Before You Begin

Question #42083

Difficulty: 8/10 | 🟢 Free

The Question

A large closed tank contains a liquid of density $\rho$ . The liquid level is at a height $H$ above a small hole in the side of the tank. The gas above the liquid is maintained at a constant gauge pressure $P_0$ . A spherical droplet of radius $r$ and density $\rho$ emerges horizontally from the hole into the surrounding air. The droplet experiences a viscous drag force $\vec{F}_d = -6\pi\eta r \vec{v}$ , where $\eta$ is the viscosity of air and $\vec{v}$ is the droplet's velocity vector. Neglecting the buoyant force from the air, what is the horizontal displacement $x$ of the droplet at the exact instant its downward vertical velocity reaches $v_y$ ? (Assume acceleration due to gravity is $g$ ).

Options

A.
$\frac{v_y}{g} \sqrt{2gH + \frac{2P_0}{\rho}}$
B.
$\frac{2r^2\rho}{9\eta} \sqrt{2gH + \frac{2P_0}{\rho}}$
C.
$\frac{v_y}{g} \sqrt{2gH}$
D.
$\frac{2r^2\rho}{9\eta} v_y$

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 on Inclined Plane|Torricelli’s Efflux Velocity|Viscosity and Stokes' Law

Why This Question Is Tricky

Common Mistakes

This question tests the transition from fluid dynamics (Torricelli's law with gauge pressure) to kinematics under non-conservative, velocity-dependent forces (Stokes' drag).

The primary trap lies in the assumption that the presence of a complex drag force ( $F = -6\pi\eta r v$ ) must explicitly manifest in the final kinematic relationship between $x$ and $v_y$ .

Students often get bogged down attempting to solve the coupled non-linear differential equations or incorrectly apply conservation of energy.

By isolating the time-dependent exponential decay factor in both orthogonal directions, the drag coefficient and mass elegantly cancel out, revealing a purely kinematic geometric constraint independent of the fluid's viscosity or the droplet's radius.

Quick Facts About This Question

Question Information

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