Range of a projectile

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. The maximum horizontal distance traveled by the projectile

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

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

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.

45 degrees goes the farthest. This is due to the nature of right triangles. Additionally, from the equation for the range :

We can see that the range will be maximum when the value of ${\displaystyle \sin 2\theta }$ is the highest (i.e. when it is equal to 1). Clearly, ${\displaystyle 2\theta }$ has to be 90 degrees. That is to say, ${\displaystyle \theta }$ is 45 degrees.

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