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A force F= -F0 e(-x/λ ) (where F0 and λ 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) = ± ( v02 + (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 vinfinity = ( v02 - ve2 )1/2 . Note that vinfinity 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- (v02 / ve2)).

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