The range is the horizontal distance traveled before the projectile hits the ground.

The distance covered in the x-direction is simply:

x = v_ox t

To find the range, use t as the time of flight, which we just calculated to be:

t

=

2 vo sin(θ) g

Plugging in v_ox = v_o cos(θ) gives:

Range

=

2 vo2 sin(θ) cos(θ) g

Using the trig. identity 2 sin(θ) cos(θ) = sin(2θ) this can be written as:

Range

=

vo2 sin(2θ) g