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Question #42407

Difficulty: 8/10 | 🟢 Free

The Question

A small projectile of initial mass $m_0$ and constant charge $+q$ is launched from the origin over a horizontal ground with initial speed $v_0$ at an angle $\theta$ to the horizontal. A uniform electric field $E$ exists in the $+x$ (horizontal) direction, and uniform gravity $g$ acts in the $-y$ (vertical) direction. During its flight, the projectile ejects uncharged dust with zero relative velocity, such that its remaining mass at time $t$ is given by $m(t) = m_0 e^{-\lambda t}$ . Which of the following expressions gives the horizontal range of the projectile? (Let $T = \frac{2v_0 \sin\theta}{g}$ be the time of flight)

Options

A.
$(v_0 \cos\theta) T + \frac{qE}{m_0 \lambda^2} \left( e^{\lambda T} - 1 - \lambda T \right)$
B.
$(v_0 \cos\theta) T + \frac{qE}{m_0 \lambda^2} \left( e^{\lambda T} - 1 \right)$
C.
$(v_0 \cos\theta) T + \frac{qE}{2 m_0} T^2$
D.
$(v_0 \cos\theta) T + \frac{qE}{m_0 \lambda^2} \left( e^{\lambda T} + 1 - \lambda T \right)$

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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

Time of Flight Logic|Radioactive Decay Law Dynamics|Differential Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the application of Newton's second law for variable mass systems.

The primary trap is assuming that a decreasing mass implies a decreasing gravitational acceleration, or failing to recognize that the constant horizontal force leads to an exponentially increasing horizontal acceleration.

Students often incorrectly use constant acceleration kinematics (Option C) by treating mass as constant, or make integration errors when integrating the velocity function $e^{\lambda t} - 1$ by forgetting the $-t$ term (Option B).

It elegantly couples standard vertical time-of-flight kinematics with non-standard, calculus-based horizontal kinematics driven by the radioactive decay law.

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