A force F = ?F0 e ^?x/? (where F0 and ? are positive constants)acts on a particle of mass m that is initially at x = x0 and movingwith velocity v0 (> 0). Show that the velocity of the particleis given by
v(x)=(v0^2+(2F0? /m)((e^-x/?)-1))^1/2
where the upper (lower) sign corresponds to the motion in thepositive (negative) x direction. Consider first the upper sign. Forsimplicity, define ve=(2F0 ? /m)^1/2 then show that the asymptoticvelocity (limiting velocity as x ? ?) is given byv?=(v0^2-ve^2)^1/2 Note that v? exists if v0 ? ve.Sketch the graphof v(x) in this case. Analyse the problem when v0 < ve by takinginto account of the lower sign in the above solution. Sketch thegraph of v(x) in this case. Show that the particle comes to rest(v(x) = 0) at a finite value of x given by xm=??ln(1-v0^2/ve^2)
