Theory Behind The Projectile Motion

Theory:
Projectile Motion

Following the prescription, the trajectory for projectile motion is derive by eliminating t from the equations for the horizontal and vertical component of the projectile motion:

x = v_0x t
y = v_0x t

We get the following formula:

y = tan O_x – [g x² / 2 (v₀ cos O_o)²]

Here O_x is the launch angle, ie

tan O_o = v_oy / v_ox

As we had mentioned previously the trajectory for projectile motion is parabolic. There are a few useful expressions which can be derived from this equation. The maximum range on level ground is the distance between y=0 solutions:

R = v_o² sin 2O / g

We see that this expression is maximal for

O_o= 45 degree where the range is

R_max = v₀² / g

Note that the smaller g the larger range, so we can throw almost 10 times further on the moon than on earth! Also note that the range decreases for steeper as well as for shallower angles which means that for any range less than max-range there are two elevations which will mach. Highest elevation always keeps the projectile in air longer so if we want to get there fast we use the lower elevation.

Another interesting characteristic of projectile motion is the max height. To determine it we need only consider the motion in the y-direction. The following equation holds for this motion with constant acceleration:

v_y² = v_0y² – 2gh_max

At maximum height we have

v_y= 0. We insert this value in the equation and solve for h to get

h _max = V_0y² / 2g

Remember that these formulae only hold for projectile motion on flat ground. For more complicated cases you will have to work the formulae for r(t) and the trajectory yourself so make sure you understand what went on in these derivations.

Created by Yeoh Ceh Hsing