Question

A particle of mass \(m\) falls from rest through a resistive medium having resistive force \(F=-kv\), where \(v\) is the velocity of the particle and \(k\) is a constant. Which of the following graphs represents velocity \(v\) versus time \(t\)? 

• A. Graph 1
• B. Graph 2
• C. Graph 3
• D. Graph 4

Hint:

Whenever resistive force is proportional to velocity, the speed approaches terminal velocity exponentially, not linearly.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
A particle falling in a resistive medium reaches a terminal velocity when the downward force of gravity equals the upward resistive force.
Step 2: Key Formula or Approach:
Equation of motion: \(m \frac{dv}{dt} = mg - kv\).
Solving the differential equation with \(v(0) = 0\): \(v(t) = \frac{mg}{k} (1 - e^{-\frac{k}{m}t})\).
Step 3: Detailed Explanation:
The resulting expression \(v(t) = v_{terminal} (1 - e^{-t/\tau})\) represents an exponential growth function.
Initially, at \(t = 0\), the velocity \(v = 0\).
As time increases, the acceleration decreases as the resistive force increases.
As \(t \to \infty\), the velocity asymptotically approaches a constant value \(v_{terminal} = \frac{mg}{k}\).
Graph 1 shows this characteristic exponential rise starting from zero and flattening out.
Step 4: Final Answer:
Graph 1 correctly represents the velocity-time relationship.