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Question #43689

Difficulty: 8/10 | 🟢 Free

The Question

A stationary nucleus in free space undergoes an annihilation event, releasing a total energy $Q$ instantaneously as an isotropic burst of gamma radiation. A perfectly absorbing, rigid, flat circular disk of mass $M$ and radius $R$ is positioned such that its center is at a distance $d$ from the nucleus. The central axis of the disk passes through the nucleus. Assuming the disk is initially at rest and free to move, what is the recoil speed $v$ of the disk immediately after the radiation is completely absorbed? (Assume $v \ll c$ , where $c$ is the speed of light in vacuum, and ignore any gravitational effects).

Options

A.
$\frac{Q}{4 M c} \left( \frac{R^2}{d^2 + R^2} \right)$
B.
$\frac{Q}{2 M c} \left( 1 - \frac{d}{\sqrt{d^2 + R^2}} \right)$
C.
$\frac{Q}{2 M c} \left( \frac{R^2}{d^2 + R^2} \right)$
D.
$\frac{Q}{4 M c} \left( 1 - \frac{d}{\sqrt{d^2 + R^2}} \right)$

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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

Q-value Calculation|Energy Density Locality|Magnitude Ratio Identity

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to distinguish between scalar energy flux and vector momentum flux in radiation pressure problems.

A common trap is to calculate the total energy intercepted by the disk using the solid angle fraction $E_{hit} = Q(1 - \cos\theta_0)/2$ , and then incorrectly assume the net momentum transferred is simply $p = E_{hit}/c$ (Option B).

This flawed approach treats the momentum as a scalar and ignores the fact that photons strike the disk at varying angles.

Students must recognize that momentum is a vector quantity; thus, they must integrate the axial component of the photon momentum ( $dp \cos\theta$ ) over the disk's geometry, leading to the correct $\sin^2\theta_0$ dependence rather than a $(1 - \cos\theta_0)$ dependence.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Nuclei - Class 12
Concept	Q-value Calculation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free