Question

A block of mass m sliding on a smooth horizontal surface with a velocity →ν meets a long horizontal spring fixed at one end and with spring constant k, as shown in figure (8-E10). Find the maximum compression of the spring. Will the velocity of the block be the same as →ν when it comes back to the original position shown?
Figure

Answer 1

The potential energy gain from the spring is  $x=v \sqrt{\frac{m}{k}}$ and the maximum spring compression should occur when the block rests. Also the block speed won't be the same when it came back to its original position.    

Explanation:

Let the Spring compression be x.

(a) The implementation of the energy conservation law,

Maximum spring compression should occur when the block rests.

Therefore change in the block's kinetic energy due to a change in its velocity from u m / s to 0 will be equal to the potential spring energy gain.

Change in Block kinetic energy $=\frac{1}{2} m v^{2}-\frac{1}{2} m 0^{2}=\frac{1}{2} m v^{2}$

Potential energy gain from spring  $=\frac{1}{2} k x^{2}=\frac{1}{2} m v^{2}$

                                                     $k x^{2}=m v^{2}$

                                                      $x^{2}=\frac{m v^{2}}{k}$

                                                       $x=v \sqrt{\frac{m}{k}}$

(b) No. The block's speed won't be the same when it returns to its original position. It'll be in the opposite direction and the magnitude will be the same if we fail to be completely elastic for all the losses due to friction and spring.

Answer 2

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Please refer the above attachment