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Question #42073

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m$ and positive charge $q$ is projected from the origin $(0,0)$ in the $xy$ -plane with an initial speed $u = \sqrt{2gL}$ at an angle $\theta = 45^\circ$ to the positive $x$ -axis. The $y$ -axis points vertically upwards. In addition to uniform gravity $g$ acting downwards, there exists an electric field $\vec{E} = E_0 \left( \frac{x}{L} \hat{i} - \frac{y}{L} \hat{j} \right)$ in the region. If the charge-to-mass ratio of the particle is $\frac{q}{m} = \frac{g}{E_0}$ , what is the $x$ -coordinate of the particle when it first returns to the $x$ -axis ( $y=0$ )?

Options

A.
$L \sinh\left(\frac{\pi}{2}\right)$
B.
$L \cosh\left(\frac{\pi}{2}\right)$
C.
$L \left( e^{\pi/2} - 1 \right)$
D.
$L \sin\left(\frac{\pi}{2}\right)$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

Projectile Path Equation|Potential vs Field Relationship|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to abandon standard kinematic equations (which assume constant acceleration) when faced with a space-dependent force field.

The primary trap is assuming the 'effective gravity' is a constant vector, which would incorrectly lead to a parabolic trajectory.

By providing a non-uniform but conservative electric field $\vec{E}(x,y)$ , the problem forces the formulation of decoupled linear differential equations ( $m\ddot{x} \propto x$ and $m\ddot{y} \propto -y$ ).

Students who attempt to use standard projectile range formulas will fail, as the acceleration is a function of position, requiring exact solutions to simple harmonic and hyperbolic differential equations.

Quick Facts About This Question

Question Information

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