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Question #42043

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The Question

A spherical radioactive particle of constant radius $R$ and initial mass $m_0$ falls vertically from rest in a viscous fluid. The mass of the particle decreases according to the decay law $m(t) = m_0 e^{-\lambda t}$ . Assume the decay process ejects the product particles such that they are left completely stationary in the lab frame immediately upon emission. The fluid exerts a drag force given by Stokes' Law, $F_d = 6\pi\eta R v$ , where $\eta$ is the fluid viscosity and $v$ is the particle's velocity. Let $\beta = \frac{6\pi\eta R}{m_0}$ . Which of the following differential equations correctly governs the velocity $v(t)$ of the particle?

Options

A.
$\frac{dv}{dt} = g - (\beta e^{\lambda t} - \lambda) v$
B.
$\frac{dv}{dt} = g - \beta e^{\lambda t} v$
C.
$\frac{dv}{dt} = g - (\beta e^{\lambda t} + \lambda) v$
D.
$\frac{dv}{dt} = g - (\beta e^{-\lambda t} - \lambda) v$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Kinematics of Vertical Free Fall|Statistical Nature of Decay|Approximation

Why This Question Is Tricky

Common Mistakes

This question tests the correct application of Newton's Second Law for variable mass systems.

A common misconception among students is to blindly apply F = ma (yielding Option B) without accounting for the momentum carried away by the ejected mass.

By specifying that the decay products are left stationary in the lab frame, the relative velocity of the ejected mass is v_rel = -v.

This introduces a thrust term +v|dm/dt| into the equation of motion, making the governing equation equivalent to F_ext = d(mv)/dt.

The problem elegantly combines radioactive decay kinetics with fluid kinematics, requiring students to correctly couple the time-dependent mass m(t) = m_0 e^(-λt) and its derivative into the variable mass differential equation.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Kinematics of Vertical Free Fall
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free