Answer
The spring with twice the spring constant should be compressed by 0.71 cm
Work Step by Step
When a spring is compressed a distance $x$, the energy stored in the spring is $U_s = \frac{1}{2}kx^2$, where $k$ is the spring constant of the spring.
Let $(U_s)_1$ be the energy stored in the spring with spring constant $k$. Let $(U_s)_2$ be the energy stored in the spring with spring constant $2k$;
$(U_s)_2 = (U_s)_1$
$\frac{1}{2}(2k)x^2 = \frac{1}{2}k(1.0~cm)^2$
$x^2 = \frac{(1.0~cm)^2}{2}$
$x = \sqrt{\frac{(1.0~cm)^2}{2}}$
$x = 0.71~cm$
The spring with twice the spring constant should be compressed by 0.71 cm.
