Answer to Question #162939 in Classical Mechanics for fari

Question

A projectile is fired at an angle αwith initial velocity v0. If it experiences the 

resistive force 𝐹𝑟 = −𝑘𝑣

2

, find the expression for distance covered x as a 

function of vx ant time. Also discuss the special cases.

Answer 1

Answer

For projectile motion

$u_x=v_0cos\alpha$

$u_y=v_0sin\alpha$

Force in x direction

$F_x=-kv_x^2=ma_x$

So acceleration

$a_x=\frac{dv_x}{dt}=-\frac{ kv_x^2}{m}$

So using integration limits

$\int_{v_0cos\alpha}^{v_x}\frac{dv_x}{v_x^2}=\int _0^t\frac{-kdt}{m}$

So we get

$[\frac{1}{v_x}]_{v_0cos\alpha}^{v_0}=-\frac{kt}{m}$

So in terms of v_x and t

$v_x=-\frac{m}{kt}\\v_x\propto\frac{1}{t}$