شارح الدرس: Average Speed العلوم

In this explainer, we will learn how to distinguish between constant speed and average speed.

We can recall that the speed of an object is a measure of the distance moved by that object per unit of time.

We may also recall that if an object moves at a constant speed, this means the value of this speed is not changing. In other words, an object moving at a constant speed travels equal distances in equal time intervals.

For example, the car shown in the diagram below travels at a constant speed.

In this case, the car travels an equal distance of 20 m in each 1 s time interval.

For an object moving at a constant speed of , which covers a distance in a time , we have the following equation that relates these three quantities:

This equation applies both to the journey as a whole and to any individual section of that journey.

It is a useful exercise to verify this for the car in the diagram above and show that the speed of the car will always be found to be 20 m/s no matter what part of the journey is used to calculate this speed.

This holds true for any object that moves at a constant speed. No matter what part of the journey we use for the calculation, for an object with a constant speed, we will always find the same value for this speed.

However, not all objects always move at a constant speed.

For a concrete example of this, let us again consider the motion of a car. Initially, that car must have been parked somewhere. In other words, at some point, it started from a state of “not moving”; we would describe this as saying that the car started from rest. An object at rest has a speed of 0 m/s.

If we have a car that starts with a speed of 0 m/s and after some amount of time is moving at a speed of 20 m/s, we know that its speed cannot have been constant. Throughout the time interval in which the speed of the car changed from 0 m/s to 20 m/s, it was neither moving at a constant speed of 0 m/s nor moving at a constant speed of 20 m/s. At different times within that interval, the speed must have had all of the values between 0 m/s and 20 m/s.

When the speed of an object is not constant, this means that it does not travel equal distances in equal intervals of time. In this case, we say that the object has a nonconstant, or changing, speed.

Whether or not the speed of an object is constant, we can always talk about an average speed of the object over a given interval.

When we talk about averages in physics, often we are referring to the averaging of several repeated measurements. In this context, we could imagine a car that drives the same distance along the same section of road multiple times. We could measure the speed of the car each time and then take the average of all these measurements to get an average speed.

However, in the context of nonconstant speeds, when we talk about an average speed, we mean something different. We are considering an object that moves some distance over some time interval, changing its speed as it does so. The average we are talking about is not the averaging over repeated measurements. Rather, it is that averaging of all of the different speeds the object has during this motion.

Consider a car that moves as follows:

We can see from the diagram that the car does not travel an equal distance in each 1 s interval. Therefore, we know that the car has a nonconstant speed.

The larger the distance traveled in a given 1 s time interval, the greater the average speed over that interval.

In this case, the car travels a greater distance in the interval between 1 s and 2 s than it does in the other two intervals, so we can say it has the greatest average speed in this second interval.

Let us have a look at a couple of example problems.

Example 1: Finding the Section of a Journey with the Greatest Average Speed

The speed of a toy car is measured by recording its position each second. Between which of the following times is the average speed of the car the greatest?

1. From when measuring starts to 1 second after measuring starts
2. From 1 second after measuring starts to 2 seconds after measuring starts
3. From 2 seconds after measuring starts to 3 seconds after measuring starts

Answer

The question is asking us to find during which of the time intervals shown the toy car has the greatest average speed. All of the time intervals shown have the same length of 1 s. This means that, to find the interval with the greatest average speed, we need to find the time interval in which the toy car moves the greatest distance.

Looking at the diagram, we can see that during the first interval, from when measuring starts to 1 s later, the car moves a distance of 1 m.

During the second interval, from 1 s to 2 s after measuring starts, the car moves a distance of more than 1 m. We can see this from the diagram as follows: At the start of the time interval, the car is aligned with a 1 m mark. At the end of the time interval, the car has gone beyond the next 1 m mark—so it must have traveled a distance that is greater than 1 m in this time interval.

During the third interval, from 2 s to 3 s after measuring starts, the car moves a distance of less than 1 m. We can see this from the diagram as it begins this interval ahead of a 1 m mark and it ends the interval aligned with the next 1 m mark.

We know that the car travels the greatest distance in the time interval between 1 s after measuring starts and 2 s after measuring starts. This means that we know it has the greatest average speed in this interval.

Therefore, our answer to the question is that the average speed of the car is greatest from 1 second after measuring starts to 2 seconds after measuring starts. This is the answer given in option B.

Example 2: Determining the Intervals of Nonconstant Speed

The toy car shown was traveling at a uniform speed before we started to measure its speed by recording its position each second. When was the car definitely not uniformly moving at a speed of 1 metre per second?

1. Between 1 second after measurement starts and 3 seconds after measurement starts
2. Between the start of measurement and 1 second after the start of measurement
3. For the whole time that the car moved

Answer

The question is asking us to work out when the toy car shown in the diagram cannot possibly have had a constant speed of 1 m/s.

It is only possible for the car to have had a speed of 1 m/s during a given 1 s time interval if it moved a total distance of 1 m during that interval. If it moved a different distance from this, it was moving a different number of metres than one during that second, so it must have had a different average speed than 1 metre per second.

Looking at the diagram, we can see that during the first interval the car does move a distance of 1 m. Therefore, it is possible that the car had a constant speed of 1 m/s during this interval.

We note that it is not a guarantee that the car had a constant speed of 1 m/s during this interval, only that it is not impossible. All we know for sure is that the average over this interval was 1 m/s; it could be the case that the speed changed during the 1 s interval.

During the next interval (between 1 s and 2 s after measuring starts), the car moves a distance that is greater than 1 m. It begins the interval lined up with a 1 m mark but ends the interval ahead of the next 1 m mark.

Therefore, during the interval between 1 s and 2 s after measuring starts, the car cannot have had an average speed of 1 m/s. Its average speed must have been greater.

During the interval between 2 s and 3 s after measuring starts, the car moves a distance that is less than 1 m. It begins the interval ahead of a 1 m mark but ends the interval lined up with the next 1 m mark.

Therefore, during the interval between 2 s and 3 s after measuring starts, the car cannot have had an average speed of 1 m/s. Its average speed must have been less.

Therefore, our answer is that the car is definitely not moving at a constant speed of 1 metre per second between 1 second after measurement starts and 3 seconds after measurement starts. This is the answer given in option A.

We have seen what is meant by the average speed of an object. We can also describe this average speed mathematically.

The average speed of an object over a given distance is the total distance moved by it divided by the total time taken to move that distance.

To see what this means, let us consider a car that moves as shown in the diagram below:

The diagram shows two intervals of time: between 0 seconds and some later time , and between and a further time . The diagram also shows the distance traveled by the car in each of those two intervals, starting from 0 metres at a time of 0 seconds. After a time , the car has traveled a distance . After a time , the car has traveled a total distance of .

Let us suppose that we want to find the average speed of this car during the second interval shown, that is, between time and time . We may do this as follows.

We know that the average speed over an interval is the total distance moved in that interval divided by the total time taken to move this distance. This means we need an expression for the duration of the second interval, as well as an expression for the distance moved in that interval.

The time interval begins at a time and ends at a time . Therefore, we can say that the length of time of this interval is .

At the end of the interval, the car has moved a total distance of from its position at 0 seconds. However, at the start of the interval, the car had already moved a distance of . So, the distance moved during the second interval is .

Then, the average speed is the distance moved during the interval, , divided by the duration of that interval, .

Mathematically, we can write this as

In this expression, is the change to the total distance traveled by the car that occurs during the interval of time .

We can write this more compactly using the following notation:

The symbol is used to indicate that we are considering a change in distance and a change in time during the given interval.

With this notation, we may define the average speed of an object mathematically as follows.

In general, and may be any distance values, marking the start and the end of whichever section of an object’s motion we are interested in. Meanwhile, is the time at which the object has traveled a distance and is the time at which the object has traveled a distance .

Sometimes we are interested in an average over an interval that begins at the point we start measuring at (i.e., at 0 metres and 0 seconds). In this case, and . Similarly, and .

Consider a car that moves as shown in the diagram below:

Suppose we want to calculate the average speed during the first second of motion.

The section begins at a time and ends at a time . At the beginning of this section, the car has traveled 0 metres from the point at which we begin measuring, so we have . At the end of this section, the car has moved 6 metres from its position when measurement begins, so we have .

We may then calculate the average speed of the car during this section as follows:

Now let us suppose we want to know the car’s average speed during the subsequent 1-second interval. This section begins at time and ends at time . At the beginning of this section, the car is 6 metres from its start position, so we have . At the end of the section, the car is 16 metres from the start position, so .

The average speed is then

We have calculated different average speeds for each of the two sections of motion shown in the diagram. This tells us that the speed of the car cannot be constant across the whole journey. We may also see this directly from the diagram, as we can tell that the car does not move equal distances in equal intervals of time—the distance moved by the car in the second interval is greater than the distance moved in the first interval.

Let us consider another car that travels the same total 16 m distance as the first, but this second car travels at a constant speed. The motion of the second car is shown in the diagram below:

Again, we may calculate the average speed over each of the 1-second intervals shown.

For the first interval, we have and , giving . The car begins at 0 metres and ends the interval at 8 metres, so we have and . This gives us .

Calculating the average speed over this interval, we have

For the second interval, we have and , which gives . The car begins this interval at 8 metres and ends it at 16 metres, so we have and . This gives .

The average speed over the second interval is given by

So, for this car that moves at a constant speed, we have found the same average speed for both of the sections of its motion.

This result holds more generally. For any object moving at a constant speed, the values of distance and time are in the same proportion for any part of the motion. This means that the constant speed equation may be applied to any section of the object’s motion and will give the same result for the speed no matter what section we consider.

Let us look at a couple more example problems.

Example 3: Comparing Average Speeds

The speed of a toy car is measured by recording its position each second. How does the average speed of the car in the first second compare to its average speed throughout the whole journey?

1. The average speed in the first second is greater than the average speed during the time the speed is measured.
2. The average speed in the first second is equal to the average speed during the time the speed is measured.
3. The average speed in the first second is less than the average speed during the time the speed is measured.

Answer

In this question, we are asked to work out how the average speed of the car in the first second compares to the average speed over the entire time that the speed is measured.

To do this, we will work out what the average speed of the car is during this first interval. We will then work out the average speed over the entire time that the speed is measured for. Finally, we will compare the two values.

During the first interval, we can see from the diagram that the car moves a distance of 1 m. This means that, for this interval, we have and .

We have used a subscript “1” to indicate that these values are for the first interval.

We can recall our equation for the average speed :

Substituting in and , we get that , the average speed over the first interval, is given by

Now let us consider the entire time over which the speed is measured.

We can see from the diagram that the speed is measured for a total time of 3 s and that the car moves a total distance of 3 m in this time. This means that we have and .

We have used a subscript “T” to indicate that these values are for the total time for which the speed is measured.

Substituting and into the equation for average speed, we get that , the average speed over the total time, is given by

So, we have found that and . In other words, both these average speeds have the same value.

Therefore, our answer to the question is that the average speed in the first second is equal to the average speed during the time the speed is measured. This is the answer given in option B.

Example 4: Working Out Which of Two Objects Has the Greater Average Speed

A blue object and an orange object move across a grid of equally spaced lines. Both objects move for 5 seconds. The arrows show the distances moved each second. Which color object has a greater average speed?

1. The blue object
2. Both objects have the same average speed.
3. The orange object

Answer

In this question, we are given a diagram showing two objects moving across a grid. We are asked to work out which object has the greater average speed.

We can see that the two objects move differently during the time shown. The orange object moves a distance of 1 square in each second; therefore, it moves at a constant speed. Meanwhile, the blue object does not move equal amounts of squares in each second of time; therefore, it does not move at a constant speed.

However, we can see that the total distance moved by each of the objects is the same. It is equal to 5 squares on the grid. We do not know how much each square is in metres, but we do know that the lines on the grid are equally spaced. Therefore, we can talk about the distance in terms of units of “squares.”

The question tells us that each object moves for 5 s.

Since each object moves for the same distance (5 squares) and takes the same time to do this (5 seconds), we know that each object must have the same average speed.

We can explicitly calculate this average speed, in units of squares per second, using the equation for average speed:

In this case, and . This gives us an average speed of

We stress that the total distance and the total time are the same for both the orange object and the blue object. So, this average speed calculation applies for both objects.

Our answer to the question is therefore that both objects have the same average speed. This is the answer given in option B.

Example 4 highlights an interesting point about average speed. The average speed of an object over a given time interval is defined using the total distance traveled and the total length of time of the interval. It does not matter how the speed varies within that time interval; all that matters is the average value.

This means that it is possible for two objects to have the same average speed as each other over a given interval of time, even if within that interval the speed of each object changes in a different way.

We have seen how we may calculate the average speed of an object given the total distance moved by that object over a total time interval of .

We can also rearrange the equation in order to make either or the subject. These rearrangements work in exactly the same way as when rearranging the equation for motion at a constant speed.

Making the subject, we have the following equation:

If we know the average speed of an object and we know how much time the object moves for, this equation allows us to calculate the total distance moved by that object. This holds true whether or not the speed of the object is constant; we only need to know the average speed over the time interval .

Making the subject, we get the following equation:

Then, if we know the average speed of an object and the total distance that the object travels, this equation enables us to calculate how much time it takes to travel this distance. As before, it does not matter whether or not the speed is constant; we only need to know the average for the distance traveled.

Let us consider a concrete example where we need to use one of these rearranged equations.

Suppose we have a runner who competes in a 400 m race and runs at an average speed of 8 m/s. We want to work out how long it takes them to complete the race.

In this case, we have an average speed of and a total distance moved of . We want to find the value of . This means we need our average speed equation rearranged so that is the subject:

Then, we can substitute in our values of and :

Finally, evaluating the right-hand side, we have that

Therefore, we have found that the runner takes a time of 50 s to complete the race.

Let us now summarize what has been learned in this explainer.