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Acceleration

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Definition

Acceleration, the second derivative of displacement, is defined to be the change of velocity per unit time at a certain instance.

A common misconception is that acceleration implies a POSITIVE change of velocity, while it could also mean a NEGATIVE one.

Formula for Acceleration

Let \textbf{v}_1 be the velocity of an object at a time t_1 and \textbf{v}_2 be the velocity of the same object at a time t_2. If acceleration, \textbf{a}, is known to be constant, then \textbf{a} = \frac{\textbf{v}_2 -\textbf{v}_1 }{t_2 - t_1} Note that velocity is a vector, so the magnitudes cannot be just subtracted in general.

If acceleration is not constant, then we can treat velocity as a function of time, v(t). Then, at a particular instance, \textbf{a} = \lim_{h\to 0} \frac{v(t+h)-v(t)}{(t+h)-t} = v'(t)

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