Before You Begin

Question #42340

Difficulty: 8/10 | 🟢 Free

The Question

A small incompressible solid sphere of mass $m$ and density $\rho$ is released from rest just below the free surface of a pool of liquid with density $\sigma$ ( $\sigma < \rho$ ) and depth $H$ . The atmospheric pressure at the free surface is $P_0$ . The liquid exerts a drag force of magnitude $F_d = k v^2$ on the sphere, where $v$ is its speed. The sphere reaches a terminal speed $v_0$ well before striking the horizontal bottom. The collision with the bottom has a coefficient of restitution $e$ . Assuming the collision time is infinitesimal, what is the maximum height $h$ the sphere rises above the bottom after the first bounce?

Options

A.
$\frac{v_0^2}{2g(1 - \sigma/\rho)} \ln(1 + e^2)$
B.
$\frac{e^2 v_0^2}{2g(1 - \sigma/\rho)}$
C.
$\frac{v_0^2}{2g(1 - \sigma/\rho)} \ln\left(1 + e^2 + \frac{P_0}{\rho g H}\right)$
D.
$\frac{v_0^2}{2g(1 - \sigma/\rho)} (1 - e^{-e^2})$

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

Coefficient of Restitution|Atmospheric Pressure Reference|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the decoupling of impulsive forces from non-conservative continuous forces (fluid drag) and the correct application of the Work-Energy theorem.

A common trap is incorrectly incorporating the atmospheric pressure $P_0$ into the work done, failing to realize that for an incompressible object, the uniform pressure field does no net work over the displacement.

Furthermore, students often erroneously apply energy conservation without integrating the quadratic drag force during the upward journey, leading to the distractor $h = e^2 v_0^2 / (2g_{\text{eff}})$ .

The problem demands rigorous separation of the terminal velocity phase, the impulsive collision phase, and the retarded upward transit.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Coefficient of Restitution
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free