What is velocity?

Your notion of velocity is probably similar to its scientific definition. You know that a large displacement in a small amount of time means a large velocity and that velocity has units of distance divided by time, such as miles per hour or kilometers per hour.

Average velocity is defined to be the change in position divided by the time of travel.

In this formula, $v_{avg}$v, start subscript, a, v, g, end subscript is the average velocity; $\Delta x$delta, x is the change in position, or displacement; and $x_f$x, start subscript, f, end subscript and $x_0$x, start subscript, 0, end subscript are the final and beginning positions at times $t_f$t, start subscript, f, end subscript and $t_0$t, start subscript, 0, end subscript, respectively. If the starting time $t_0$t, start subscript, 0, end subscript is taken to be zero, then the average velocity is written as below:

Note: $t$t is shorthand for $\Delta t$delta, t.

Notice that this definition indicates that velocity is a vector because displacement is a vector. It has both magnitude and direction. The International System of Units (SI) unit for velocity is meters per second or $\dfrac{\text{m}}{\text{s}}$start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, but many other units such as $\dfrac{\text{km}}{\text{hr}}$start fraction, start text, k, m, end text, divided by, start text, h, r, end text, end fraction, $\dfrac{\text{mi}}{\text{hr}}$start fraction, start text, m, i, end text, divided by, start text, h, r, end text, end fraction (also written as mph), and $\dfrac{\text{cm}}{\text{s}}$start fraction, start text, c, m, end text, divided by, start text, s, end text, end fraction are commonly used. Suppose, for example, an airplane passenger took 5 seconds to move −4 meters, where the negative sign indicates that displacement is toward the back of the plane. His average velocity can be written as below:

The minus sign indicates the average velocity is also toward the rear of the plane.

The average velocity of an object does not tell us anything about what happens to it between the starting point and ending point, however. For example, we cannot tell from average velocity whether the airplane passenger stops momentarily or backs up before he goes to the back of the plane. To get more details, we must consider smaller segments of the trip over smaller time intervals. For instance, in the figure below, we see that the total trip displacement, $\Delta x _ \text{tot}$delta, x, start subscript, start text, t, o, t, end text, end subscript, consists of 4 segments, $\Delta x_\text a$delta, x, start subscript, start text, a, end text, end subscript, $\Delta x_\text b$delta, x, start subscript, start text, b, end text, end subscript, $\Delta x_\text c$delta, x, start subscript, start text, c, end text, end subscript, and $\Delta x_\text d$delta, x, start subscript, start text, d, end text, end subscript.

The smaller the time intervals considered in a motion, the more detailed the information. Carrying this process to its logical conclusion, we are left with an infinitesimally small interval. Over such an interval, the average velocity becomes the instantaneous velocity, or the velocity at a specific moment. A car’s speedometer, for example, shows the magnitude—but not the direction—of the instantaneous velocity of the car. Police give tickets based on instantaneous velocity, but when calculating how long it will take to get from one place to another on a road trip, you need to use average velocity. Instantaneous velocity, $v$v, is simply the average velocity at a specific instant in time or over an infinitesimally small time interval.

Mathematically, finding instantaneous velocity, $v$v, at a precise instant $t$t can involve taking a limit, a calculus operation beyond the scope of this article. However, under many circumstances, we can find precise values for instantaneous velocity without calculus.

In everyday language, most people use the terms speed and velocity interchangeably. In physics, however, they do not have the same meaning, and they are distinct concepts. One major difference is that speed has no direction. Thus, speed is a scalar. Just as we need to distinguish between instantaneous velocity and average velocity, we also need to distinguish between instantaneous speed and average speed.

Instantaneous speed is the magnitude of instantaneous velocity. For example, suppose the airplane passenger at one instant had an instantaneous velocity of $−3.0 \dfrac{\text m}{\text{s}}$−, 3, point, 0, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, the negative meaning toward the rear of the plane. At that same time his instantaneous speed was $3.0 \dfrac{\text {m}}{\text{s}}$3, point, 0, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction. Or suppose that at a particular instant during a shopping trip, your instantaneous velocity is $40 \dfrac{\text {km}}{\text{hr}}$40, start fraction, start text, k, m, end text, divided by, start text, h, r, end text, end fraction due north. Your instantaneous speed at that instant would be $40 \dfrac{\text {km}}{\text{hr}}$40, start fraction, start text, k, m, end text, divided by, start text, h, r, end text, end fraction—the same magnitude but without a direction. Average speed, however, is very different from average velocity. Average speed is the distance traveled divided by elapsed time. So, while the magnitudes of the instantaneous speed and velocity are always identical, the magnitudes of average speed and velocity can be very different.

Since distance traveled can be greater than the magnitude of displacement, the average speed can be greater than the magnitude of the average velocity. For example, if you drive to a store and return home in half an hour and your car’s odometer shows the total distance traveled was 6 km, then your average speed was $12 \dfrac{\text {km}}{\text{hr}}$12, start fraction, start text, k, m, end text, divided by, start text, h, r, end text, end fraction. Your average velocity, however, was zero because your displacement for the round trip is zero—Displacement is change in position and, thus, is zero for a round trip. Thus average speed is not simply the magnitude of average velocity.

Another way of visualizing the motion of an object is to use a graph. A plot of position or of velocity as a function of time can be very useful. For example, for this trip to the store, the position, velocity, and speed-vs.-time graphs are displayed in Figure 3. Note that these graphs depict a very simplified model of the trip. We are assuming that speed is constant during the trip, which is unrealistic given that we’ll probably stop at the store. But for simplicity’s sake, we will model it with no stops or changes in speed. We are also assuming that the route between the store and the house is a perfectly straight line.

An iguana with a poor sense of spatial awareness is walking back and forth in the desert. First the iguana walks 12 meters to the right in a time of 20 seconds. Then the iguana runs 16 meters to the left in a time of 8 seconds.

To find the average speed we take the total distance traveled divided by the time interval.

To find the average velocity we take the displacement $\Delta x$delta, x divided by the time interval.

A hungry dolphin is swimming horizontally back and forth looking for food. The motion of the dolphin is given by the position graph shown below.

Determine the following for the dolphin:
a. average velocity between time $t=0 \text{ s}$t, equals, 0, start text, space, s, end text to $t=6\text{ s}$t, equals, 6, start text, space, s, end text
b. average speed between $t=0 \text{ s}$t, equals, 0, start text, space, s, end text to $t=6\text{ s}$t, equals, 6, start text, space, s, end text
c. instantaneous velocity at time $t=1\text{ s}$t, equals, 1, start text, space, s, end text
d. instantaneous speed at time $t=4\text{ s}$t, equals, 4, start text, space, s, end text

Part A: Average velocity is defined to be the displacement per time.

Part B: Average speed is defined to be the distance traveled per time. The distance is the sum of the total path length traveled by the dolphin, so we just add up all the distances traveled by the dolphin for each leg of the trip.

Part C: Instantaneous velocity is the velocity at a given moment and will be equal to the slope of the graph at that moment. To find the slope at $t=1\text{s}$t, equals, 1, start text, s, end text we can determine the "rise over run" for any two points on the graph between $t=0\text{s}$t, equals, 0, start text, s, end text and $t=3\text{s}$t, equals, 3, start text, s, end text (since the slope is constant between those times). Choosing the times $t=2\text{s}$t, equals, 2, start text, s, end text and $t=0\text{s}$t, equals, 0, start text, s, end text, we find the slope as follows,

Part D: Instantaneous speed is the speed at a given moment in time and will be equal to the magnitude of the slope. Since the slope at $t=4\text{s}$t, equals, 4, start text, s, end text is equal to zero, the instantaneous speed at $t=4\text{s}$t, equals, 4, start text, s, end text is also equal to zero.