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, \[\textbf{a} = \lim_{h\to 0} \frac{v(t+h)-v(t)}{(t+h)-t} = v'(t)\]

Useful Formulae

Position and its time derivatives are often used in kinematics. For example, the following four equations relate the position $x$, velocity $v$, and (constant) acceleration $a$ by magnitude: \begin{align*} x&=x_0+v_0t+\frac{1}{2}at^2 \\ \Delta x&=\left(\frac{v+v_0}{2}\right)t \\ v^2&=v_0^2+2a\Delta x \\ \overline{v}&=\frac{v+v_0}{2}. \end{align*} By the chain rule, one can also show \[a=v\frac{\text{d} v}{\text{d}x}.\] Lastly, we have the famous formula of Newton relating the force and acceleration experienced by a massive object: \[\mathbf{F}=m\mathbf{a}.\]

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