A block oc mass 'm' is moving on a smooth horizational surface with speed `v_(0)` towards a vertical wall as shown. A massless spring of stiffness 'K' is attached to the wall such that the block compresses the spring as it moves closer to the wall. The natural length of the spring is `I_(o)` (i) What is the speed of the block, as the speing has been c ompressed by a distance x? (ii) What is the macimum copression produced in the spring?

Explanation

The various forces action on the block during its motion are (a) Weight, mg, downwards (b) Normal reaction N upwards (c ) Spring force, kx, leftward. The work done by the weight and normal reaction is zero
The work done by the speing force is `W_(sp)=-underset(0)overset(x)intkxdx`

negative sign appears as block moves towards right, whereas force act feftward.
(i) By work-energy principle, `W=Deltak`
`implies-underset(0)overset(x)intkxdx=1/2mv_(1)^(2)-1/2mv^(2)`
`implies(mv_(1)^(2)-mv^(2))/(2)=(-kx^(2))/(2)`
`thereforev_(1)=sqrt(v^(2)-k/mx^(2))`
(ii) The compression in the spring increases till the block moves towards teh wall. At the instant speed of the block reduces to zero, the compression is maximum. After that, compression will start decresing again.
So when `x=x_(max)v_(f)=0`
`implies0=sqrt(v^(2)-k/mx_(max)^(2))`
`or x_(max)=sqrt((mv^(2))/(k))`