A force F = F0 e ^x/ (where F0 and are positive constants) acts on a

A force F = −F0 e ^−x/λ (where F0 and λ are positive constants) acts on a particle of mass m that is initially at x = x0 and moving with velocity v0 (> 0). Show that the velocity of the particle is given by

v(x)=(v0^2+(2F0λ /m)((e^-x/λ)-1))^1/2

where the upper (lower) sign corresponds to the motion in the positive (negative) x direction. Consider first the upper sign. For simplicity, define ve = (2F0 λ /m)^1/2 then show that the asymptotic velocity (limiting velocity as x → ∞) is given by v∞=(v0^2-ve^2)^1/2 Note that v∞ exists if v0 ≥ ve. Sketch the graph of v(x) in this case. Analyse the problem when v0 < ve by taking into account of the lower sign in the above solution. Sketch the graph 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=−λ ln(1-v0^2/ve^2)