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Question #42072

Difficulty: 8/10 | 🟢 Free

The Question

A vessel is in the shape of a paraboloid of revolution, such that its cross-sectional area at a height $y$ from its bottom is given by $A(y) = k y$ , where $k$ is a positive constant. The vessel is completely filled with an ideal, non-viscous liquid up to a height $H$ . It is placed on a horizontal table of height $h$ from the ground. A small hole of cross-sectional area $a$ ( $a \ll kH$ ) is opened at the bottom of the vessel, and the liquid shoots out horizontally. What is the time-averaged horizontal range of the liquid on the ground until the vessel completely empties?

Options

A.
$\frac{3}{2} \sqrt{hH}$
B.
$\frac{4}{3} \sqrt{hH}$
C.
$\frac{5}{4} \sqrt{hH}$
D.
$\sqrt{hH}$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Projectile Path Equation|Torricelli’s Efflux Velocity|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the ability to couple fluid dynamics (Torricelli's law and continuity equation) with kinematics (projectile motion) and perform a time-average over a variable that is geometrically constrained.

A common trap is averaging the range over the height $y$ instead of time $t$ , which yields the incorrect result $\frac{4}{3}\sqrt{hH}$ .

Students must recognize that the time differential $dt$ is non-linearly related to the height differential $dy$ due to the varying cross-sectional area $A(y) = ky$ and the varying efflux velocity $v \propto \sqrt{y}$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Projectile Path Equation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free