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Question #42667

Difficulty: 8/10 | 🟢 Free

The Question

A cart of initial mass $m_0 = 2 \text{ kg}$ is moving with a velocity $v_0 = 4 \text{ m/s}$ along a smooth horizontal surface. At $x = 0$ , it enters a region where it experiences a conservative restoring force $F = -kx$ , where $k = 4 \text{ N/m}$ . The region is filled with a stationary dust cloud of uniform linear mass density $\lambda = 1.5 \text{ kg/m}$ . As the cart moves, it sweeps up all the dust it encounters, which sticks to the cart. What is the maximum distance $x_{max}$ the cart travels into the dust cloud before coming to rest momentarily?

Options

A.
$2.0 \text{ m}$
B.
$2\sqrt{2} \text{ m}$
C.
$4.0 \text{ m}$
D.
$2.5 \text{ m}$

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

Equipotential Surface Geometry|Conservative Field Path Independence|Variable Mass Systems|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the misapplication of the Work-Energy Theorem in variable mass systems.

Students often incorrectly assume that the work done by a conservative force equals the change in kinetic energy ( $\Delta K = -\Delta U$ ), leading to the trap answer $2\sqrt{2}$ m by equating initial kinetic energy to spring potential energy.

However, the accretion of stationary dust represents a continuous perfectly inelastic collision.

Mechanical energy is not conserved; it is continuously dissipated as heat during the mass-sweeping process.

By forcing the use of momentum-based equations of motion ( $\frac{dp}{dt} = F_{ext}$ ), the problem discriminates between students who memorize static energy formulas and those who understand the fundamental momentum flux and internal work in variable mass dynamics.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Electrostatic Potential and Capacitance - Class 12
Concept	Equipotential Surface Geometry
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free