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# What is velocity?

Velocity or speed? Instantaneous or average? Keep building your physics vocabulary.

## What does velocity mean?

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.
${v}_{avg}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}=\frac{{x}_{f}-{x}_{0}}{{t}_{f}-{t}_{0}}$
In this formula, ${v}_{avg}$ is the average velocity; $\mathrm{\Delta }x$ is the change in position, or displacement; and ${x}_{f}$ and ${x}_{0}$ are the final and beginning positions at times ${t}_{f}$ and ${t}_{0}$, respectively. If the starting time ${t}_{0}$ is taken to be zero, then the average velocity is written as below:
${v}_{avg}=\frac{\mathrm{\Delta }x}{t}$
Note: $t$ is shorthand for $\mathrm{\Delta }t$.
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 $\frac{\text{m}}{\text{s}}$, but many other units such as $\frac{\text{km}}{\text{hr}}$, $\frac{\text{mi}}{\text{hr}}$ (also written as mph), and $\frac{\text{cm}}{\text{s}}$ 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, $\mathrm{\Delta }{x}_{\text{tot}}$, consists of 4 segments, $\mathrm{\Delta }{x}_{\text{a}}$, $\mathrm{\Delta }{x}_{\text{b}}$, $\mathrm{\Delta }{x}_{\text{c}}$, and $\mathrm{\Delta }{x}_{\text{d}}$.
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, $v$, is simply the average velocity at a specific instant in time or over an infinitesimally small time interval.
Mathematically, finding instantaneous velocity, $v$, at a precise instant $t$ 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.

## What does speed mean?

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 $-3.0\frac{\text{m}}{\text{s}}$, the negative meaning toward the rear of the plane. At that same time his instantaneous speed was $3.0\frac{\text{m}}{\text{s}}$. Or suppose that at a particular instant during a shopping trip, your instantaneous velocity is $40\frac{\text{km}}{\text{hr}}$ due north. Your instantaneous speed at that instant would be $40\frac{\text{km}}{\text{hr}}$—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 $12\frac{\text{km}}{\text{hr}}$. 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.

### What do solved examples involving velocity and speed look like?

#### Example 1: Disoriented iguana

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.
What was the average speed and average velocity of the iguana for the entire trip?
Assume that rightward is the positive direction.
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 $\mathrm{\Delta }x$ divided by the time interval.

#### Example 2: Hungry dolphin

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 to
b. average speed between to
c. instantaneous velocity at time
d. instantaneous speed at time
Part A: Average velocity is defined to be the displacement per time.
${v}_{avg}=-\frac{4}{3}\frac{\text{m}}{\text{s}}\phantom{\rule{1em}{0ex}}\text{(Calculate and celebrate.)}$
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.
${v}_{avg}=\frac{8}{3}\frac{\text{m}}{\text{s}}\phantom{\rule{1em}{0ex}}\text{(calculate and celebrate)}$
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 $t=1\text{s}$ we can determine the "rise over run" for any two points on the graph between $t=0\text{s}$ and $t=3\text{s}$ (since the slope is constant between those times). Choosing the times $t=2\text{s}$ and $t=0\text{s}$, we find the slope as follows,
${v}_{\text{instantaneous}}=\text{slope}=\frac{{x}_{2}-{x}_{0}}{{t}_{2}-{t}_{0}}$
${v}_{\text{instantaneous}}=-4\frac{\text{m}}{\text{s}}$
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 $t=4\text{s}$ is equal to zero, the instantaneous speed at $t=4\text{s}$ is also equal to zero.

## Want to join the conversation?

• Can an object have a northbound velocity and southbound acceleration
• Another example: You are driving northbound and you hit the brakes. The velocity is northbound and the acceleration is southbound. The southbound acceleration adds a southbound change in velocity to the northbound velocity over time the brakes are applied. When the car comes to a stop the brakes are no longer resisting a movement so the the acceleration goes to 0 and the velocity can then stay at 0.
• What if the line of the graph is curved? I know how to find the instantaneous speed given the function, but how do you find it given only the graph?
• choose a time (t) at which you want to find the instantaneous speed.
Draw a tangent to the curve at that point (t)
Find the gradient of the tangent.

Its the same as differentiating and plugging in the value of t

OK??
• Speed and Velocity are almost the same with the exception that speed is a scalar quantity and velocity is a vector quantity. Is this true?
• Not entirely actually. Speed takes into account the entire distance travelled over a time period, while velocity is the displacement over a time period. This usually doesn't matter unless the direction the particle is travelling switches
• example 2: Hungry Dolphin.
I thought that to find instantaneous velocity I would just look at where the line intersects 1 second. This gave me the answer of 4 instead of -4. Could anyone explain to me why it is -4?
thanks, Jack.
• The answer is -4 because the slope is negative.
• what is terminal velocity?
• Terminal velocity is reached when gravity and air resistance balance, keeping an object in free fall from accelerating.
• for the example 1 where we used the disoriented iguana, how is the time interval 28 seconds?
I thought the formula for change in time was final time - intial time.
shouldn't it be 8-20=-12s?
• You don't need a formula here, they tell you he went one way for 20 seconds and the the other way for 8 seconds. That's 28 seconds.
If you were using a stop watch, the initial time would be 0, the time when he turns around would be 20, and the final time would be 28. 28-0 = 28.
• How would you calculate velocity if the displacement is zero? For instance, a car traveling on a circular track that begins and ends at the same position.
• size circumference of circular track is displacement. Starting position and ending at same position count as lap. Wherefore let *C = pi*d* where C is circumference, pi = 3.1415 and d is diameter of circle. So now you can calculate velocity like this:
v = C / t or v = (pi * d) / t
• An average in algebra is where you add numbers, say, 80, 40 and 30, and you divide that number by three, one for each quantity. So, if you can lay out where someone is running or driving a car, and they stay at a constant rate except for three speed changes, can you just add up the three speeds and divide by three? So,
52mph 70mph 40mph
|______|________|_______|
52+70+40=162
162/3
54mph average?