A block of mass `m` is attached to the frame by a light spring of stiffness `k`. The frame and block are initially at rest with `x=x_(0)`, the uncompressed length of the spring. If the frame is given a constant horizontal acceleration `a_(0)` towards left, determine the maximum velocity `(v_(ret))_(max)` of the block relative to the frame (block is free to move inside frame). Ignore any friction.

Explanation

Draw the `FBD` of block in the frame of accelerated frame. In this figure, block has maximum velocity when it is in equilibrium with respect to accelerated frame
Using work energy theorem
`(1)/(2)kx^(2)+(1)/(2)mv_(0)^(2)=ma_(0)x`
`rArr (1)/(2)kx(m^(2)a_(0)^(2))/(k^(2))+(1)/(2)mv_(0)^(2)=(m^(2)a_(0)^(2))/(k)`
`rArr v_(0)^(2)=(ma_(0)^(2))/(k)rArr v_(0)=a_(0)sqrt((m)/(k))`