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Question #42033

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is driven along a straight horizontal axis by an engine that delivers a constant power $P$ . The particle experiences a resistive drag force of magnitude $k v^2$ , where $v$ is the instantaneous velocity and $k$ is a positive constant. If the particle starts from rest at the origin, what is the distance $x$ covered by the particle when it attains a velocity $v_0$ (where $v_0 < (P/k)^{1/3}$ )?

Options

A.
$\frac{m}{3k} \ln\left( \frac{P}{P - k v_0^3} \right)$
B.
$\frac{m}{2k} \ln\left( \frac{P}{P - k v_0^2} \right)$
C.
$\frac{m}{3k} \ln\left( \frac{P + k v_0^3}{P} \right)$
D.
$\frac{m}{k} \ln\left( \frac{P}{P - k v_0^3} \right)$

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

The Velocity-Position Identity|Power-Velocity Relationship|Differential Equations

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from a temporal dynamic framework to a spatial quadrature using the identity $a = v(dv/dx)$ .

A common trap is attempting to integrate $F = m(dv/dt)$ directly, which leads to a complex time-domain expression $dt = m v / (P - k v^3) dv$ that causes 'integration paralysis' since finding $x(t)$ from $v(t)$ becomes analytically intractable.

By recognizing that the question asks for distance $x$ as a function of velocity $v$ , the student must bypass time entirely.

Furthermore, it traps students who incorrectly treat power $P$ as a constant force rather than applying $F = P/v$ .

Quick Facts About This Question

Question Information

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