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Question #61292

Difficulty: 8/10 | 🟢 Free

The Question

A particle moves in the $xy$ -plane such that its $x$ -coordinate increases at a constant rate $u > 0$ . Its geometric trajectory $y=f(x)$ is determined by the differential equation $x \frac{dy}{dx} - y = x^2 \left[ \frac{x}{a} \right]$ for $x \ge a$ , with the initial condition $y(a) = 0$ , where $a > 0$ is a constant and $[ \cdot ]$ denotes the greatest integer function. What is the $y$ -component of the particle's velocity exactly at the instant when its $x$ -coordinate reaches $2a$ ?

Options

A.
3au
B.
5au
C.
4au
D.
\text{Does not exist}

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Physical Interpretation|Geometric Interpretation|Greatest Integer Function Traps

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This problem tests the physical interpretation of velocity derived from a geometric trajectory containing a Greatest Integer Function (GIF).

Students often mechanically substitute $x=2a$ into the differential equation, evaluating $[2a/a] = 2$ to incorrectly find $v_y = 5au$ , or assume the left-hand limit to get $3au$ .

The trap lies in failing to recognize that while the position function $y(x)$ is continuous (as it is an integral of a step function), it has a sharp corner at $x=2a$ .

Consequently, the spatial derivative $dy/dx$ has a jump discontinuity, meaning the physical velocity vector is undefined at that exact instant.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Limits and Derivatives - Class 11
Concept	Physical Interpretation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free