The velocity ($v$)-time ($t$) plot of the motion of a body is shown below: The acceleration ($a$)-time ($t$) g

Question

The velocity ($v$)-time ($t$) plot of the motion of a body is shown below: The acceleration ($a$)-time ($t$) graph that best suits this motion is:

Accepted Answer

1.  **Principle:** The acceleration of a particle is defined as the rate of change of velocity with respect to time ($a = \frac{dv}{dt}$). Geometrically, this corresponds to the **slope** of the velocity-time ($v-t$) graph .
2.  **Analysis:** To determine the correct acceleration-time ($a-t$) graph from a given $v-t$ graph, one must analyze the slope of the $v-t$ curve at various intervals:
    *   **Positive Slope:** If the $v-t$ line goes upwards (velocity increases), the acceleration is positive.
    *   **Negative Slope:** If the $v-t$ line goes downwards (velocity decreases), the acceleration is negative.
    *   **Zero Slope:** If the $v-t$ line is horizontal (constant velocity), the acceleration is zero.
    *   **Constant Slope:** If the $v-t$ graph consists of straight lines, the acceleration is constant during those intervals (represented by horizontal lines on the $a-t$ graph).
3.  **Conclusion:** The correct $a-t$ graph will depict the values of these slopes corresponding to the time intervals in the $v-t$ plot.

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