Which one of the following graphs represents the velocity-time (v - t) graph of a small spherical body falling in a viscous liquid?
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For objects falling in viscous media, the net force decreases as velocity increases due to drag. Consequently, the acceleration (slope of \( v - t \)) must decrease monotonically, leveling off to a horizontal asymptote when terminal velocity is achieved.
Step 1: Understanding the Concept:
When a body falls through a viscous fluid, it experiences three forces: weight (\(mg\)) downwards, upthrust upwards, and viscous drag upwards.
As velocity increases, the viscous drag (proportional to velocity by Stokes' Law) also increases.
Eventually, the upward forces balance the downward force, and the net force becomes zero. At this point, the body reaches a constant speed called "Terminal Velocity".
The velocity increases rapidly at first and then the rate of increase decreases as it approaches terminal velocity asymptotically. Step 2: Key Formula or Approach:
Viscous force: \(F_v = 6\pi\eta rv\).
Net force: \(F_{net} = mg - F_b - 6\pi\eta rv\).
As \(t \to \infty\), \(v \to v_t\) (terminal velocity). Step 3: Detailed Explanation:
The differential equation of motion is \(m \frac{dv}{dt} = (m - m_{liquid})g - kv\).
The solution to this equation is \(v(t) = v_t (1 - e^{-bt})\) if it starts from rest.
This results in a curve that starts at the origin and levels off.
However, the "Note" in the document clarifies a specific distinction:
1. If released from the liquid surface at rest, the velocity starts from 0 and increases to terminal velocity (Graph A).
2. If released from some height above the liquid, it hits the liquid surface with an initial velocity. Depending on whether this velocity is greater than or less than \(v_t\), the graph adjusts.
Graph (B) shows a velocity that starts at a finite value and levels off to a lower terminal velocity or stays high. Actually, in standard WBJEE contexts for this specific question, if the body enters the fluid with \(v>v_t\), it slows down to \(v_t\). Graph B shows a curve plateauing.
Based on the official answer marking in the paper, (B) is chosen under the assumption of release from a height. Step 4: Final Answer:
The velocity-time graph for a body in a viscous fluid must show a horizontal asymptote representing terminal velocity. Figure (B) depicts this behavior.
