Velocity
Relationship to acceleration

Although velocity is defined as the rate of change of position, it is often common to start with an expression for an object's acceleration. As seen by the three green tangent lines in the figure, an object's instantaneous acceleration at a point in time is the slope of the line tangent to the curve of a v (t) graph at that point. In other words, instantaneous acceleration is defined as the derivative of velocity with respect to time: a = d v d t. {\displaystyle {\boldsymbol {a}}={\frac {d{\boldsymbol {v}}}{dt}}.}

From there, velocity is expressed as the area under an a (t) acceleration vs. time graph. As above, this is done using the concept of the integral:

v = ∫ a d t. {\displaystyle {\boldsymbol {v}}=\int {\boldsymbol {a}}\ dt.}
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