projectile.txt

The equation to find the distance over the ground (range), R, of a projectile fired over level ground in a vacuum is as follows:

R=(v^2 sin 2theta)/g
Where v is the magnitude of the initial velocity of the projectile, theta is the angle of elevation, and g is the gravitational field strength, 9.8 m/s^2 on earth.

How do we derive this equation?

The coordinates of a projectile in 2D space at a particular moment in time can be expressed as an ordered pair, (d(t), h(t)),
where d(t) is the horizontal distance from the starting point, and h(t) is the vertical height from the starting elevation.

h(t) can be defined as follows, assuming h(0)=0:
h(t)=-1/2gt^2 + v_y t

d(t) can be defined as follows, again assuming d(0)=0:
d(t)=v_x t

We want to find the distance that the projectile traveled when it hit the ground. To do so, we first need to find the time when it hit the ground,
that is, we need to find t such that h(t)=0.

h(t)=-1/2gt^2 + v_y t
   0=t(-1/2gt + v_y)
   t=0, -1/2gt + v_y=0
              -1/2gt=-v_y
                  gt=2v_y
                   t=2v_y/g

So the projectile hits the ground at t=2v_y/g. To find the distance from the starting point that it lands, we sub this expression for t
into d(t), which gives us R.

R = v_x t
  = v_x 2v_y/g
  = (2 v_x v_y)/g

The horizontal and verical components of the initial velocity can be expressed as the ordered pair (v_x, v_y),
but can also be written as the ordered pair (v cos theta, v sin theta).

R = (2 v cos(theta) v sin (theta))/g
R = (v^2 2sin(theta)cos(theta))/g

The final step is to use the trig identity sin 2theta = 2sin(theta)cos(theta)

R = (v^2 sin 2theta)/g

From here we can see that the maximum range R for constant v and g will occur when sin 2theta = 1.
Solving this for theta gives us the angle that allows for the maximum range.

0 < theta < pi.
sin 2theta = 1.

Let m=2theta.
0 < m < 2pi.
sin m = 1.
m = pi/2.

theta = pi/4. (Or theta = 45 degrees.)

Note that for a constant v and g, there are two angles theta that result in the same distance for R. Both of these values
for theta are between 0 and 90 degrees, but one is greater than 45 degrees and one is less than 45 degrees. This means
that there are two possible angles for the projectile to be fired at for it to reach the desired target.
(Except for the case when theta = 45 degrees exactly.) Mortars and long-range artillery typically fire their shells
at angles of elevation greater than 45 degrees, because it allows the shell to avoid more obstacles.