Range of a projectile

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In physics, a projectile launched with specific initial conditions will have a range. It may be more predictable assuming a flat Earth with a uniform gravity field, and no air resistance. The horizontal ranges of a projectile are equal for two complementary angles of projection with the same velocity.

The following applies for ranges which are small compared to the size of the Earth. For longer ranges see sub-orbital spaceflight. The maximum horizontal distance travelled by the projectile, neglecting air resistance, can be calculated as follows:^[1]

${\displaystyle d={\frac {v\cos \theta }{g}}\left(v\sin \theta +{\sqrt {v^{2}\sin ^{2}\theta +2gy_{0}}}\right)}$

where

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

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

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