A block of mass m travelling at a speed v compresses a spring by a...

According to the law of conservation of mechanical energy, the decrease in kinetic energy of the block is equal to the potential energy stored in the spring.

Initial kinetic energy of the block is $K_i = \dfrac{1}{2}mv^2$

Final kinetic energy of the block is $K_f = \dfrac{1}{2}m\left(\dfrac{v}{2}\right)^2 = \dfrac{1}{8}mv^2$

Potential energy stored in the spring is $U = \dfrac{1}{2}kx^2$

Equating the change in kinetic energy to the potential energy:

$K_i - K_f = U$

$\dfrac{1}{2}mv^2 - \dfrac{1}{8}mv^2 = \dfrac{1}{2}kx^2$

$\dfrac{3}{8}mv^2 = \dfrac{1}{2}kx^2$

$k = \dfrac{3mv^2}{4x^2}$