Question

Which of the velocity-time \( (v-t) \) graph(s) can possibly represent one-dimensional motion of a particle?

• A. Fig A
• B. Fig B
• C. Fig C
• D. Fig D

Hint:

To quickly test if any coordinate graph (like \( v-t \), \( s-t \), or \( a-t \)) is physically possible, draw a vertical line representing a single instant of time. If the line crosses the curve more than once, the graph is physically impossible because a particle cannot be in two states or positions at the exact same instant.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks us to determine which of the given velocity-time \( (v-t) \) graphs can represent a physically possible one-dimensional motion of a particle.

Step 2: Key Formula or Approach:
For any real, physical one-dimensional motion:
- Time \( t \) is a continuously and monotonically increasing variable (it only moves forward).
- At any unique instant of time \( t \), a particle can have only one unique value of velocity. Thus, velocity must be a single-valued function of time.
- Graphically, this means the curve must pass the "vertical line test" (any vertical line drawn parallel to the velocity axis must intersect the curve at most once).

Step 3: Detailed Explanation:
Let's analyze each graph:
- Graph (A): Every vertical line intersects the curve at exactly one point. Velocity is a single-valued function of time, which is physically possible. Thus, (A) is correct.
- Graph (B): The curve loops back on itself. A vertical line can intersect the curve at more than one point, meaning the particle would possess multiple different velocities at a single instant of time. This is physically impossible. Thus, (B) is incorrect.
- Graph (C): Similarly, the curve is S-shaped/looped, failing the vertical line test. It indicates multiple velocities at the same instant of time, which is physically impossible. Thus, (C) is incorrect.
- Graph (D): Every vertical line intersects the curve at exactly one point. Velocity is single-valued and decreases smoothly over time, which is physically possible. Thus, (D) is correct.

Step 4: Final Answer:
The graphs that can represent physical one-dimensional motion are (A) and (D).