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 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.

Through 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

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 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 average speed movement is rarely used. However, it deserves attention, since it allows us to introduce the concept of instantaneous speed.

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, accelerates to a speed of 10 m / s in a second, 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/s. 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

• a \u003d (V - V0) / t \u003d (0 - 5) / 5 \u003d - 1 m / s².

Average speed V cf \u003d (V0 + V) / 2 \u003d 5 / 2 \u003d 2.5 m / s.

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 are 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:

Thus

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.

All tasks in which there is movement of objects, their movement or rotation, are somehow connected with speed.

This term characterizes the movement of an object in space over a certain period of time - the number of units of distance per unit of time. He is a frequent "guest" of both sections of mathematics and physics. The original body can change its location both uniformly and with acceleration. In the first case, the speed is static and does not change during the movement, in the second, on the contrary, it increases or decreases.

How to find speed - uniform motion

If the speed of the movement of the body remained unchanged from the beginning of the movement to the end of the path, then we are talking about moving with constant acceleration - uniform movement. It can be straight or curved. In the first case, the trajectory of the body is a straight line.

Then V=S/t, where:

• V is the desired speed,
• S - distance traveled (total path),
• t is the total time of movement.

How to find speed - acceleration is constant

If an object was moving with acceleration, then its speed changed as it moved. In this case, the expression will help to find the desired value:

V \u003d V (beginning) + at, where:

• V (beginning) - the initial speed of the object,
• a is the acceleration of the body,
• t is the total travel time.

How to find speed - uneven motion

In this case, there is a situation when the body passes different parts of the path in different times.
S(1) - for t(1),
S(2) - for t(2), etc.

On the first section, the movement took place at a “tempo” V(1), on the second - V(2), and so on.

To find out the speed of an object moving all the way (its average value), use the expression:

How to find speed - rotation of an object

In the case of rotation, we are talking about the angular velocity, which determines the angle through which the element rotates per unit of time. The desired value is denoted by the symbol ω (rad / s).

• ω = Δφ/Δt, where:

Δφ – passed angle (angle increment),
Δt - elapsed time (movement time - time increment).

• If the rotation is uniform, the desired value (ω) is associated with such a concept as the period of rotation - how long will it take for our object to make 1 complete revolution. In this case:

ω = 2π/T, where:
π is a constant ≈3.14,
T is the period.

Or ω = 2πn, where:
π is a constant ≈3.14,
n is the frequency of circulation.

• With the known linear speed of the object for each point on the path of motion and the radius of the circle along which it moves, to find the speed ω, the following expression will be required:

ω = V/R, where:
V is the numerical value of the vector quantity (linear velocity),
R is the radius of the body's trajectory.

How to find speed - approaching and moving away points

In such tasks, it would be appropriate to use the terms approach speed and distance speed.

If the objects are heading towards each other, then the speed of approach (retreat) will be as follows:
V (approach) = V(1) + V(2), where V(1) and V(2) are the velocities of the corresponding objects.

If one of the bodies catches up with the other, then V (closer) = V(1) - V(2), V(1) is greater than V(2).

How to find speed - movement on a body of water

If events unfold on the water, then the speed of the current (i.e., the movement of water relative to a fixed shore) is added to the object’s own speed (movement of the body relative to the water). How are these concepts related?

In the case of moving downstream, V=V(own) + V(tech).
If against the current - V \u003d V (own) - V (flow).

Uneven movement is considered to be a movement with a changing speed. The speed can change direction. It can be concluded that any movement NOT along a straight path is non-uniform. For example, the movement of a body in a circle, the movement of a body thrown into the distance, etc.

The speed can vary by numerical value. This movement will also be uneven. A special case of such motion is uniformly accelerated motion.

Sometimes there is an uneven movement, which consists of alternating different kind movements, for example, at first the bus accelerates (the movement is uniformly accelerated), then it moves uniformly for some time, and then stops.

Instant Speed

It is possible to characterize uneven movement only by speed. But the speed is always changing! Therefore, we can only talk about the speed at a given instant of time. When traveling by car, the speedometer shows you the instantaneous speed of movement every second. But in this case, the time should be reduced not to a second, but to consider a much smaller period of time!

average speed

What is average speed? It is wrong to think that it is necessary to add up all the instantaneous speeds and divide by their number. This is the most common misconception about average speed! The average speed is all the way divided by the elapsed time. And it is not defined in any other way. If we consider the movement of the car, we can estimate its average speeds in the first half of the way, in the second, all the way. The average speeds may be the same, or they may be different in these sections.

At average values, a horizontal line is drawn on top.

Average movement speed. Average ground speed

If the movement of the body is not rectilinear, then the path traveled by the body will be greater than its displacement. In this case, the average travel speed is different from the average ground speed. Ground speed is a scalar.

The main thing to remember

1) Definition and types of uneven movement;
2) The difference between the average and instantaneous speeds;
3) The rule for finding the average speed of movement

Often you need to solve a problem where the entire path is divided into equal sections, average speeds are given for each section, it is required to find the average speed for the entire path. The wrong decision will be if you add up the average speeds and divide by their number. Below is a formula that can be used to solve such problems.

The instantaneous speed can be determined using the motion graph. The instantaneous velocity of a body at any point on the graph is determined by the slope of the tangent to the curve at the corresponding point. Instantaneous speed - the tangent of the slope of the tangent to the graph of the function.

Exercises

While driving a car, speedometer readings were taken every minute. Is it possible to determine the average speed of the car from these data?

It is impossible, since in the general case the value of the average speed is not equal to the arithmetic mean of the values instant speeds. But the path and time are not given.

What is the speed of the alternating motion shown by the car's speedometer?

close to instantaneous. Close, since the time interval should be infinitely small, and when taking readings from the speedometer, it is impossible to judge time in this way.

In what case are the instantaneous and average speeds equal to each other? Why?

With uniform motion. Because the speed does not change.

The speed of the hammer on impact is 8m/s. What is the speed: average or instantaneous?