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Question #42066

Difficulty: 8/10 | 🟢 Free

The Question

Two particles, $A$ and $B$ , are projected simultaneously in a uniform gravitational field $\vec{g} = -g \hat{j}$ . Particle $A$ is projected from the origin $(0,0)$ with initial velocity $\vec{u}_A = u_0 \hat{i} + u_0 \hat{j}$ . Particle $B$ is projected from the point $(d, 0)$ with initial velocity $\vec{u}_B = 2u_0 \hat{j}$ . Both particles experience an air drag force $\vec{F} = -m \gamma \vec{v}$ , where $m$ is the mass of the particle, $\vec{v}$ is its velocity, and $\gamma$ is a positive constant. If $d = \frac{3u_0}{\gamma}$ , what is the distance of closest approach between the two particles?

Options

A.
$\sqrt{5} \frac{u_0}{\gamma}$
B.
$\frac{3}{\sqrt{2}} \frac{u_0}{\gamma}$
C.
$\sqrt{10} \frac{u_0}{\gamma}$
D.
$\frac{5}{2} \frac{u_0}{\gamma}$

Check Your Answer

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

Relative Velocity in 2D|Collision Impulses|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the understanding of relative motion under velocity-dependent non-conservative forces.

The primary trap is assuming that the relative trajectory, which is a straight line, extends infinitely as it would in standard projectile motion.

Students often calculate the perpendicular distance from the origin to the line of relative motion (yielding $\frac{3}{\sqrt{2}} \frac{u_0}{\gamma}$ ), failing to realize that the drag force exponentially decays the relative velocity.

This imposes a strict upper bound on the maximum relative displacement.

Consequently, the geometric closest point on the line is never reached, and the true distance of closest approach is the asymptotic limit of the trajectory.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Plane - Class 11 Part 1
Concept	Relative Velocity in 2D
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free