C

Step 1: The horizontal motion of the shell can be analyzed separately from the vertical motion since they are independent of one another.
Step 2: The horizontal range (R) observed by the person on the shore can be derived using the relative velocities involved. The shell is subject to the horizontal speed of the ship (v_2) and the vertical speed of the shell (v_1).
Step 3: The time of flight (t) for the shell when fired vertically can be derived from the equation of motion. The total time of flight can be expressed as \( t = \frac{2v_1}{g} \) where g is the acceleration due to gravity.
Step 4: The horizontal distance covered by the shell is given by \( R = v_{horizontal} \times t \). Here, \( v_{horizontal} = v_2 \) (the speed of the ship) and we have \( t = \frac{2v_1}{g} \).
Step 5: Substituting t, we get \( R = v_2 \times \frac{2v_1}{g} \).
Step 6: This matches with Option C, which states the horizontal range as \( \frac{2v_1v_2}{g} \). Hence, the correct answer is Option C.