University Physics gives it as the range equation.

$$R = \frac{v_0^2 \sin 2\theta}{g}$$

The range of a projectile formula from a height is longer, because the object keeps travelling while it falls the extra distance h₀. The Projectile Range Calculator uses this form whenever the launch height is above zero.

$$R = \frac{v_0 \cos\theta \left( v_0 \sin\theta + \sqrt{v_0^2 \sin^2\theta + 2 g h_0} \right)}{g}$$

The maximum range of a projectile is at 45° on level ground. The reason is in the first formula. The sine of 2θ can never exceed 1, and it reaches 1 when 2θ is 90°. At that angle the range is simply the speed squared divided by gravity.

$$R_{max} = \frac{v_0^2}{g}$$

The range is at its minimum, zero, at both ends of the scale. A 90° launch goes straight up and comes straight down. A 0° launch from the ground lands at once. Between them, any two angles that add to 90° give the same range, such as 15° and 75°.

Maximum Range by Launch Speed

Because the speed is squared, doubling it gives four times the distance. The table shows the farthest a projectile can travel from ground level on Earth at each speed, all at 45°.

Gravity sits under the line, so a weaker pull stretches every figure. The default launch that covers 40.36 m on Earth would reach 103.69 m on Mars and 237.05 m on the Moon, and only 18.19 m on Jupiter. The fourth card of the Projectile Range Calculator lists these for any entry.

Range Without an Angle

Some problems give the launch velocity as two parts and no angle. The range can be calculated directly from them. With horizontal velocity v_x and vertical velocity v_y, the time of flight comes from the vertical part and the distance from the horizontal one.

$$R = \frac{v_x \left( v_y + \sqrt{v_y^2 + 2 g h_0} \right)}{g}$$

From the ground this shortens to 2 v_x v_y ÷ g. A launch with 16 m/s across and 12 m/s up lands 39.16 m away from ground level, or 41.06 m away from a 1.5 m start. The Projectile Range Calculator also reports that these parts equal 20 m/s at 36.87°.

If the flight time is already known, the sum is shorter still. The distance travelled by a projectile is its horizontal velocity times the time in the air, since nothing slows the sideways motion. At 16 m/s for 2.57 s, that is the same 41.06 m.

Range From the Trajectory Equation

A path is often written as y = ax − bx², with x and y in meters. The range is where the height returns to zero. From ground level that happens at x = a ÷ b. With a = 0.7 and b = 0.02, the range is 35 m.

$$R = \frac{a + \sqrt{a^2 + 4 b h_0}}{2 b}$$

A launch height adds a constant to the equation and the root above handles it. Used as a range equation calculator, the Projectile Range Calculator takes a, b and that height. The same coefficients from 1.5 m up give 37.03 m. The coefficient a is the tangent of the launch angle, so this path leaves at 34.99°. On Earth, b = 0.02 per meter then means a launch speed of 19.11 m/s.

Range on a Slope

When the ground rises or falls away, the range is measured along the slope. With the launch angle θ and the slope angle α both taken from the horizontal, the distance along the surface is given below. Uphill slopes are positive and downhill slopes are negative.

$$R = \frac{2 v_0^2 \cos\theta \sin(\theta – \alpha)}{g \cos^2\alpha}$$

The slope setting of the Projectile Range Calculator applies this line. At 20 m/s and 35° up a 15° slope, the range along the slope is 24.5 m. Down a 15° slope, the same launch travels 54.87 m. The best angle is no longer 45°. It is 45° plus half the slope angle, so it rises on an uphill and drops on a downhill.

Pairs of angles still match on a slope. Two launches land at the same point when their angles add up to 90° plus the slope angle. On the 15° uphill, 35° and 70° both reach 24.5 m, with flight times of 1.44 s and 3.46 s.

How Launch Height and Aim Change Range

The third card of the Projectile Range Calculator shows how sensitive the range is. For the default launch, one extra degree adds 0.42 m, one extra m/s of speed adds 3.84 m, and one extra meter of launch height adds 1.29 m. Speed is the setting that matters most.

Height also shifts the best angle. The higher the launch point, the flatter the throw that goes farthest, because the object has more time to travel while it falls. From 20 m up, the default 35° is already the best angle.

A landing spot above or below the launch on a separate surface, such as a ramp jump, is handled by the Car Jump Distance Calculator. A level launch from a height is the special case covered by the Horizontal Projectile Motion Calculator.

Projectile Range Questions

How do you find the speed of a projectile from its range?

Turn the level-ground formula around. The launch speed is the square root of the range times gravity, divided by the sine of twice the angle. A ball that lands 40 m away after leaving at 35° was launched at 20.43 m/s. During the flight the speed is the combination of the fixed horizontal part and the changing vertical part.

What are the three motion formulas behind the range?

They are the constant-acceleration equations. Final velocity equals initial velocity plus acceleration times time. Distance equals initial velocity times time plus half the acceleration times time squared. Final velocity squared equals initial velocity squared plus twice the acceleration times distance.

The range uses the second one twice. Vertically, with gravity as the acceleration, it gives the time of flight. Horizontally, with no acceleration, it gives the distance. The step-by-step panel of the Projectile Range Calculator shows both lines with the numbers filled in.

What are examples of projectile motion?

A thrown baseball, a kicked football, a basketball shot, a long jumper in the air and water leaving a hose are all projectiles. Each one moves under gravity alone once released. Range matters in every case, whether the aim is distance, as in a javelin throw, or landing on a target.

Does air resistance change the range?

It shortens it, and it lowers the best angle below 45°. The Projectile Range Calculator leaves air out, so its result is the upper limit. The Projectile Motion Calculator can add drag on a ball and compare the two ranges. The peak of the flight is covered by the Maximum Height Calculator.