How to determine the average speed if the speed is known. What is the formula for calculating average speed?

Very simple! You need to divide the entire path by the time that the object of movement was on the way. In other words, one can define average speed as the arithmetic mean of all the velocities 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

With the second part of the way small problems: Speed is expressed in hours and driving time is expressed 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 possible to find the entire path covered by the traveler special work: 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 to us how to find the average speed of movement: given 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 you the 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.

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. Problem solving skills can be useful in real life, and the tasks themselves can be found in exams, so remember 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.
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).

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 example:
  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))).
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).

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 example:
  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.

The concept of speed is one of the main concepts in kinematics.
Many people probably know that speed is physical quantity, showing how fast (or how slowly) a moving body moves in space. Of course we are talking about displacement in the chosen reference system. Do you know, however, that not one, but three concepts of speed are used? There is speed in this moment time, called instantaneous speed, and there are two concepts of average speed for 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.

Through B indicates the position of the point at the moment t + Δt(it is fixed by the radius vector ( t + Δt)). Let be 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; it is characterized only by a numerical value.
The vector shown in figure b

called displacement material point 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 uniform motion around the 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.

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 your 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 measurement, but in international system speed units are measured in meters per second (m/s). Convert minutes to seconds: (5 minutes) x (60 seconds/minute) = 300 seconds.

Even if in scientific task time is given in hours or other units, it is better to calculate the speed first 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).

The average speed is the speed that is obtained if the entire path is divided by the time during which the object covered this path. Average speed formula:

V cf \u003d S / t.

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

In order not to be confused with hours and minutes, we translate all minutes into hours: 15 min. = 0.4 hour, 36 min. = 0.6 hour. Substitute the numerical values in the last formula:

V cf \u003d (20 * 0.4 + 0.5 * 6 + 0.6 * 15) / (0.4 + 0.5 + 0.6) \u003d (8 + 3 + 9) / (0.4 + 0.5 + 0.6) = 20 / 1.5 = 13.3 km/h

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

How to find the average speed of movement with acceleration

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

In other words, if the car, starting off, in a second accelerated to a speed of 10 m / s, then its acceleration is equal to 10 m per second per second a = 10 m / s². If in the next second the car stopped, then its acceleration is also equal to 10 m / s², only with a minus sign: a \u003d -10 m / s².

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 the acceleration, t is the time during which this acceleration was observed. Plus or minus in the formula is set depending on whether the speed increased or decreased.

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

Vav = (V0 + V) / 2.

Finding the average speed: task

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

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