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Question #42341

Difficulty: 8/10 | 🟢 Free

The Question

A uniform rigid rod of mass $M = 4m$ and length $L$ rests on a smooth horizontal surface. A small ball of mass $m$ moving with a velocity $v_0$ perpendicular to the rod strikes it elastically ( $e=1$ ) at a distance $d = L/4$ from its center of mass. Which of the following represents the velocity of the ball immediately after the collision? (Assume the standard sign convention where the initial velocity direction is positive).

Options

A.
$-\frac{13}{19} v_0$
B.
$-\frac{9}{19} v_0$
C.
$-\frac{9}{23} v_0$
D.
$-\frac{15}{17} v_0$

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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

Coefficient of Restitution|Parallel and Perpendicular Axes|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the coupled application of the impulse-momentum theorem and the coefficient of restitution in rigid body collisions.

A common lethal trap is calculating the angular velocity by incorrectly applying the parallel axis theorem to the moment of inertia ( $I_P = I_{cm} + Md^2$ ) when taking torque about the impact point, forgetting that the $L = I\omega$ relation about an arbitrary non-fixed point requires an additional $\vec{r} \times \vec{p}_{cm}$ term.

Another frequent error is omitting the translational velocity of the center of mass when formulating the velocity of separation for the $e=1$ constraint.

The correct approach isolates the CM frame for independent linear and angular impulse equations before superimposing them kinematically at the contact point.

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