Velocity-Time Graphs for One-Dimensional Motion
For one-dimensional motion, the velocity ($v$) of a particle must be a unique function of time ($t$). This means that for any specific instant in time, there can only be one corresponding velocity value. Graphically, this implies that any vertical line drawn on a velocity-time ($v-t$) graph must intersect the curve at most once.
Analysis of Graphs
- Graph 1 (Image A): This graph depicts velocity as a function of time. For every time value shown, there is only one velocity value. This is physically possible for one-dimensional motion.
- Graph 2 (Image B): This graph shows multiple velocity values for a single time value (at the peak of the curve). This violates the definition of a function and cannot represent the motion of a single particle.
- Graph 3 (Image C): Similar to Graph 2, this graph is not a valid function, as it shows multiple velocity values for certain time instances, particularly where the velocity appears to change direction abruptly.
- Graph 4 (Empty): This option is represented by an empty tag. If this represents a particle at rest ($v=0$ for all $t$), it is a valid case of one-dimensional motion.
Conclusion
A valid $v-t$ graph must represent velocity as a unique function of time. Graph 1 satisfies this condition. Assuming the empty Graph 4 represents a particle at rest ($v=0$), it also represents valid one-dimensional motion. Graphs 2 and 3 are invalid because they do not represent velocity as a unique function of time.
Therefore, Graph 1 and Graph 4 can possibly represent one-dimensional motion.
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