Q. A tank of height $H$ is fully filled with water. If the water rushing from a hole made in the tank below the free surface, strikes the floor at maximum horizontal distance, then the depth of the hole from the free surface must be
A) $\left( \frac{3}{4} \right)H$
B) $\left( \frac{2}{3} \right)H$
C) $\left( \frac{1}{4} \right)H$
D) $\left( \frac{1}{2} \right)H$
Explanation
(d) As hole is made in the tank below the free surface of water, so water rushing from this hole follows a parabolic path.
The velocity of efflux liquid,
$v = \sqrt{2gh}$
$Time\ t = \sqrt{\frac{2(H - h)}{g}}$
Horizontal range, $R = vt$
$\ R = \left( 2gh \times \frac{2(H - h)}{g} \right)^{\frac{1}{2}}$
$ie,\ \ R^{2} = 4h(H - h) = 4\left( Hh - h^{2} \right)$
The range is maximum if
$\frac{dR}{dh} = 0$
$or\ \ \frac{2RdR}{dh} = 4(H - 2h)$
$or\ \ \ \ 0 = (H - 2h)$
$or\ \ \ \ h = \frac{H}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.