A 20 kg body is released from rest so as to slide in between vertical rails and compresses a vertical spring `(k=1920(N)/(m)`) placed at a distance `h=1.0`m from the strating position of the body. The rails offer a frictional force of 40 N opposing the motion of body. Find (a) the velocity v of the body just before striking with the spring, (b) the maximum compression of the spring and (c ) the distance `h^(')` through which the body is rebounded up.

Explanation

(a) Between (1) and (2), loss in PE. `=`gain in K.E.`+`work done against friction
`mgh=(1)/(2)mv^2+fh`
`20xx10xx1=(1)/(2)xx20v^2+40xx1impliesv4(m)//(s)`
(b) Between (2) and (3) loss in P.E. `+` loss in K.E.`=`gain in spring energy `+` work done against friction
`mgx+(1)/(2)xxmv^2=(1)/(2)kx^2+fx`
`20xx10xx x+(1)/(2)xx20xx4^2=(1)/(2)xx1920x^2+40x`
`960x^2-160x-160=0implies6x^2-x-1=0`
`x=(1+-sqrt(1+4xx6xx1))/(12)`
`=(1+-5)/(12)=(1)/(2)m`
(c ) Between (3) and (4): `(1)/(2)kx^2=mgh^(')+fh^(')`
`=(1)/(2)xx1920((1)/(2))^2=(20xx10+40)h^(')`
`h^(')=1.0m`