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Question #42057

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is initially at rest at $t=0$ . It is subjected to a time-varying external driving force $F_{\text{ext}}(t) = m a_0 e^{-\frac{k}{m}t}$ , where $a_0$ and $k$ are positive constants. The particle simultaneously experiences a drag force $F_d = -kv$ , where $v$ is its instantaneous velocity. What is the magnitude of the maximum instantaneous power dissipated by the drag force during the motion?

Options

A.
$\frac{m^2 a_0^2}{k e^2}$
B.
$\frac{m^2 a_0^2}{k}$
C.
$\frac{m^2 a_0^2}{4 k}$
D.
$\frac{2 m^2 a_0^2}{k e}$

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

Area Under Acceleration-Time|Power-Velocity Relationship|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to couple kinematics with dynamics under time-varying and velocity-dependent forces.

The primary trap lies in treating the external acceleration independently of the drag force.

Students commonly integrate the given acceleration $a(t)$ directly to find velocity (i.e., $v = \int a dt$ ), ignoring that the drag force fundamentally alters the differential equation of motion to $dv/dt + (k/m)v = a(t)$ .

By setting the external force's decay constant equal to the drag coefficient per unit mass ( $k/m$ ), the problem requires solving a linear first-order ODE via an integrating factor, leading to a resonance-like $t e^{-kt/m}$ velocity profile before applying the $P = F \cdot v$ relationship.

Quick Facts About This Question

Question Information

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