Question

A particle is falling under gravity. Air resistance on particle is F = \(-\)kv. Find correct option :

• A.
• B.
• C.
• D.

Hint:

In resistive media, look for the "saturation" or "leveling off" effect in graphs. Acceleration decreases over time, so the slope of the v-t graph must decrease.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
When an object falls under gravity with air resistance proportional to velocity, it initially accelerates.
As velocity increases, the resistive force increases until it balances gravity. This velocity is called terminal velocity.
Step 2: Key Formula or Approach:
Newton's second law: \(m \frac{dv}{dt} = mg - kv\).
This leads to the velocity equation: \(v(t) = \frac{mg}{k}(1 - e^{-kt/m})\).
Step 3: Detailed Explanation:
The velocity starts from zero (\(t=0, v=0\)).
As \(t\) increases, the term \(e^{-kt/m}\) decreases toward zero.
The velocity \(v\) increases but with a decreasing rate (decreasing slope).
Eventually, the velocity approaches a constant value \(v_{terminal} = \frac{mg}{k}\) as \(t \rightarrow \infty\).
The graph representing this behavior is an exponential curve that levels off horizontally at the terminal velocity. Graph (2) represents this correctly.
Step 4: Final Answer:
Option (2) is the correct graph.