Answer to Question #84742 in Mechanics | Relativity for Ash

Question

How do I calculate the range of a projectile when it is subjected to air resistance? What formulas do I use?

Answer 1

With air resistance it's not that easy as in the case without air resistance. Under the influence of air resistance the body is subjected to these two forces: force of gravity and force of air resistance (drag force). Split our initial speed into two projections - horizontal

$$v_x(0)=v_0cos\theta$$

and vertical

$$v_z(0)=v_0sin\theta$$

:

$$m\frac{\text{d}v_x}{\text{d}t}=-bv_x,$$ $$m\frac{\text{d}v_z}{\text{d}t}=-mg-bv_z.$$

where

$$b$$

- a positive coefficient for air resistance. It depends on shape of the given body, its frontal cross-sectional area, the speed of the object.

Integration gives:

$$x(t)=\frac{v_0m}{b}\cdot cos\theta\cdot (1-e^{-\frac{tb}{m}}).$$

This result, however, contains

$$b$$

which can strongly affect the range. Also it contains time

$$t$$

. This result does not imply considering rotation during the motion. So for the given body with certain sizes and known properties of air it is only possible to calculate the range numerically.