Projectile Motion with Air Resistance

[Science Proff]

Guys, Hello,
I am a high school student who has just started basic calculus and would like to know how to find the motion of the projectile taking air resistance into account. (The range, time, etc)(I have learned about projectile motion without air resistance) Please help me.

 

[jambaugh]

Accounting for air resistance in ballistic flight is complex. The equation of motion without drag comes from the solution of a differential equation F=ma or:
$$m\frac{d^2\mathbf{x}}{dt^2} = \mathbf{F}_g$$
It is simple in the case of projectile motions without air resistance near the Earth's surface because the force is constant. Once you throw drag in there you have the force as a typically complicated function of the velocity. There are certain simplifications one can make. In Stokes' drag one assumes the drag force is proportionate to the velocity (and of course in the opposite direction) this keeps the equation linear:
$$m\frac{d^2 \mathbf{x}}{dt^2} = \mathbf{F}_g - D \frac{d \mathbf{x}}{dt}^2$$
where D is the drag coefficient which depends on the properties of the fluid (air) and the shape and size of the object.

See: http://en.wikipedia.org/wiki/Drag_(physics)"