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Question #42406

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and positive charge $q$ is projected from the origin at $t=0$ with an initial speed $u$ at an angle $\theta$ to the horizontal. The particle moves in a region where a uniform downward gravitational field $g$ and a time-varying vertical electric field $\vec{E}(t) = E_0 \sin(\omega t) \hat{j}$ are present. The $y$ -axis is taken vertically upwards. Given that $qE_0 < mg$ , which of the following transcendental equations correctly determines the time of flight $T$ of the particle?

Options

A.
$\frac{1}{2}gT^2 - \left(u \sin\theta + \frac{qE_0}{m\omega}\right)T + \frac{qE_0}{m\omega^2}\sin(\omega T) = 0$
B.
$\frac{1}{2}gT^2 - (u \sin\theta) T + \frac{qE_0}{m\omega^2}\sin(\omega T) = 0$
C.
$\frac{1}{2}gT^2 - \left(u \sin\theta - \frac{qE_0}{m\omega}\right)T - \frac{qE_0}{m\omega^2}\sin(\omega T) = 0$
D.
$\frac{1}{2}gT^2 - (u \sin\theta) T - \frac{qE_0}{m\omega^2}(1 - \cos(\omega T)) = 0$

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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

Time of Flight Logic|Trajectory of Charged Particle|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to analyze kinematics under a time-varying force, specifically invalidating the standard projectile motion formulas.

The primary trap lies in the integration of the time-varying acceleration $a_y(t)$ .

Students frequently forget to account for the lower limit of integration when evaluating $\int_0^t \sin(\omega t') dt'$ , missing the constant term $\frac{qE_0}{m\omega}$ in the velocity expression.

This omission leads to an incorrect linear term in the final displacement equation (Option B).

The problem enforces rigorous application of calculus in physics, separating students who rely on memorized range formulas from those who build equations from first principles.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Time of Flight Logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free