Answer
(a) $930N/m$
(b) $1.7cm$
Work Step by Step
(a) We can find the required spring constant as follows:
$K=\frac{mv_{\circ}^2}{x^2}$
We plug in the known values to obtain:
$K=\frac{(1.4)(0.62)^2}{(0.024)^2}$
$K=930N/m$
(b) We can find the new required maximum compression as follows:
$\frac{x_{new}}{x_{old}}=\frac{\sqrt{m_{new}v_{new}^2/K}}{\sqrt{m_{old}v_{old}^2/K}}=\sqrt{\frac{(2m_{old})(\frac{1}{2}v_{old})^2}{m_{old}v_{old}^2}}={\frac{1}{\sqrt 2}}$
This can be rearranged as:
$x_{new}=\frac{x_{old}}{\sqrt 2}$
$\implies x_{new}=\frac{2.4cm}{\sqrt 2}=1.7cm$
