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Question #42040

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ is released from rest from a height $h$ above a horizontal platform. Upon impact, the platform exerts an upward time-varying impulsive force $F(t) = k t^2$ on the particle, where $t$ is the time elapsed since the instant of first contact and $k$ is a positive constant. The particle remains in contact with the platform until the platform returns to its initial equilibrium position (i.e., the net displacement of the particle during the contact interval becomes zero). Assuming the effect of gravity during the extremely short contact time is negligible, what is the maximum height $H$ reached by the particle after it rebounds?

Options

A.
$h$
B.
$3h$
C.
$4h$
D.
$9h$

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

Kinematics of Vertical Free Fall|Impulse-Momentum Theorem|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the transition from kinematics to the integral form of Newton's Second Law for a time-dependent impulsive force.

The primary trap is the assumption that a collision with a platform must be energy-conserving or dissipative, leading students to intuitively guess $H \le h$ .

Furthermore, students often confuse time-dependent forces $F(t)$ with position-dependent forces $F(x)$ , incorrectly attempting to apply the Work-Energy theorem directly without evaluating the kinematic integral $x(t)$ .

By forcing the boundary condition $x(\tau) = 0$ , the problem requires solving the differential equation of motion to reveal that the active platform does net positive work on the particle, resulting in an effective coefficient of restitution $e=3$ .

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