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Question #42034

Difficulty: 8/10 | 🟢 Free

The Question

An alpha particle of mass $m$ is fired directly towards a heavy stationary nucleus. The repulsive Coulomb force on the alpha particle is given by $F = \frac{k}{x^2}$ , where $x$ is the distance from the nucleus and $k$ is a positive constant. The particle passes the point $x = 3d$ with a velocity $v_0$ directed towards the nucleus. Assuming $v_0 \ll c$ (where $c$ is the speed of light), what is the square of its velocity, $v^2$ , when it reaches the point $x = d$ ?

Options

A.
$v_0^2 - \frac{4k}{3md}$
B.
$v_0^2 - \frac{4k}{9md}$
C.
$v_0^2 - \frac{4k}{md}$
D.
$\left(v_0^2 - \frac{4k}{3md}\right) \left(1 - \frac{v_0^2}{c^2}\right)^{-1/2}$

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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

The Velocity-Position Identity|Coulombic Central Force|Calculus (Integration)

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to transition from constant acceleration kinematics to the correct application of the Work-Energy theorem or direct integration of Newton's Second Law for a position-dependent force.

The primary 'lethal' trap is the kinematic habit of assuming constant acceleration; evaluating acceleration at the initial point ( $x=3d$ ) and using $v^2 = u^2 + 2a\Delta x$ leads directly to the incorrect option B.

Furthermore, the inclusion of a relativistic factor in option D serves as a secondary distractor, preying on mathematically over-zealous students who might unnecessarily invoke Special Relativity despite the explicit non-relativistic constraint ( $v_0 \ll c$ ).

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	The Velocity-Position Identity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free