How to determine the average speed if the speed is known. How to find average speed

2 . The skier passed the first section 120 m long in 2 minutes, and he passed the second section 27 m long in 1.5 minutes. Find average speed skier's movements all the way.

3 . Moving along the highway, the cyclist traveled 20 km in 40 minutes, then he overcame a 600 m long country road in 2 minutes, and he traveled the remaining 39 km 400 m along the highway in 78 minutes. What is the average speed for the entire journey?

4 . The boy walked 1.2 km in 25 minutes, then rested for half an hour, and then ran another 800 m in 5 minutes. What was his average speed for the entire journey?

Level B

1 . About what speed - average or instantaneous - in question in the following cases:

a) a bullet flies out of a rifle at a speed of 800 m/s;

b) the speed of the Earth around the Sun is 30 km/s;

c) a maximum speed limiter of 60 km/h is installed on the road section;

d) a car drove past you at a speed of 72 km/h;

e) the bus covered the distance between Mogilev and Minsk at a speed of 50 km/h?

2 . An electric train travels 63 km from one station to another in 1 hour 10 minutes at an average speed of 70 km/h. How long do stops take?

3 . The self-propelled mower has a working width of 10 m. Determine the area of the field mown in 10 minutes if the average speed of the mower is 0.1 m/s.

4 . On a horizontal section of the road, the car traveled at a speed of 72 km/h for 10 minutes, and then drove uphill at a speed of 36 km/h for 20 minutes. What is the average speed for the entire journey?

5 . For the first half of the time, when moving from one point to another, the cyclist rode at a speed of 12 km/h, and for the second half of the time (due to a tire puncture) he walked at a speed of 4 km/h. Determine the average speed of the cyclist.

6 . The student traveled 1/3 of the total time on a bus at a speed of 60 km/h, another 1/3 of the total time on a bicycle at a speed of 20 km/h, the rest of the time he traveled at a speed of 7 km/h. Determine the average speed of the student.

7 . The cyclist was traveling from one city to another. He traveled half the way at a speed of 12 km/h, and the other half (due to a tire puncture) he walked at a speed of 4 km/h. Determine its average speed.

8 . A motorcyclist traveled from one point to another at a speed of 60 km/h and traveled back at a speed of 10 m/s. Determine the average speed of the motorcyclist for the entire journey.

9 . The student traveled 1/3 of the way on a bus at a speed of 40 km/h, another 1/3 of the way on a bicycle at a speed of 20 km/h, and covered the last third of the way at a speed of 10 km/h. Determine the average speed of the student.

10 . A pedestrian walked part of the way at a speed of 3 km/h, spending 2/3 of the time of his movement on this. The rest of the time he walked at a speed of 6 km / h. Determine the average speed.

11 . The speed of the train uphill is 30 km/h and downhill is 90 km/h. Determine the average speed for the entire section of the path if the descent is twice as long as the ascent.

12 . Half the time when moving from one point to another, the car moved at a constant speed of 60 km / h. At what constant speed must he move for the remaining time if the average speed is 65 km/h?

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).

Find the total displacement, i.e. the distance and direction between the start and end points of the path. As an example, consider a body moving at a constant speed in one direction.

For example, a rocket was launched in a northerly direction and moved for 5 minutes at a constant speed of 120 meters per minute. To calculate the total displacement, use the formula s = vt: (5 minutes) (120 m/min) = 600 m (North).
If the problem is given constant acceleration, use the formula s = vt + ½at 2 (the next section describes a simplified way to work with constant acceleration).

Find the total travel time. In our example, the rocket travels for 5 minutes. Average speed can be expressed in any unit of measure, but in the international system of units, speed is measured in meters per second (m/s). Convert minutes to seconds: (5 minutes) x (60 seconds/minute) = 300 seconds.

Even if in a scientific problem time is given in hours or other units, it is better to first calculate the speed and then convert it to m/s.

Calculate the average speed. If you know the value of the displacement and the total travel time, you can calculate the average speed using the formula v av = Δs/Δt. In our example, the average rocket speed is 600 m (North) / (300 seconds) = 2 m/s (North).

Be sure to indicate the direction of travel (for example, "forward" or "north").
In the formula vav = ∆s/∆t the symbol "delta" (Δ) means "change of magnitude", that is, Δs/Δt means "change of position to change of time".
The average speed can be written as v avg or as v with a horizontal bar over it.

Solution over challenging tasks, for example, if the body is rotating or the acceleration is not constant. In these cases, the average speed is still calculated as the ratio of total displacement to total time. It doesn't matter what happens to the body between the start and end points of the path. Here are some examples of problems with the same total displacement and total time (and therefore the same average speed).

Anna walks west at a speed of 1 m/s for 2 seconds, then instantly accelerates to 3 m/s and continues walking west for 2 seconds. Its total displacement is (1 m/s)(2 s) + (3 m/s)(2 s) = 8 m (westward). Total travel time: 2s + 2s = 4s. Her average speed: 8 m / 4 s = 2 m/s (west).
Boris walks west at 5 m/s for 3 seconds, then turns around and walks east at 7 m/s for 1 second. We can think of eastward movement as "negative movement" westward, so the total movement is (5 m/s)(3 s) + (-7 m/s)(1 s) = 8 meters. The total time is 4 s. The average speed is 8 m (west) / 4 s = 2 m/s (west).
Julia walks 1 meter north, then walks 8 meters west, and then walks 1 meter south. The total travel time is 4 seconds. Draw a diagram of this movement on paper and you will see that it ends 8 meters west of the starting point, that is, the total movement is 8 m. The total travel time was 4 seconds. The average speed is 8 m (west) / 4 s = 2 m/s (west).

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. Introduced detailed analysis tasks for finding the average speed of a body from a tutor in mathematics and physics.

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.

Finally, let's solve the problem from the All-Russian Olympiad for schoolchildren in physics, which took place last year, which is related to the topic of our today's lesson.

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

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 a high school mechanics course 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 the "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 is 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: observant readers did not go unnoticed practical significance using the right algorithm. The correctly calculated average speed in the given examples turned out to be slightly lower " 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.

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, having made mentally simple calculations, understands that he will need 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 how to find the average speed of movement: given the direct distance between extreme points way or precisely calculating From the above, it is clear to everyone: one should conduct a calculation, taking into account precisely 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, the full time is taken for calculation, 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 at different times of the day.