# Range of a projectile

The path of this projectile launched from a height y0 has a range d.

In physics, assuming a flat Earth with a uniform gravity field, and no air resistance, a projectile launched with specific initial conditions will have a predictable range.

The following applies for ranges which are small compared to the size of the Earth. For longer ranges see sub-orbital spaceflight.

• g: the gravitational acceleration—usually taken to be 9.81 m/s2 (32 f/s2) near the Earth's surface
• θ: the angle at which the projectile is launched
• v: the velocity at which the projectile is launched
• y0: the initial height of the projectile
• d: the total horizontal distance travelled by the projectile

When neglecting air resistance, the range of a projectile will be

$d = \frac{v \cos \theta}{g} \left( v \sin \theta + \sqrt{v^2 \sin^2 \theta + 2gy_0} \right)$

If (y0) is taken to be zero, meaning the object is being launched on flat ground, the range of the projectile will then simplify to

$d = \frac{v^2 \sin 2 \theta}{g}$

## Ideal projectile motion

Ideal projectile motion states that there is no air resistance and no change in gravitational acceleration. This assumption simplifies the mathematics greatly, and is a close approximation of actual projectile motion in cases where the distances travelled are small. Ideal projectile motion is also a good introduction to the topic before adding the complications of air resistance.

### Derivations

Consider a projectile which is given a initial velocity "v°" at an angle "θ" with the horizontal.Let the  range of the projectile "R".
For obtaining an expression the range of the projectile we consider horizontal motion


The horizontal motion the data is given as follow

    Initial velocity= v°x=vo cosθ
Final velocity=0
Displacement=R


We know that s=vt

      R=v°T
R=v°cosθ (2v°/g)sinθ
R=v°cosθ -2 v°/g sinθ
R=vo²/g cosθ 2 sinθ
R=vo²/g 2sinθ cosθ
<2sinθ cosθ=sin(2θ)>
R=vo²sin(2θ)/g

this is equation for range of the projectile

## Actual projectile motion

In addition to air resistance, which slows a projectile and reduces its range, many other factors also have to be accounted for when actual projectile motion is considered.

### Projectile characteristics

Generally speaking, a projectile with greater volume faces greater air resistance, reducing the range of the projectile. This can be modified by the projectile shape: a tall and wide, but short projectile will face greater air resistance than a low and narrow, but long, projectile of the same volume. The surface of the projectile also must be considered: a smooth projectile will face less air resistance than a rough-surfaced one, and irregularities on the surface of a projectile may change its trajectory if they create more drag on one side of the projectile than on the other. Mass also becomes important, as a more massive projectile will have more kinetic energy, and will thus be less affected by air resistance. The distribution of mass within the projectile can also be important, as an unevenly weighted projectile may spin undesirably, causing irregularities in its trajectory due to the magnus effect.

If a projectile is given rotation along its axes of travel, irregularities in the projectile's shape and weight distribution tend to be canceled out. See rifling for a greater explanation.

### Firearm barrels

For projectiles that are launched by firearms and artillery, the nature of the gun's barrel is also important. Longer barrels allow more of the propellant's energy to be given to the projectile, yielding greater range. Rifling, while it may not increase the average (arithmetic mean) range of many shots from the same gun, will increase the accuracy and precision of the gun.