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Question #42046

Difficulty: 8/10 | 🟢 Free

The Question

An incompressible, inviscid fluid of density $\rho$ flows steadily through a horizontal pipe of varying cross-section. The fluid velocity is $v_0$ at $x=0$ and $2v_0$ at $x=L$ . A small particle of density $3\rho$ is released from rest at $x=0$ . Neglecting viscous drag and added mass effects, and assuming the only horizontal force on the particle is due to the fluid's macroscopic pressure gradient, what is the velocity of the particle at $x=L$ ?

Options

A.
$v_0$
B.
$\sqrt{3}v_0$
C.
$\frac{v_0}{\sqrt{3}}$
D.
$2v_0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Relative Velocity in 1D|Bernoulli’s Energy Balance

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of the physical origin of forces in a fluid and the coupling of spatial derivatives in kinematics.

A common trap is assuming the particle simply adopts the fluid's velocity or acceleration, ignoring its own inertia.

Students often fail to realize that the pressure gradient, derived from Bernoulli's equation ( $\nabla P = -\rho v \nabla v$ ), acts as a buoyant-like driving force.

By setting up the non-linear differential equation $m v_p (dv_p/dx) = -V (dP/dx)$ , the problem elegantly links fluid dynamics with particle kinematics, penalizing those who blindly apply $v = u + at$ or assume constant acceleration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Relative Velocity in 1D
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free