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Question #42056

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $M$ is projected vertically upward from the ground with an initial velocity $v_0$ . The block is attached to one end of a uniform heavy flexible string of linear mass density $\lambda$ . The string is initially coiled loosely on the ground. Assuming the string is sufficiently long and unfolds smoothly without tangling, what is the maximum height $h$ reached by the block? (Let $g$ be the acceleration due to gravity).

Options

A.
$\frac{1}{\lambda} \left[ \left( M^3 + \frac{3\lambda M^2 v_0^2}{2g} \right)^{\frac{1}{3}} - M \right]$
B.
$\frac{1}{\lambda} \left[ \left( M^2 + \frac{\lambda M v_0^2}{g} \right)^{\frac{1}{2}} - M \right]$
C.
$\frac{1}{\lambda} \left[ \left( M^3 + \frac{3\lambda M^2 v_0^2}{g} \right)^{\frac{1}{3}} - M \right]$
D.
$\frac{v_0^2}{2g}$

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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

Turning Point Identification|Tension in Massless vs Massive Strings|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the ability to model a continuous inelastic collision within a variable mass system and identify the correct position-dependent restoring force gradient.

Students commonly fall into the 'Lethal Collision' trap by incorrectly applying the principle of conservation of mechanical energy (Option B), assuming the work done by gravity is the only energy sink.

However, as the massive string is lifted, stationary links are abruptly jerked into motion, resulting in continuous kinetic energy loss.

By forcing the use of the momentum equation for variable mass, the student must integrate the position-dependent tension to find the true turning point, invalidating the constant tension or energy conservation shortcuts.

Quick Facts About This Question

Question Information

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