Motion graphs and derivatives

In mechanics, the derivative of the position vs. time graph of an object is equal to the velocity of the object. In the International System of Units, the position of the moving object is measured in meters relative to the origin, while the time is measured in seconds. Placing position on the y-axis and time on the x-axis, the slope of the curve is given by:

Here ${\displaystyle s}$ is the position of the object, and ${\displaystyle t}$ is the time. Therefore, the slope of the curve gives the change in position (in metres) divided by the change in time (in seconds), which is the definition of the average velocity (in meters per second ${\displaystyle ({\begin{matrix}{\frac {m}{s}}\end{matrix}})}$) for that interval of time on the graph. If this interval is made to be infinitesimally small, such that ${\displaystyle {\Delta s}}$ becomes ${\displaystyle {ds}}$ and ${\displaystyle {\Delta t}}$ becomes ${\displaystyle {dt}}$, the result is the instantaneous velocity at time ${\displaystyle t}$, or the derivative of the position with respect to time.

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