How is the average speed of the car. How to find the average speed of a car after driving in different modes

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 this is 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:

In this way

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 decision. 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.

Very simple! You need to divide the entire path by the time that the object of movement was on the way. Expressed differently, we can define the average speed as the arithmetic mean of all the speeds of the object. But there are some nuances in solving problems in this area.

For example, to calculate the average speed, the following version of the problem is given: the traveler first walked at a speed of 4 km per hour for an hour. Then a passing car "picked up" him, and he drove the rest of the way in 15 minutes. And the car was moving at a speed of 60 km per hour. How to determine the average traveler's speed?

You should not just add 4 km and 60 and divide them in half, this will be the wrong solution! After all, the paths traveled on foot and by car are unknown to us. So, first you need to calculate the entire path.

The first part of the path is easy to find: 4 km per hour X 1 hour = 4 km

There are minor problems with the second part of the journey: the speed is expressed in hours, and the travel time is in minutes. This nuance often makes it difficult to find the right answer when questions are posed, how to find the average speed, path or time.

Express 15 minutes in hours. For this 15 minutes: 60 minutes = 0.25 hours. Now let's calculate what way the traveler did on a ride?

60 km/h X 0.25 h = 15 km

Now it will not be difficult to find the entire path covered by the traveler: 15 km + 4 km = 19 km.

The travel time is also fairly easy to calculate. This is 1 hour + 0.25 hours = 1.25 hours.

And now it is already clear how to find the average speed: you need to divide the entire path by the time that the traveler spent to overcome it. That is, 19 km: 1.25 hours = 15.2 km/h.

There is such an anecdote in the subject. A man hurrying on asks the owner of the field: “Can I go to the station through your site? I'm a bit late and would like to shorten my path by going straight ahead. Then I will definitely make it to the train, which leaves at 16:45!” “Of course you can shorten your path by passing through my meadow! And if my bull notices you there, then you will even have time for that train that leaves at 16 hours and 15 minutes.

This comical situation, meanwhile, is directly related to such a mathematical concept as the average speed of movement. After all, a potential passenger is trying to shorten his path for the simple reason that he knows the average speed of his movement, for example, 5 km per hour. And the pedestrian, knowing that the detour along the asphalt road is 7.5 km, mentally simple calculations, understands that it will take him an hour and a half on this road (7.5 km: 5 km / h = 1.5 hour).

He, leaving the house too late, is limited in time, and therefore decides to shorten his path.

And here we are faced with the first rule that dictates to us how to find the average speed of movement: taking into account the direct distance between the extreme points of the path, or precisely calculating From the above, it is clear to everyone: one should calculate, taking into account exactly the trajectory of the path.

Shortening the path, but not changing its average speed, the object in the face of a pedestrian receives a gain in time. The farmer, assuming the average speed of the “sprinter” running away from the angry bull, also makes simple calculations and gives his result.

Motorists often use the second, important, rule for calculating the average speed, which concerns the time spent on the road. This relates to the question of how to find the average speed in case the object has stops along the way.

In this option, usually, if there are no additional clarifications, for the calculation they take full time including stops. Therefore, a car driver can say that his average speed in the morning on a free road is much higher than the average speed in rush hour, although the speedometer shows the same figure in both cases.

Knowing these figures, an experienced driver will never be late anywhere, having assumed in advance what his average speed of movement in the city will be. different time days.

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 instance, artificial satellite Earth overcomes about 8 km in 1 second, and tectonic plates, on which the continents are located, according to the measurements of 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, and general principle no.

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 problem is often used in 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.

Remember that speed is given by both a numerical value and a direction. Velocity describes the rate of change in the position of a body, as well as the direction in which this body is moving. For example, 100 m/s (to the south).

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 way at a speed ...", or "the pedestrian walked the first third of the way 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 values 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. V 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: observant readers did not go unnoticed practical significance using the correct algorithm. The correctly calculated average speed in the given examples turned out to be slightly lower " average temperature"on the highway. Therefore, a false algorithm for systems that record speeding would mean a greater number of erroneous traffic police decisions sent in "letters of happiness" to drivers.