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Instantaneous velocity

The instantaneous velocity of an object is the limit average velocity as the time interval approaches zero. At any particular time t, it can be calculated as the derivative of the position with respect to time: v = lim Δ t → 0 Δ s Δ t = d s d t. {\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {s}}}{\Delta t}}={\frac {d{\boldsymbol {s}}}{dt}}.}

From this derivative equation, in the one-dimensional case it can be seen that the area under a velocity vs. time (v vs. t graph) is the displacement, s. In calculus terms, the integral of the velocity function v (t) is the displacement function s (t). In the figure, this corresponds to the yellow area under the curve. s = ∫ v d t. {\displaystyle {\boldsymbol {s}}=\int {\boldsymbol {v}}\ dt.}

Although the concept of an instantaneous velocity might at first seem counter-intuitive, it may be thought of as the velocity that the object would continue to travel at if it stopped accelerating at that moment.