How is the average speed found? How to find average speed

There are average values, the incorrect definition of which has become a joke or a parable. Any incorrect calculations are commented on with a common, generally understood reference to such an obviously absurd result. For example, the phrase “average temperature in the hospital” will make everyone smile with sarcastic understanding. However, the same experts often, without thinking, add up the speeds on individual sections of the route and divide the calculated sum by the number of these sections in order to get an equally meaningless answer. Let us recall from the high school mechanics course how to find the average speed in the correct, not absurd, way.

Analogue of "average temperature" in mechanics

In what cases do the tricky conditions of a problem push us to a hasty, thoughtless answer? If they talk about “parts” of the path, but do not indicate their length, this alarms even a person who is little experienced in solving such examples. But if the problem directly indicates equal intervals, for example, “for the first half of the path the train followed at a speed...”, or “the pedestrian walked the first third of the path at a speed...”, and then describes in detail how the object moved at the remaining equal intervals. areas, that is, the ratio is known S 1 = S 2 = ... = S n and exact speed values v 1, v 2, ... v n, our thinking often misfires unforgivably. 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 turns out to be incorrect.

Simple “formulas” for calculating quantities during uniform motion

Both 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), "formula" path;
t=S/v(2), "formula" for calculating movement time ;
v=S/t(3), “formula” for determining the average speed on a section of track S traversed in time t.

That is, to find the desired value v using relation (3), we need to know the other two exactly. It is 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 entire movement time? t.

Mathematical Hidden Error Detection

In the example we are solving, the distance traveled by the body (train or pedestrian) will be equal to the product nS n(since we n once we add up equal sections of the path, in the given examples - halves, n=2, or thirds, n=3). We know nothing about the total time of movement. How to determine the average speed if the denominator of the fraction (3) is not explicitly specified? Let us use relation (2), for each section of the path we determine t n = S n: v n. Amount We will write the time intervals calculated in this way under the line of the fraction (3). It is clear that in order to get rid of the "+" signs, you need to give everything 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 train problem after reduction by S n we have v av = nv 1 v 2: v 1 + v 2, n = 2 (4) . For the case of a pedestrian, the question of how to find the average speed is even more difficult to solve: v av = 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 confirm with one’s fingers that determining the arithmetic mean is the wrong way to do calculations vWed, let’s make the example more concrete by replacing abstract letters with numbers. For the train, let's take the speeds 40 km/h And 60 km/h(wrong answer - 50 km/h). For a pedestrian - 5 , 6 And 4 km/h(average - 5 km/h). It is easy to verify by substituting the values in relations (4) and (5) that the correct answers are for the locomotive 48 km/h and for a person - 4.(864) km/h(periodic decimal fraction, the result is not very mathematically beautiful).

When the arithmetic mean does not fail

If the problem is formulated as follows: “For equal intervals of time, the body first moved with speed v 1, then v 2, 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. We will let the reader see this for himself by summing up equal time intervals in the denominator and using in the numerator v avg 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 turning to the fraction v av = S: t.

Algorithm for all occasions

In order to definitely avoid mistakes, when deciding 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 travel time;
divide the first result by the second, the unknown quantities not specified in the problem (subject to the correct formulation of the conditions) are reduced.

The article discusses the simplest cases when the initial data are given for equal shares of time or equal sections of the path. In the general case, the ratio of chronological intervals or distances traveled by a body can be very arbitrary (but at the same time mathematically defined, expressed as a specific integer or fraction). Rule for referring to ratio v av = S: t absolutely universal and never fails, no matter how complex algebraic transformations have to be performed at first glance.

Finally, we note: the practical significance of using the right algorithm has not gone unnoticed by observant readers. The correctly calculated average speed in the examples given turned out to be slightly lower than the “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 “chain letters” to drivers.

Remember that speed is given by both a numerical value and a direction. Velocity describes how quickly a body's position changes, as well as the direction in which that body is moving. For example, 100 m/s (south).

Find the total displacement, that is, the distance and direction between the starting and ending 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 a 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 measurement, 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 of measurement, it is better to first calculate the speed and then convert it to m/s.

Calculate the average speed. If you know the displacement value and the total travel time, you can calculate the average speed using the formula v av = Δs/Δt. In our example, the average speed of the rocket 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 v av = Δs/Δt the symbol "delta" (Δ) means "change in magnitude", that is, Δs/Δt means "change in position to change in time".
The average speed can be written as v av or as v with a horizontal bar on top.

Solving more complex problems, 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 starting and ending 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 1 m/s for 2 seconds, then instantly accelerates to 3 m/s and continues to walk west for 2 seconds. Its total displacement is (1 m/s)(2 s) + (3 m/s)(2 s) = 8 m (to the west). Total travel time: 2 s + 2 s = 4 s. 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 consider the movement to the east as a "negative movement" to the west, so the total movement is (5 m/s)(3 s) + (-7 m/s)(1 s) = 8 meters. The total time is 4 s. 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, so the total movement is 8 m. The total travel time was 4 seconds. Average speed is 8 m (west) / 4 s = 2 m/s (west).

2 . The skier covered the first section, 120 m long, in 2 minutes, and the second, 27 m long, he covered in 1.5 minutes. Find the average speed of the skier along the entire route.

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

Level B

1 . What kind of speed - average or instantaneous - are we talking about 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) on the road section there is a maximum speed limiter of 60 km/h;

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

3 . A self-propelled mower has a cutting width of 10 m. Determine the area of the field mowed 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 drove 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 along the entire journey?

5 . For the first half of the time, when moving from one point to another, a cyclist rode at a speed of 12 km/h, and for the second half of the time (due to a punctured tire) 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, and the rest of the time at a speed of 7 km/h. Determine the average speed of the student.

7 . A cyclist was traveling from one city to another. He drove half the way at a speed of 12 km/h, and the second half (due to a punctured tire) he walked at a speed of 4 km/h. Determine the average speed of its movement.

8 . The motorcyclist moved from one point to another at a speed of 60 km/h, and covered the return journey at a speed of 10 m/s. Determine the average speed of the motorcyclist for the entire period of movement.

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

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

11 . The speed of the train on the ascent is 30 km/h, and on the descent – 90 km/h. Determine the average speed along the entire route 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 should he move for the remaining time if the average speed is 65 km/h?

Average speed is the speed that is obtained if the entire path is divided by the time it takes the object to cover this path. Average speed formula:

V av = S/t.

S = S1 + S2 + S3 = v1*t1 + v2*t2 + v3*t3
V av = S/t = (v1*t1 + v2*t2 + v3*t3) / (t1 + t2 + t3)

To avoid confusion with hours and minutes, we convert all minutes to hours: 15 minutes. = 0.4 hour, 36 min. = 0.6 hour. Substitute the numerical values into the last formula:

V av = (20*0.4 + 0.5*6 + 0.6*15) / (0.4 + 0.5 + 0.6) = (8 + 3 + 9) / (0.4 + 0.5 + 0.6) = 20 / 1.5 = 13.3 km/h

Answer: average speed V av = 13.3 km/h.

How to find the average speed of an accelerating motion

If the speed at the beginning of the movement differs from the speed at the end, such movement is called accelerated. Moreover, the body does not always actually move faster and faster. If the movement slows down, they still say that it is moving with acceleration, only the acceleration will be negative.

In other words, if a car, moving away, accelerated to a speed of 10 m/sec in a second, then its acceleration a is equal to 10 m per second per second a = 10 m/sec². If in the next second the car stops, then its acceleration is also equal to 10 m/sec², only with a minus sign: a = -10 m/sec².

The speed of movement with acceleration at the end of the time interval is calculated by the formula:

V = V0 ± at,

where V0 is the initial speed of movement, a is acceleration, t is the time during which this acceleration was observed. A plus or minus is placed in the formula depending on whether the speed increased or decreased.

The average speed over a period of time t is calculated as the arithmetic mean of the initial and final speeds:

V av = (V0 + V) / 2.

Finding the average speed: problem

The ball was pushed along a flat plane with an initial speed V0 = 5 m/sec. After 5 sec. the ball stopped. What are the acceleration and average speed?

Final speed of the ball V = 0 m/sec. The acceleration from the first formula is equal to

a = (V - V0)/ t = (0 - 5)/ 5 = - 1 m/sec².

Average speed V av = (V0 + V) / 2 = 5 /2 = 2.5 m/sec.