Question #162939

A projectile is fired at an angle Ξ±with initial velocity v0. If it experiences the 

resistive force πΉπ‘Ÿ = βˆ’π‘˜π‘£

2

, find the expression for distance covered x as a 

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


1
Expert's answer
2021-02-15T07:28:10-0500

Answer

For projectile motion

ux=v0cosΞ±u_x=v_0cos\alpha

uy=v0sinΞ±u_y=v_0sin\alpha

Force in x direction

Fx=βˆ’kvx2=maxF_x=-kv_x^2=ma_x

So acceleration

ax=dvxdt=βˆ’kvx2ma_x=\frac{dv_x}{dt}=-\frac{ kv_x^2}{m}


So using integration limits

∫v0cosΞ±vxdvxvx2=∫0tβˆ’kdtm\int_{v_0cos\alpha}^{v_x}\frac{dv_x}{v_x^2}=\int _0^t\frac{-kdt}{m}

So we get


[1vx]v0cosΞ±v0=βˆ’ktm[\frac{1}{v_x}]_{v_0cos\alpha}^{v_0}=-\frac{kt}{m}

So in terms of vx and t

vx=βˆ’mktvx∝1tv_x=-\frac{m}{kt}\\v_x\propto\frac{1}{t}




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