# Projectile Range Calculator

By Hanna Pamuła, PhD candidate
Last updated: Oct 05, 2021

With this projectile range calculator, you'll quickly find out how far the object can be thrown. All you need to do is enter the three parameters of projectile motion - velocity, angle, and height from which the projectile is launched. In no time you'll find the horizontal displacement of your object. Experiment with the calculator and discover yourself which angle guarantees the maximum distance of a projectile - or scroll down and learn more about projectile range formulas.

## Projectile range formulas

Let's split the equations into two cases: when we launch the projectile from the ground and when the object is thrown from some initial height (e.g. table, building, bridge). If you want to find the range for horizontal speed only (so angle = 0), you can use this projectile range calculator or go directly to horizontal projectile motion calculator page. Air resistance is neglected in all calculations.

1. Launch from the ground (initial height = 0)

To find the formula for the range of the projectile, let's start from the equation of motion. The projectile range is the distance traveled by the object when it returns to the ground (so y=0):

`0 = V₀ * t * sin(α) - g * t² / 2`

From that equation, we'll find `t`, which is the time of flight to the ground:

`t = 2 * V₀ * sin(α) / g`

Also, we know that the maximum distance of the projectile can be found from simple relation `d = V * t`. Velocity in our case is the horizontal velocity `Vx = V0 * cos(α)`, and time to reach the ground is a value we've already calculated:

`d = V * t = V₀ * cos(α) * 2 * V₀ * sin(α) / g `

`d = 2 * V₀² * cos(α) * sin(α) / g `

Knowing the trigonometric identity `sin(2 * x) = 2 * sin(x) * cos(x)`, we can write the final formula as:

`d = V₀² * sin(2 * α) / g ` Trajectories of projectiles launched at different elevation angles but the same speed (g = 10 m/s²). By Cmglee - Own work, CC BY-SA 3.0, Link

Did you found out when distance is maximal? Yes, it's when the sin(2α) has the highest possible value:

`sin(2*α) = 1 => 2*α = 90° => ``α = 45°`

2. Launch from an elevation (initial height > 0)

In that case, the time spent flying upwards is shorter than the time when object is falling down (time from reaching the maximum height to striking the ground). The formula for projectile range may be then written as:

`d = V₀ * cos(α) * [V₀ * sin(α) + √((V₀ * sin(α))² + 2 * g * h)] / g`

## Projectile range calculator example

Let's find out the range of the hypothetical volcanic ballistic projectiles 🌋:

1. Enter the projectile velocity. Assume that a rock was ejected from the volcano with a speed of 30 m/s.
2. Type in the angle. Let's say that the angle was equal to 25°.
3. Fill in the box of initial height. Our volcano was quite small and the rock lands on the side of the volcano at 100 m lower altitude than the launching point.
4. The projectile range calculator shows the answer! The maximal distance is 162.87 m for our volcano example. That's a lot!

In reality, the projectile motion is a much complicated phenomenon. The air resistance reduces a range of a projectile - and that's dependent on the object volume, shape, surface smoothness, mass, etc.

Hanna Pamuła, PhD candidate
Velocity
ft/s
Angle of launch
deg
Initial height
ft
Range
Range
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