Velocity

How fast is the point moving? Using calculus, its velocity, $v$, is defined as the derivative of $y$ with respect to time:

$$v = { dy(t) \over dt } .$$ (8.1)

Using numerical computations, $v$ is approximately equal to $\Delta y / \Delta t$, in which $\Delta t$ denotes a small change in time and

$$\Delta y = y(t + \Delta t) - y(t).$$ (8.2)

In other words, $\Delta y$ is the change in $y$ from the start to the end of the time change. The velocity $v$ can be used to estimate the change in $y$ over $\Delta t$ as

$$\Delta y \approx v \Delta t .$$ (8.3)

The approximation quality improves as $\Delta t$ becomes smaller and $v$ itself varies less during the time from $t$ to $t + \Delta t$.

We can write $v(t)$ to indicate that velocity may change over time. The position can be calculated for any time $t$ from the velocity using integration as^8.1

$$y(t) = y(0) + \int_0^t v(s) ds ,$$ (8.4)

which assumes that $y$ was known at the starting time $t = 0$. If $v(t)$ is constant for all time, represented as $v$, then $y(t) = y(0) + vt$. The integral in (8.4) accounts for $v(t)$ being allowed to vary.