Which of the velocity-time \( (v-t) \) graph(s) can possibly represent one-dimensional motion of a particle?
Show 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.
Step 1: Understanding the Concept:
For a velocity-time graph to represent the physical motion of a particle, it must satisfy certain physical constraints.
1. Time is a monotonically increasing independent variable; it cannot go backward.
2. A single particle cannot have two different velocities at the same instant of time (it would require an impossible jump in position or state).
3. The graph must be a "function" where for every value of \(t\), there is exactly one value of \(v\). Step 2: Key Formula or Approach:
Vertical Line Test: Draw a vertical line at any time \(t\) on the axis. If it intersects the graph at more than one point, the graph does not represent a physical motion. Step 3: Detailed Explanation:
Looking at the four provided graphs:
Graph (A): A vertical line intersects the curve at only one point for every value of \(t\). Velocity is well-defined and continuous. Possible.
Graph (B): The graph is a closed loop (ellipse/circle). A vertical line drawn through the middle of the loop intersects the curve at two different points. This implies the particle has two different velocities at the same time, which is impossible.
Graph (C): The graph has a "C" shape or similar overlap. Again, for a range of \(t\), there are two corresponding velocities. Impossible.
Graph (D): The graph represents a decaying function. For every \(t\), there is only one value of \(v\). Velocity is well-defined. Possible.
Therefore, only graphs (A) and (D) are physically valid for one-dimensional motion. Step 4: Final Answer:
By applying the single-valued function constraint, graphs (A) and (D) are identified as valid physical representations, while (B) and (C) are discarded due to time-domain overlap.
