Velocity at a Moment and Instantaneous Rate of Change

How fast is something moving right now, at this instant? Answering this question rigorously requires limits. This section develops the concept of instantaneous velocity from average velocity, and then generalizes to the idea of an instantaneous rate of change of any function.

Concept	Quick Reference
Average velocity	$v_{avg} = \frac{f (t_{0} + Δ t) - f (t_{0})}{Δ t}$
Instantaneous velocity	$v (t_{0}) = lim_{Δ t \to 0} \frac{f (t_{0} + Δ t) - f (t_{0})}{Δ t}$
Instantaneous speed	$speed = \| v (t_{0}) \|$
Rate of change of $f$ at $x_{0}$	$lim_{Δ x \to 0} \frac{f (x_{0} + Δ x) - f (x_{0})}{Δ x}$

Introduction

We often wish to know how quickly phenomena occur or in which direction a change is taking place, whether it involves an increase or a decrease. In other words, we are often interested in finding the rate of change.

Perhaps the most common example is formulating a precise definition of velocity for a moving object. The simplest case is rectilinear motion, where an object travels along a straight line. In this section, we examine the concept of velocity (the rate of change of the object's position) for this type of motion, and then discuss rate of change more generally.

Average Velocity

If a car travels a distance of 240 kilometers in 3 hours, we say it has traveled at the rate of 80 kilometers an hour. But we know that this does not necessarily mean that the speedometer registers 80 km/hr all the time. 80 km/hr is its average speed (or average velocity).

In general, suppose that an object is moving along a straight line. Let's choose one direction as positive and the opposite as negative, and one point as the origin $O$ . Let $s$ be the object's position (its coordinate) on this straight line, and $s = f (t)$ be a function giving the position of the object at time $t$ .

Number line showing position s of an object moving along a straight line

The object's average velocity from time $t_{0}$ to time $t_{1}$ is found by dividing displacement (change in position) $Δ s = f (t_{1}) - f (t_{0})$ by $Δ t = t_{1} - t_{0}$ :

\begin{aligned} v_{\text{avg}} &= \frac{f(t_1) - f(t_0)}{t_1 - t_0} \\ &= \frac{\Delta s}{\Delta t}. \end{aligned}

Because $t_{1} = t_{0} + Δ t$ , we can rewrite this as

v_{avg} = \frac{f (t_{0} + Δ t) - f (t_{0})}{Δ t} .

Graph showing displacement Delta s over time interval Delta t for average velocity

Velocity vs. Speed Velocity and speed have two distinct meanings. The **average velocity** is calculated by dividing change in position (displacement)

Δ s

by change in time

Δ t

v_{avg} = \frac{displacement}{travel time} = \frac{final position - initial position}{travel time} .

Velocity is a quantity with a sign, meaning it can be positive or negative:

final position < initial position \Rightarrow v_{avg} < 0.

The **average speed** is calculated by dividing the distance traveled by the elapsed time:

average speed = \frac{distance traveled}{travel time} .

Speed is always nonnegative. Suppose you travel to a city that is 150 km away and return in 4 hours. Since you travel

150 + 150 = 300

km, your average speed is

300 / 4 = 75

km/hr. However, your average velocity is zero because your final position is the same as your initial position, and your total displacement is zero.

Assume you drop a stone from rest and air resistance is negligible. If $s$ denotes the stone's position (distance fallen) after $t$ seconds, then

s = \frac{1}{2} g t^{2},

where $g$ is the gravitational acceleration constant. If $s$ is measured in feet and $t$ in seconds, $g \approx 32$ , giving $s = 16 t^{2}$ . Find the average velocity of the stone: (a) during the first 3 seconds of fall; (b) during the 1-second interval between second 2 and second 3.

Solution

(a) The position of the stone 3 seconds after release is

s (3) = 16 (3)^{2} = 144 ft .

The average velocity during the first 3 seconds is \begin{aligned} v_{\text{avg}} &= \frac{s(3) - s(0)}{3 - 0} \\ &= \frac{144 - 0}{3} \\ &= 48 \text{ ft/s}. \end{aligned} (b) The average velocity from second 2 to second 3 is \begin{aligned} v_{\text{avg}} &= \frac{s(3) - s(2)}{3 - 2} \\ &= \frac{16(3)^2 - 16(2)^2}{1} \\ &= 16(9 - 4) \\ &= 80 \text{ ft/s}. \end{aligned}

Instantaneous Velocity

For a small time interval

Δ t

, velocity does not change very much. Therefore, we can approximate the velocity at a single instant by the average velocity over this short time span. As

Δ t

approaches zero, this defines the instantaneous velocity at time

t_{0}

v (t_{0}) = lim_{Δ t \to 0} \frac{f (t_{0} + Δ t) - f (t_{0})}{Δ t} .

How can we define the velocity at a single instant $t_{0}$ ? (The velocity at an instant is called instantaneous velocity, or simply velocity.) Our intuition suggests that the instantaneous velocity is approximately equal to the average velocity if the "averaging time" $Δ t$ is small. This approximation gets better and better as $Δ t$ gets smaller and smaller. So for an object with position function $s = f (t)$ , the instantaneous velocity at $t_{0}$ is

v (t_{0}) = lim_{Δ t \to 0} v_{avg} = lim_{Δ t \to 0} \frac{Δ s}{Δ t}

v (t_{0}) = lim_{Δ t \to 0} \frac{f (t_{0} + Δ t) - f (t_{0})}{Δ t} .

The above limit is called the derivative of the position function with respect to time.

The absolute value of $v (t_{0})$ is called the instantaneous speed:

speed = | v (t_{0}) | .

The stone's position at time $t$ is given by $s = f (t) = 16 t^{2}$ . What is the instantaneous velocity of the stone at $t = 3$ s?

Solution

First, let's estimate it numerically. The table below shows average velocities over shorter and shorter intervals near

t = 3

Table of average velocities for stone falling, showing approach to 96 ft/s as Delta t approaches 0

It appears that the average velocity approaches 96 ft/s as

Δ t \to 0

, and thus the instantaneous velocity at

t_{0} = 3

seconds is 96 ft/s. Now let's confirm it algebraically: \begin{aligned} \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t} &= \lim_{\Delta t \to 0} \frac{f(3 + \Delta t) - f(3)}{\Delta t} \\ &= \lim_{\Delta t \to 0} \frac{16(3 + \Delta t)^2 - 16(3)^2}{\Delta t} \\ &= \lim_{\Delta t \to 0} \frac{16(9 + 6\Delta t + (\Delta t)^2) - 144}{\Delta t} \\ &= \lim_{\Delta t \to 0} \frac{96\Delta t + 16(\Delta t)^2}{\Delta t} \\ &= \lim_{\Delta t \to 0} [96 + 16\Delta t] \\ &= 96 \text{ ft/s}. \end{aligned}

Rate of Change

If, instead of position, we represent another variable in terms of time with $f (t)$ , then the average rate of change of $f$ between $t_{0}$ and $t_{0} + Δ t$ is given by

\frac{f (t_{0} + Δ t) - f (t_{0})}{Δ t},

and the instantaneous rate of change of $f$ at time $t_{0}$ is defined as

lim_{Δ t \to 0} \frac{f (t_{0} + Δ t) - f (t_{0})}{Δ t} .

More generally, if two quantities $x$ and $y$ are related by the functional relationship $y = f (x)$ , the (instantaneous) rate of change of $y$ with respect to $x$ at $x_{0}$ is defined as the following limit:

lim_{Δ x \to 0} \frac{f (x_{0} + Δ x) - f (x_{0})}{Δ x},

provided that this limit exists.

Often we drop "instantaneous" and simply say "rate of change of $y$ with respect to $x$ ."

Frequently Asked Questions

What is the difference between average velocity and instantaneous velocity?

Average velocity measures the total displacement divided by the total time over an interval. Instantaneous velocity is the limit of average velocity as the time interval shrinks to zero. It captures the velocity at a single moment in time.

Why can instantaneous velocity be negative?

Velocity is a signed quantity. A negative velocity means the object is moving in the negative direction (opposite to the chosen positive direction). For example, if rightward is positive and the object moves leftward, its velocity is negative. Speed, on the other hand, is always nonnegative.

Is rate of change always about time?

No. Rate of change is a general concept: it measures how fast one quantity changes relative to another. While velocity is the rate of change of position with respect to time, we can equally talk about the rate of change of temperature with respect to distance, or profit with respect to units sold. The mathematical definition is the same in all cases.

Introduction

Average Velocity

Instantaneous Velocity

Rate of Change

Frequently Asked Questions

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