Physics Equations/Kinematics - Wikiversity

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By definition, velocity involves two different positions at two different times. However, we may take the limit that these differences are very small and define the instantaneous velocity. v ( t ) = lim Δ t → 0 Δ x Δ t = lim Δ t → 0 x ( t f ) − x ( t i ) t f − t i {\displaystyle v(t)=\lim _{\Delta t\to 0}{\frac {\Delta x}{\Delta t}}=\lim _{\Delta t\to 0}{\frac {x(t_{f})-x(t_{i})}{t_{f}-t_{i}}}}

A connection to differential calculus is seen by rewriting t i {\displaystyle t_{i}} and t f {\displaystyle t_{f}} as t and t+ Δ t {\displaystyle \Delta t} , so that x f = x ( t + Δ t ) {\displaystyle x_{f}=x(t+\Delta t)} :

v ( t ) = lim Δ t → 0 x ( t + Δ t ) − x ( t ) Δ t = d x d t {\displaystyle v(t)=\lim _{\Delta t\to 0}{\frac {x(t+\Delta t)-x(t)}{\Delta t}}={\frac {dx}{dt}}}

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