At school, each of us came across a problem similar to the following. If the car moved part of the way at one speed, and the next segment of the road at another, how to find the average speed?
What is this value and why is it needed? Let's try to figure this out.
Speed in physics is a quantity that describes the amount of distance traveled per unit of time. That is, when they say that the speed of a pedestrian is 5 km / h, this means that he travels a distance of 5 km in 1 hour.
The formula for finding speed looks like this:
V=S/t, where S is the distance traveled, t is the time.
There is no single dimension in this formula, since it describes both extremely slow and very fast processes.
For example, artificial satellite Earth overcomes about 8 km in 1 second, and the tectonic plates on which the continents are located, according to scientists, diverge by only a few millimeters per year. Therefore, the dimensions of the speed can be different - km / h, m / s, mm / s, etc.
The principle is that the distance is divided by the time required to overcome the path. Do not forget about the dimension if complex calculations are carried out.
In order not to get confused and not make a mistake in the answer, all values are given in the same units of measurement. If the length of the path is indicated in kilometers, and some part of it is in centimeters, then until we get unity in dimension, we will not know the correct answer.
constant speed
Description of the formula.
The simplest case in physics is uniform motion. The speed is constant, does not change throughout the journey. There are even speed constants, summarized in tables - unchanged values. For example, sound propagates in air at a speed of 340.3 m/s.
And light is the absolute champion in this regard, it has the highest speed in our Universe - 300,000 km / s. These values do not change from the starting point of the movement to the end point. They depend only on the medium in which they move (air, vacuum, water, etc.).
Uniform motion often occurs to us in Everyday life. This is how a conveyor works in a plant or factory, a funicular on mountain routes, an elevator (with the exception of very short periods of start and stop).
The graph of such a movement is very simple and is a straight line. 1 second - 1 m, 2 seconds - 2 m, 100 seconds - 100 m. All points are on the same straight line.
uneven speed
Unfortunately, this is ideal both in life and in physics is extremely rare. Many processes run uneven speed, then speeding up, then slowing down.
Let's imagine the movement of an ordinary intercity bus. At the beginning of the journey, it accelerates, slows down at traffic lights, or even stops altogether. Then it goes faster outside the city, but slower on the rises, and accelerates again on the descents.
If you depict this process in the form of a graph, you get a very intricate line. It is possible to determine the speed from the graph only for a specific point, but there is no general principle.
You will need a whole set of formulas, each of which is suitable only for its section of the drawing. But there is nothing terrible. To describe the movement of the bus, the average value is used.
You can find the average speed of movement using the same formula. Indeed, we know the distance between the bus stations, measured the travel time. By dividing one by the other, find the desired value.
What is it for?
Such calculations are useful to everyone. We plan our day and travel all the time. Having a dacha outside the city, it makes sense to find out the average ground speed when traveling there.
This will make it easier to plan your holiday. By learning to find this value, we can be more punctual, stop being late.
Let's return to the example proposed at the very beginning, when the car traveled part of the way at one speed, and another part at a different one. This type of task is very often used in the school curriculum. Therefore, when your child asks you to help him solve a similar issue, it will be easy for you to do it.
Adding the lengths of the sections of the path, you get the total distance. By dividing their values by the speeds indicated in the initial data, it is possible to determine the time spent on each of the sections. Adding them together, we get the time spent on the whole journey.
This article is about how to find the average speed. The definition of this concept is given, and two important particular cases of finding the average speed are considered. A detailed analysis of tasks for finding the average speed of a body from a tutor in mathematics and physics is presented.
Determination of average speed
medium speed the movement of the body is called the ratio of the path traveled by the body to the time during which the body moved:
Let's learn how to find it on the example of the following problem:
Please note that in this case this value did not coincide with the arithmetic mean of the speeds and , which is equal to:
m/s.
Special cases of finding the average speed
1. Two identical sections of the path. Let the body move the first half of the way with the speed , and the second half of the way — with the speed . It is required to find the average speed of the body.
2. Two identical movement intervals. Let the body move at a speed for a certain period of time, and then began to move at a speed for the same period of time. It is required to find the average speed of the body.
Here we got the only case when the average speed of movement coincided with the arithmetic average speeds and on two sections of the path.
Let's solve the problem in the end All-Russian Olympiad schoolchildren in physics, which took place last year, which is related to the topic of our lesson today.
| The body moved with, and the average speed of movement was 4 m/s. It is known that for the last few seconds the average velocity of the same body was 10 m/s. Determine the average speed of the body for the first s of movement. |
The distance traveled by the body is: 
m. You can also find the path that the body has traveled for the last since its movement: m. Then for the first since its movement, the body has overcome the path in m. Therefore, the average speed on this section of the path was: 
m/s.
They like to offer tasks for finding the average speed of movement at the Unified State Examination and the OGE in physics, entrance exams, and olympiads. Every student should learn how to solve these problems if he plans to continue his education at the university. A knowledgeable friend, a school teacher or a tutor in mathematics and physics can help to cope with this task. Good luck with your physics studies!
Sergey Valerievich
Tasks for average speed (hereinafter referred to as SC). We have already considered tasks for rectilinear motion. I recommend to look at the articles "" and "". Typical tasks for average speed are a group of tasks for movement, they are included in the exam in mathematics, and such a task may well be in front of you at the time of the exam itself. Problems are simple and quickly solved.
The meaning is this: imagine an object of movement, such as a car. It passes certain sections of the path at different speeds. The whole journey takes some time. So: the average speed is such a constant speed with which the car would cover a given distance in the same time. That is, the formula for the average speed is as follows:
If there were two sections of the path, then
If three, then respectively:
* In the denominator, we summarize the time, and in the numerator, the distances traveled for the corresponding time intervals.
The car drove the first third of the track at a speed of 90 km/h, the second third at a speed of 60 km/h, and the last third at a speed of 45 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.
As already mentioned, it is necessary to divide the entire path by the entire time of movement. The condition says about three sections of the path. Formula:
Denote the whole let S. Then the car drove the first third of the way:
The car drove the second third of the way:
The car drove the last third of the way:
Thus
Decide for yourself:
The car drove the first third of the track at a speed of 60 km/h, the second third at a speed of 120 km/h, and the last third at a speed of 110 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.
The first hour the car drove at a speed of 100 km/h, the next two hours at a speed of 90 km/h, and then for two hours at a speed of 80 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.
The condition says about three sections of the path. We will search for the SC by the formula:
The sections of the path are not given to us, but we can easily calculate them:
The first section of the path was 1∙100 = 100 kilometers.
The second section of the path was 2∙90 = 180 kilometers.
The third section of the path was 2∙80 = 160 kilometers.
Calculate speed:
Decide for yourself:
For the first two hours the car was traveling at a speed of 50 km/h, the next hour at a speed of 100 km/h, and then for two hours at a speed of 75 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.
The car drove the first 120 km at a speed of 60 km/h, the next 120 km at a speed of 80 km/h, and then 150 km at a speed of 100 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.
It is said about three sections of the path. Formula:
The length of the sections is given. Let's determine the time that the car spent on each section: 120/60 hours were spent on the first section, 120/80 hours on the second section, and 150/100 hours on the third. Calculate speed:
Decide for yourself:
The first 190 km the car drove at a speed of 50 km/h, the next 180 km - at a speed of 90 km/h, and then 170 km - at a speed of 100 km/h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.
Half the time spent on the road, the car was traveling at a speed of 74 km / h, and the second half of the time - at a speed of 66 km / h. Locate the vehicle's SK throughout the journey. Give your answer in km/h.
*There is a problem about a traveler who crossed the sea. The guys have problems with the solution. If you do not see it, then register on the site! The registration (login) button is located in the MAIN MENU of the site. After registration, log in to the site and refresh this page.
The traveler crossed the sea on a yacht with average speed 17 km/h. He flew back on a sports plane at a speed of 323 km / h. Find the traveler's average speed for the entire journey. Give your answer in km/h.
Sincerely, Alexander.
P.S: I would be grateful if you tell about the site in social networks.
There are average values, the incorrect definition of which has become an anecdote or a parable. Any incorrectly made calculations are commented on by a commonly understood reference to such a deliberately absurd result. Everyone, for example, will cause a smile of sarcastic understanding of the phrase "average temperature in the hospital." However, the same experts often, without hesitation, add up the speeds on separate sections of the path and divide the calculated sum by the number of these sections in order to get an equally meaningless answer. Recall from the course of mechanics high school how to find the average speed in the right way and not in an absurd way.
Analogue of "average temperature" in mechanics
In what cases do the cunningly formulated conditions of the problem push us to a hasty, thoughtless answer? If it is said about "parts" of the path, but their length is not indicated, this alarms even a person who is not very experienced in solving such examples. But if the task directly indicates equal intervals, for example, "the train followed the first half of the path at a speed ...", or "the pedestrian walked the first third of the path at a speed ...", and then it details how the object moved on the remaining equal areas, that is, the ratio is known S 1 \u003d S 2 \u003d ... \u003d S n and exact speeds v 1, v 2, ... v n, our thinking often gives an unforgivable misfire. The arithmetic mean of the speeds is considered, that is, all known values v add up and divide into n. As a result, the answer is wrong.
Simple "formulas" for calculating quantities in uniform motion
And for the entire distance traveled, and for its individual sections, in the case of averaging the speed, the relations written for uniform motion are valid:
- S=vt(1), the "formula" of the path;
- t=S/v(2), "formula" for calculating the time of movement ;
- v=S/t(3), "formula" for determining the average speed on the track section S passed during the time t.
That is, to find the desired value v using relation (3), we need to know exactly the other two. It is precisely when solving the question of how to find the average speed of movement that we first of all must determine what the entire distance traveled is S and what is the whole time of movement t.
Mathematical detection of latent error
In the example we are solving, the path traveled by the body (train or pedestrian) will be equal to the product nS n(because we n once we add up equal sections of the path, in the examples given - halves, n=2, or thirds, n=3). We do not know anything about the total travel time. How to determine the average speed if the denominator of the fraction (3) is not explicitly set? We use relation (2), for each section of the path we determine t n = S n: v n. Amount the time intervals calculated in this way will be written under the line of the fraction (3). It is clear that in order to get rid of the "+" signs, you need to give all S n: v n to a common denominator. The result is a "two-story fraction". Next, we use the rule: the denominator of the denominator goes into the numerator. As a result, for the problem with the train after the reduction by S n we have v cf \u003d nv 1 v 2: v 1 + v 2, n \u003d 2 (4) . For the case of a pedestrian, the question of how to find the average speed is even more difficult to solve: v cf \u003d nv 1 v 2 v 3: v 1v2 + v 2 v 3 + v 3 v 1,n=3(5).
Explicit confirmation of the error "in numbers"
In order to "on the fingers" confirm that the definition of the arithmetic mean is an erroneous way when calculating vWed, we concretize the example by replacing abstract letters with numbers. For the train, take the speed 40 km/h And 60 km/h(wrong answer - 50 km/h). For the pedestrian 5 , 6 And 4 km/h(average - 5 km/h). It is easy to see, by substituting the values in relations (4) and (5), that the correct answers are for the locomotive 48 km/h and for a human 4,(864) km/h(periodic decimal, the result is mathematically not very beautiful).
When the arithmetic mean fails
If the problem is formulated as follows: "For equal intervals of time, the body first moved with a speed v1, then v2, v 3 and so on", a quick answer to the question of how to find the average speed can be found in the wrong way. Let the reader see for himself by summing equal periods of time in the denominator and using in the numerator v cf relation (1). This is perhaps the only case when an erroneous method leads to a correct result. But for guaranteed accurate calculations, you need to use the only correct algorithm, invariably referring to the fraction v cf = S: t.
Algorithm for all occasions
In order to avoid mistakes for sure, when solving the question of how to find the average speed, it is enough to remember and follow a simple sequence of actions:
- determine the entire path by summing the lengths of its individual sections;
- set all the way;
- divide the first result by the second, the unknown values not specified in the problem are reduced in this case (subject to the correct formulation of the conditions).
The article considers the simplest cases when the initial data are given for equal parts of the time or equal sections of the path. In the general case, the ratio of chronological intervals or distances covered by the body can be the most arbitrary (but mathematically defined, expressed as a specific integer or fraction). The rule for referring to the ratio v cf = S: t absolutely universal and never fails, no matter how complicated at first glance algebraic transformations have to be performed.
Finally, we note that for observant readers, the practical significance of using the correct algorithm has not gone unnoticed. Correctly calculated average speed in the above examples turned out to be slightly lower than the "average temperature" on the track. Therefore, a false algorithm for systems that record speeding would mean more erroneous traffic police regulations sent in "letters of happiness" to drivers.
