# Acceleration

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