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.
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
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, an artificial satellite of the 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 movement is often encountered 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 take place at an uneven speed, sometimes accelerating, sometimes 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.
To calculate average speed, use a simple formula: Speed = Distance traveled Time (\displaystyle (\text(Speed))=(\frac (\text(Distance traveled))(\text(Time)))). But in some tasks two speed values are given - on different parts of the distance traveled or at different time intervals. In these cases, you need to use other formulas to calculate the average speed. The skills for solving such problems can be useful in real life, and the problems themselves can be encountered in exams, so memorize the formulas and understand the principles of solving problems.
Steps
One path value and one time value
- the length of the path traveled by the body;
- the time it took the body to travel this path.
- For example: a car traveled 150 km in 3 hours. Find the average speed of the car.
-
Formula: where v (\displaystyle v)- average speed, s (\displaystyle s)- distance traveled, t (\displaystyle t)- the time it took to travel.
Substitute the distance traveled into the formula. Substitute the path value for s (\displaystyle s).
- In our example, the car has traveled 150 km. The formula will be written like this: v = 150 t (\displaystyle v=(\frac (150)(t))).
-
Plug in the time into the formula. Substitute the time value for t (\displaystyle t).
- In our example, the car drove for 3 hours. The formula will be written as follows:.
-
Divide the path by the time. You will find the average speed (usually it is measured in kilometers per hour).
- In our example:
v = 150 3 (\displaystyle v=(\frac (150)(3)))
Thus, if a car traveled 150 km in 3 hours, then it was moving at an average speed of 50 km/h.
- In our example:
-
Calculate the total distance travelled. To do this, add up the values of the traveled sections of the path. Substitute the total distance traveled into the formula (instead of s (\displaystyle s)).
- In our example, the car has traveled 150 km, 120 km and 70 km. Total distance traveled: .
-
T (\displaystyle t)).
- . Thus, the formula will be written as:.
-
- In our example:
v = 340 6 (\displaystyle v=(\frac (340)(6)))
Thus, if a car traveled 150 km in 3 hours, 120 km in 2 hours, 70 km in 1 hour, then it was moving at an average speed of 57 km/h (rounded).
- In our example:
Multiple speeds and multiple times
-
Look at these values. Use this method if the following quantities are given:
Write down the formula for calculating the average speed. Formula: v = s t (\displaystyle v=(\frac (s)(t))), where v (\displaystyle v)- average speed, s (\displaystyle s)- total distance travelled, t (\displaystyle t) is the total time it took to travel.
-
Calculate the common path. To do this, multiply each speed by the corresponding time. This will give you the length of each section of the path. To calculate the total path, add the values of the path segments traveled. Substitute the total distance traveled into the formula (instead of s (\displaystyle s)).
- For instance:
50 km/h for 3 h = 50 × 3 = 150 (\displaystyle 50\times 3=150) km
60 km/h for 2 h = 60 × 2 = 120 (\displaystyle 60\times 2=120) km
70 km/h for 1 h = 70 × 1 = 70 (\displaystyle 70\times 1=70) km
Total distance covered: 150 + 120 + 70 = 340 (\displaystyle 150+120+70=340) km. Thus, the formula will be written as: v = 340 t (\displaystyle v=(\frac (340)(t))).
- For instance:
-
Calculate the total travel time. To do this, add the values of the time for which each section of the path was covered. Plug the total time into the formula (instead of t (\displaystyle t)).
- In our example, the car drove for 3 hours, 2 hours and 1 hour. The total travel time is: 3 + 2 + 1 = 6 (\displaystyle 3+2+1=6). Thus, the formula will be written as: v = 340 6 (\displaystyle v=(\frac (340)(6))).
-
Divide the total distance by the total time. You will find the average speed.
- In our example:
v = 340 6 (\displaystyle v=(\frac (340)(6)))
v = 56 , 67 (\displaystyle v=56,67)
Thus, if a car was moving at a speed of 50 km/h for 3 hours, at a speed of 60 km/h for 2 hours, at a speed of 70 km/h for 1 hour, then it was moving at an average speed of 57 km/h ( rounded).
- In our example:
By two speeds and two identical times
-
Look at these values. Use this method if the following quantities and conditions are given:
- two or more speeds with which the body moved;
- a body moves at certain speeds for equal periods of time.
- For example: a car traveled at a speed of 40 km/h for 2 hours and at a speed of 60 km/h for another 2 hours. Find the average speed of the car for the entire journey.
-
Write down the formula for calculating the average speed given two speeds at which a body moves for equal periods of time. Formula: v = a + b 2 (\displaystyle v=(\frac (a+b)(2))), where v (\displaystyle v)- average speed, a (\displaystyle a)- the speed of the body during the first period of time, b (\displaystyle b)- the speed of the body during the second (same as the first) period of time.
- In such tasks, the values of time intervals are not important - the main thing is that they are equal.
- Given multiple velocities and equal time intervals, rewrite the formula as follows: v = a + b + c 3 (\displaystyle v=(\frac (a+b+c)(3))) or v = a + b + c + d 4 (\displaystyle v=(\frac (a+b+c+d)(4))). If the time intervals are equal, add up all the speed values and divide them by the number of such values.
-
Substitute the speed values into the formula. It doesn't matter what value to substitute for a (\displaystyle a), and which one instead of b (\displaystyle b).
- For example, if the first speed is 40 km/h and the second speed is 60 km/h, the formula would be: .
-
Add up the two speeds. Then divide the sum by two. You will find the average speed for the entire journey.
- For instance:
v = 40 + 60 2 (\displaystyle v=(\frac (40+60)(2)))
v = 100 2 (\displaystyle v=(\frac (100)(2)))
v=50 (\displaystyle v=50)
Thus, if the car was traveling at 40 km/h for 2 hours and at 60 km/h for another 2 hours, the average speed of the car for the entire journey was 50 km/h.
- For instance:
The concept of speed is one of the main concepts in kinematics.
Many people probably know that speed is a physical quantity that shows how fast (or how slowly) a moving body moves in space. Of course, we are talking about moving in the chosen reference system. Do you know, however, that not one, but three concepts of speed are used? There is a speed at a given moment of time, called instantaneous speed, and there are two concepts of average speed over a given period of time - the average ground speed (in English speed) and the average speed of movement (in English velocity).
We will consider a material point in the coordinate system x, y, z(Fig. a).
Position A points at time t characterize by coordinates x(t), y(t), z(t), representing the three components of the radius vector ( t). The point moves, its position in the selected coordinate system changes over time - the end of the radius vector ( t) describes a curve called the trajectory of the moving point.
The trajectory described for the time interval from t before t + Δt shown in figure b. 
Across B indicates the position of the point at the moment t + Δt(it is fixed by the radius vector ( t + Δt)). Let Δs is the length of the curvilinear trajectory under consideration, i.e. the path traveled by the point in the time from t before t + Δt.
The average ground speed of a point for a given period of time is determined by the ratio 
It's obvious that v p− scalar value; it is characterized only by a numerical value.
The vector shown in figure b
is called the displacement of a material point in time from t before t + Δt.
The average speed of movement for a given period of time is determined by the ratio 
It's obvious that v cf− vector quantity. vector direction v cf coincides with the direction of movement Δr.
Note that in the case of rectilinear motion, the average ground speed of the moving point coincides with the modulus of the average speed in displacement.
The movement of a point along a rectilinear or curvilinear trajectory is called uniform if, in relation (1), the value vп does not depend on Δt. If, for example, we reduce Δt 2 times, then the length of the path traveled by the point Δs will decrease by 2 times. In uniform motion, a point travels a path of equal length in equal time intervals.
Question:
Can we assume that with a uniform motion of a point from Δt does not also depend on the vector cp of the average velocity with respect to displacement?
Answer:
This can be considered only in the case of rectilinear motion (in this case, we recall that the modulus of the average speed for displacement is equal to the average ground speed). If the uniform motion is performed along a curvilinear trajectory, then with a change in the averaging interval Δt both the modulus and the direction of the average velocity vector along the displacement will change. With uniform curvilinear motion equal time intervals Δt will correspond to different displacement vectors Δr(and hence different vectors v cf).
True, in the case of uniform motion along a circle, equal time intervals will correspond to equal values of the displacement modulus |r|(and therefore equal |v cf |). But the directions of displacements (and hence the vectors v cf) and in this case will be different for the same Δt. This is seen in the figure 
Where a point uniformly moving along a circle describes equal arcs in equal intervals of time AB, BC, CD. Although the displacement vectors 1
, 2
, 3
have the same modules, but their directions are different, so there is no need to talk about the equality of these vectors.
Note
Of the two average speeds in problems, the average ground speed is usually considered, and the average travel speed is used quite rarely. However, it deserves attention, since it allows us to introduce the concept of instantaneous speed.
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 "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 first third of the path the pedestrian walked 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(a periodic decimal, the result is mathematically not very pretty).
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 a greater number of erroneous traffic police decisions sent in "letters of happiness" to drivers.