The velocity time graph of a particle starting from the...

The velocity time graph of a particle starting from the origin at time $\mathrm{t}=0$ and moving along a straight line is as shown. If $\mathrm{A}_{1}, \mathrm{~A}_{2}$ and $\mathrm{A}_{3}$ are the shaded areas and $\mathrm{A}_{2}>2 \mathrm{~A}_{1}$ and $\mathrm{A}_{3}<\mathrm{A}_{1}$, the number of times the particle crosses the origin is (a) 0 (b) 1 (c) 2 (d) 3

Key Concepts

Velocity-Time Graph

A velocity-time graph represents the velocity of an object as it changes over time. The graph’s vertical axis shows velocity and its horizontal axis shows time. This type of graph is essential for understanding motion in kinematics, as it visually demonstrates acceleration, deceleration, and constant velocity phases.

Displacement from Areas Under the Curve

The displacement of an object during a time interval is given by the area under its velocity-time graph over that interval. Areas above the time-axis indicate motion in one direction, while areas below indicate motion in the opposite direction. Calculating the net displacement involves adding positive areas and subtracting negative areas, which is crucial for determining whether and how the object returns to its starting point.

Zero Crossing (Origin Crossing) in Motion

Crossing the origin in the context of motion means that the object's net displacement becomes zero at some point, indicating that it has returned to its starting position. In velocity-time analysis, this is deduced when the cumulative areas (displacements) sum to zero, implying a sign change in the total displacement.

Interpreting Comparative Areas

Comparing different shaded regions (areas) in a velocity-time graph helps in understanding the relative magnitudes of displacement during different intervals. When areas representing displacement in reverse directions have defined relationships, it can be inferred whether the moving object has changed direction and crossed its starting point. Such comparisons are used to establish criteria for counting the number of times an object returns to the origin.

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