Instantaneous Velocity

Key Questions

• Provided that the graph is of distance as a function of time, the slope of the line tangent to the function at a given point represents the instantaneous velocity at that point.

In order to get an idea of this slope, one must use limits. For an example, suppose one is given a distance function $x = f \left(t\right)$, and one wishes to find the instantaneous velocity, or rate of change of distance, at the point ${p}_{0} = \left({t}_{0} , f \left({t}_{0}\right)\right)$, it helps to first examine another nearby point, ${p}_{1} = \left({t}_{0} + a , f \left({t}_{0} + a\right)\right)$, where $a$ is some arbitrarily small constant. The slope of the secant line passing through the graph at these points is:

$\frac{f \left({t}_{0} + a\right) - f \left({t}_{0}\right)}{a}$

As ${p}_{1}$ approaches ${p}_{0}$ (which will occur as our $a$ decreases), our above $\mathrm{di} f f e r e n c e q u o t i e n t$ will approach a limit, here designated $L$, which is the slope of the tangent line at the given point. At that point, a point-slope equation using our above points can provide a more exact equation.

If instead one is familiar with differentiation, and the function is both continuous and differentiable at the given value of $t$, then we can simply differentiate the function. Given that most distance functions are polynomial functions, of the form $x = f \left(t\right) = a {t}^{n} + b {t}^{n - 1} + c {t}^{n - 2} + \ldots + y t + z ,$ these can be differentiated using the power rule which states that for a function $f \left(t\right) = a {t}^{n} , \frac{\mathrm{df}}{\mathrm{dt}}$ (or $f ' \left(t\right)$) = $\left(n\right) a {t}^{n - 1}$.

Thus for our general polynomial function above, $x ' = f ' \left(t\right) = \left(n\right) a {t}^{n - 1} + \left(n - 1\right) b {t}^{n - 2} + \left(n - 2\right) c {t}^{n - 3} + \ldots + y$ (Note that since $t = {t}^{1}$ (as any number raised to the first power equals itself), reducing the power by 1 leaves us with ${t}^{0} = 1$, hence why the final term is simply $y$. Note also that our $z$ term, being a constant, did not change with respect to $t$ and thus was discarded in differentiation).

This $f ' \left(t\right)$ is the derivative of the distance function with respect to time; thus, it measures the rate of change of distance with respect to time, which is simply the velocity.

• Instantaneous velocity is the velocity at which an object is travelling at exactly the instant that is specified.

If I travel north at exactly 10m/s for exactly ten seconds, then turn west and travel exactly 5m/s for another ten seconds exactly, my average velocity is roughly 5.59m/s in a (roughly) north-by-northwest direction. However, my instantaneous velocity is my velocity at any given point: at exactly five seconds into my trip, my instantaneous velocity is 10m/s north; at exactly fifteen seconds in, it's 5m/s west.