A

Step 1: Use the conservation of energy principle. The initial kinetic energy (KE) of the block is \( KE_i = \frac{1}{2} mv^2 \).
Step 2: After compressing the spring, the speed of the block becomes one fourth, so the final speed (v_f) is \( v_f = \frac{v}{4} \). The final kinetic energy (KE_f) is \( KE_f = \frac{1}{2} m \left(\frac{v}{4}\right)^2 = \frac{1}{2} m \frac{v^2}{16} = \frac{mv^2}{32} \).
Step 3: The energy stored in the spring when compressed by a distance x is given by: \( PE = \frac{1}{2} k x^2 \), where k is the spring constant.
Step 4: According to the conservation of energy, the initial kinetic energy is equal to the final kinetic energy plus the potential energy stored in the spring: \( \frac{1}{2} mv^2 = \frac{mv^2}{32} + \frac{1}{2} k x^2 \).
Step 5: Rearranging the equation gives: \( \frac{1}{2} mv^2 - \frac{mv^2}{32} = \frac{1}{2} k x^2 \).
Step 6: Simplifying the left-hand side: \( \frac{16mv^2}{32} - \frac{mv^2}{32} = \frac{15mv^2}{32} \).
Step 7: Thus, we have: \( \frac{15mv^2}{32} = \frac{1}{2} k x^2 \).
Step 8: Multiply both sides by 2: \( \frac{15mv^2}{16} = k x^2 \).
Step 9: Now, solving for k gives us: \( k = \frac{15mv^2}{16x^2} \).
Therefore, the spring constant of the spring is \( k = \frac{15mv^2}{16x^2} \).