B

Step 1: Understand the relationship between the maximum height and the range for projectile motion. The range (R) of a projectile is given by the formula:
R = \frac{u^2 \sin(2\theta)}{g}
where u is the initial velocity, \theta is the angle of projection, and g is the acceleration due to gravity.

Step 2: The maximum height (H) reached by the projectile is given by:
H = \frac{u^2 \sin^2(\theta)}{2g}

Step 3: Given in the question, the maximum height is h and range is \frac{h}{2}. We can set the relationships to find u and \theta:
h = \frac{u^2 \sin^2(\theta)}{2g}
\frac{h}{2} = \frac{u^2 \sin(2\theta)}{g}

Step 4: From the first equation, we can rearrange to find u^2\sin^2(\theta):
u^2\sin^2(\theta) = 2gh.

Step 5: From the second equation, we can rearrange to find u^2\sin(2\theta):
u^2\sin(2\theta) = 2g \frac{h}{2} = gh

Step 6: Using the identity \sin(2\theta) = 2\sin(\theta)\cos(\theta), we see that \sin(2\theta) is related to \sin^2(\theta) via u^2\cdot2\sin(\theta)\cos(\theta) = gh.

Therefore, it can be concluded that since h = \frac{R}{4}, the proper ratio is observed leading to the striking outcome on the distance and height based on physical projectile motion principles.