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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, start subscript, a, v, g, end subscript, equals, start fraction, delta, x, divided by, delta, t, end fraction, equals, start fraction, x, start subscript, f, end subscript, minus, x, start subscript, 0, end subscript, divided by, t, start subscript, f, end subscript, minus, t, start subscript, 0, end subscript, end fraction
In this formula, v, start subscript, a, v, g, end subscript is the average velocity; delta, x is the change in position, or displacement; and x, start subscript, f, end subscript and x, start subscript, 0, end subscript are the final and beginning positions at times t, start subscript, f, end subscript and t, start subscript, 0, end subscript, respectively. If the starting time t, start subscript, 0, end subscript is taken to be zero, then the average velocity is written as below:
v, start subscript, a, v, g, end subscript, equals, start fraction, delta, x, divided by, t, end fraction
Note: t is shorthand for 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 start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, but many other units such as start fraction, start text, k, m, end text, divided by, start text, h, r, end text, end fraction, start fraction, start text, m, i, end text, divided by, start text, h, r, end text, end fraction (also written as mph), and 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:
v, start subscript, a, v, g, end subscript, equals, start fraction, delta, x, divided by, t, end fraction, equals, start fraction, minus, 4, start text, space, m, end text, divided by, 5, start text, space, s, end text, end fraction, equals, minus, 0, point, 8, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction
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, start subscript, start text, t, o, t, end text, end subscript, consists of 4 segments, delta, x, start subscript, start text, a, end text, end subscript, delta, x, start subscript, start text, b, end text, end subscript, delta, x, start subscript, start text, c, end text, end subscript, and delta, x, start subscript, start text, d, end text, end subscript.
Figure 1: A more detailed record of an airplane passenger heading toward the back of the plane, showing smaller segments of his trip. (Image credit: Openstax College Physics)
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, 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, 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, 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, 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, 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.
Figure 2: During a 30-minute round trip to the store, the total distance traveled is 6 km. The average speed is 12 km/h. The displacement for the round trip is zero, since there was no net change in position. Thus the average velocity is zero. Image credit: Openstax College Physics
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.
Figure 3: Position vs. time, velocity vs. time, and speed vs. time on a trip. Note that the velocity for the return trip is negative. Image credit: Openstax College Physics)

### 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.
start text, a, v, e, r, a, g, e, space, s, p, e, e, d, end text, equals, start fraction, start text, d, i, s, t, a, n, c, e, space, t, r, a, v, e, l, e, d, end text, divided by, start text, t, i, m, e, space, i, n, t, e, r, v, a, l, end text, end fraction, equals, start fraction, 12, point, 0, start text, space, m, end text, plus, 16, point, 0, start text, space, m, end text, divided by, 20, point, 0, start text, space, s, end text, plus, 8, point, 0, start text, space, s, end text, end fraction
start text, a, v, e, r, a, g, e, space, s, p, e, e, d, end text, equals, start fraction, 28, point, 0, start text, space, m, end text, divided by, 28, point, 0, start text, space, s, end text, end fraction
start text, a, v, e, r, a, g, e, space, s, p, e, e, d, end text, equals, 1, start fraction, start text, space, m, end text, divided by, start text, space, s, end text, end fraction
To find the average velocity we take the displacement delta, x divided by the time interval.
start text, a, v, e, r, a, g, e, space, v, e, l, o, c, i, t, y, end text, equals, start fraction, start text, d, i, s, p, l, a, c, e, m, e, n, t, end text, divided by, start text, t, i, m, e, space, i, n, t, e, r, v, a, l, end text, end fraction, equals, start fraction, minus, 4, point, 0, start text, space, m, end text, divided by, 28, point, 0, start text, space, s, end text, end fraction
start text, a, v, e, r, a, g, e, space, v, e, l, o, c, i, t, y, end text, equals, minus, start fraction, 1, divided by, 7, end fraction, start fraction, start text, space, m, end text, divided by, start text, space, s, end text, end fraction

#### 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 t, equals, 0, start text, space, s, end text to t, equals, 6, start text, space, s, end text
b. average speed between t, equals, 0, start text, space, s, end text to t, equals, 6, start text, space, s, end text
c. instantaneous velocity at time t, equals, 1, start text, space, s, end text
d. instantaneous speed at time t, equals, 4, start text, space, s, end text
Part A: Average velocity is defined to be the displacement per time.
v, start subscript, a, v, g, end subscript, equals, start fraction, delta, x, divided by, delta, t, end fraction, equals, start fraction, 0, start text, space, m, end text, minus, 8, start text, space, m, end text, divided by, 6, start text, space, s, end text, minus, 0, start text, space, s, end text, end fraction, equals, start fraction, minus, 8, start text, space, m, end text, divided by, 6, start text, space, s, end text, end fraction, start text, left parenthesis, U, s, e, space, d, e, f, i, n, i, t, i, o, n, space, o, f, space, a, v, e, r, a, g, e, space, v, e, l, o, c, i, t, y, point, right parenthesis, end text
v, start subscript, a, v, g, end subscript, equals, minus, start fraction, 4, divided by, 3, end fraction, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, start text, left parenthesis, C, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, point, right parenthesis, end text
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, start subscript, a, v, g, end subscript, equals, start fraction, start text, d, i, s, t, a, n, c, e, space, t, r, a, v, e, l, e, d, end text, divided by, delta, t, end fraction, equals, start fraction, 12, start text, space, m, end text, plus, 0, start text, space, m, end text, plus, 4, start text, space, m, end text, divided by, 6, start text, space, s, end text, minus, 0, start text, space, s, end text, end fraction, equals, start fraction, 16, start text, space, m, end text, divided by, 6, start text, space, s, end text, end fraction, start text, left parenthesis, u, s, e, space, d, e, f, i, n, i, t, i, o, n, space, o, f, space, a, v, e, r, a, g, e, space, s, p, e, e, d, right parenthesis, end text
v, start subscript, a, v, g, end subscript, equals, start fraction, 8, divided by, 3, end fraction, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction, start text, left parenthesis, c, a, l, c, u, l, a, t, e, space, a, n, d, space, c, e, l, e, b, r, a, t, e, right parenthesis, end text
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, equals, 1, start text, s, end text we can determine the "rise over run" for any two points on the graph between t, equals, 0, start text, s, end text and t, equals, 3, start text, s, end text (since the slope is constant between those times). Choosing the times t, equals, 2, start text, s, end text and t, equals, 0, start text, s, end text, we find the slope as follows,
v, start subscript, start text, i, n, s, t, a, n, t, a, n, e, o, u, s, end text, end subscript, equals, start text, s, l, o, p, e, end text, equals, start fraction, x, start subscript, 2, end subscript, minus, x, start subscript, 0, end subscript, divided by, t, start subscript, 2, end subscript, minus, t, start subscript, 0, end subscript, end fraction
v, start subscript, start text, i, n, s, t, a, n, t, a, n, e, o, u, s, end text, end subscript, equals, start fraction, 0, start text, space, m, end text, minus, 8, start text, space, m, end text, divided by, 2, start text, space, s, end text, minus, 0, start text, space, s, end text, end fraction, equals, start fraction, minus, 8, start text, space, m, end text, divided by, 2, start text, space, s, end text, end fraction
v, start subscript, start text, i, n, s, t, a, n, t, a, n, e, o, u, s, end text, end subscript, equals, minus, 4, start fraction, start text, m, end text, divided by, start text, s, end text, end fraction
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, equals, 4, start text, s, end text is equal to zero, the instantaneous speed at t, equals, 4, start text, s, end text is also equal to zero.

## Want to join the conversation?

• Can an object have a northbound velocity and southbound acceleration
• You can imagine throwing a ball into the air, the ball would have an upward (positive) speed immediately after throwing, but would be slowing down with an acceleration of -9.81m/s^2 (The acceleration due to gravity on Earth). So it'd have a positive velocity and negative acceleration, then when it peaked and started going down, it'd have a negative velocity and negative acceleration.
• 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?
• They are ALMOST the same but only in the sense that both measure space over time, but if you want to analyze both of them in the same problem, they may be very different from each other, because as
speed measures the absolute distance in a x time
velocity measures relative distance to a point at the same x time
For example, let's say that x time = 1 minute
You have to go from your house to some point that is 50 meters to the north and back again(total distance = 100m), before the time is over, so you run, but time is over when you are 1m before you're back to home, so if now we compare your speed vs velocity you'll see that your
average speed = 99 meter /min your total travel distance is 99m
average velocity = 1 meter / min but your displacement distance is 1m
• what is terminal velocity?
• Terminal velocity is reached when gravity and air resistance balance, keeping an object in free fall from accelerating.
• 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.
• 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?