Speed \u200b\u200bderived from time distance. The derivative of the coordinate in time is the speed

The procedure that we have just performed is so often in mathematics, which for the values \u200b\u200bof ε and x, a special designation was invented: ε is indicated as Δt, and x - as ΔS. The value of Δt means "small additives to T", and it is understood that this additives can be done less. The icon Δ in no case does not mean multiplication on some size, just as Sin θ does not mean S · I · N · 0. This is simply some additives by time, and the Δ icon reminds us of its special character. Well, if Δ is not a multiplier, it cannot be reduced with respect to ΔS / ΔT. It is like a sin θ / sin 2θ expression to reduce all the letters and get 1/2. In these new designations, the rate is equal to the limit of the ratio ΔS / Δt with Δt, striving for zero, i.e.

This is essentially formula (8.3), but now it is clearly clear that everything changes here, but, in addition, it reminds which values \u200b\u200bchange.
There is another law that is performed with good accuracy. It says: a change in distance is equal to the speed multiplied by the time interval for which this change occurred, that is, ΔS \u003d υΔt. This rule is strictly valid only when the speed does not change during the Δt interval, and this, generally speaking, occurs only when Δt is quite small. In such cases, DS \u003d υDT is usually written, where the time interval ΔT is meant under DT, provided that it is small. If the Δt interval is large enough, the speed during this time may change and the expression ΔS \u003d υΔt will be approximate. However, if we write DT, then it is understood that the time interval is unlimitedly small and in this sense the expression ds \u003d υdt is accurate. In new designations, the expression (8.5) has

The value of DS / DT is called "derivative S software T" (such a name is reminded of what changes), and the complex process of finding a derivative is also called, in addition; Differentiation. If DS and DT appear separately, and not in the form of a DS / DT ratio, then they are the names of differentials. To better introduce you to new terminology, I will say that in the previous paragraph we found a derivative from the 5T 2 function, or simply derived from 5t 2. It turned out to be 10t. When you're more accustomed to new words, the thought itself will be clearer. For training, let's find a derivative of a more complex function. Consider the expression S \u003d AT 3 + BT + C, which can describe the movement of the point. The letters A, B, C, as well as in the usual square equation, denote constant numbers. We need to find the speed of the movement described by this formula at any time t. Consider for this moment T + ΔT, and some additive ΔS will be added to S, and we find how ΔS is expressed through Δt. Insofar as

But we need not the amount ΔS, but the ratio ΔS / ΔT. After division on Δt we get expression

which after the aspiration Δt to zero will turn into

This is the process of taking a derivative, or differentiation of functions. In fact, it is somewhat lighter than it seems at first glance. Note that if in the decompositions like the previous one, the members are proportional to (Δt) 2 or (Δt) 3 or even higher degrees, they can be immediately deleted, because they will still turn into zero when we will be ΔT Ascertaining to zero. After a small workout, you will immediately see what you need to leave, and what to immediately discard. There are many rules and formulas for differentiation of various types of functions. They can either remember or use special tables. A small list of such rules is provided in Table. 8.3.

Until now, the concept of the derivative we associated with the geometric representation of the schedule of the function. However, it would be a gross mistake to limit the role of the concept of a derivative one task about

determining tilt tangent to this curve. An even more important, from a scientific point of view, the task is to calculate the rate of change of any value varying over time. It is from this side that Newton and approached differential calculus. In particular, Newton sought to analyze the velocity phenomenon, considering the time and position of the moving particle as variable values \u200b\u200b(according to Newton's expression, "Fluuents"). When some particles move along the x axis, its movement is quite defined, the function indicating the position of the particle x at any time t. "Uniform movement" with a constant speed along the x axis is determined by a linear function where is the position of the particle at the initial moment

The movement of the particle on the plane is already described by two functions.

which determine its coordinates as a function of time. In particular, the uniform movement corresponds to two linear functions.

where two "components" of a constant speed, and a and c - the coordinates of the initial position of the particle (the particle trajectory is a straight line, the equation of which

it turns out by exclusion from the two ratios above.

If the particle moves in the vertical plane x, y under the action of alone gravity, then its movement (this is proved in elementary physics) is determined by two equations

where permanent values \u200b\u200bdepend on the state of the particle at the initial moment, the acceleration of gravity, equal to approximately 9.81, if the time is measured in seconds, and the distance is in meters. The trajectory of motion, obtained by excluding from two data equations, is parabola

unless otherwise the trajectory is the segment of the vertical axis.

If the particle is forced to move along some of this curve (just as the train moves along the rails), then it can be determined by the function (time function equal to the length of the arc calculated along this curve from some initial point to the particle position at the point p at time for example, If we are talking about a single circle, the function determines the uniform rotational motion at this circle with a speed with.

The exercise. To draw the trajectories of the flat movements given by the equations: in the parabolic movement described above, assume the initial position of the particle (at the beginning of the coordinates and read the coordinates of the highest point of the trajectory. Find the time and value x that corresponding to the secondary intersection of the trajectory with the axis

The first goal that Newton put himself was to find the velocity of the particle moving unevenly. Consider the movement of the particle along a straight line specified by the function if the movement was uniform, i.e. it was performed at a constant speed, then this speed could be found by taking two points of time and the corresponding parties' positions and accounted for

For example, if measured in hours, and; In kilometers, the difference will be the number of kilometers traveled in 1 hour, speed (kilometers per hour). Saying that the speed is the magnitude constant, meaning only that the difference attitude

does not change at any values \u200b\u200bbut if the movement is uneven (which has, for example, a place with a free falling of the body, the speed of which increases as it increases), then the ratio (3) does not give the speed values \u200b\u200bat the time A represents what is considered to be called average speed In the time of time to get speed at the time you need to calculate the limit of the average

speeds when striving in this way, together with Newton, we define the speed like this:

In other words, the speed is derived from the "traveled path" (coordinates of the particles on a straight line) in time, or the "instantaneous rate of change" of the path with respect to the time - as opposed to the average velocity of the change determined by the formula (3).

The rate of change of speed itself is called acceleration. Acceleration is simply derived from the derivative; It is usually indicated by the symbol and is called the second derivative of the function

Galileosamethel that the vertical distance x, passing by the free fall of the body for a period of time is expressed by the formula

Turning to physical applications of the derivative, we will use several other symbols for those adopted in physics.

First, the designation of functions is changing. In fact, what functions are we going to differentiate? These functions serve physical quantities depending on time. For example, the coordinate of the body x (t) and its speed V (T) can be given by formulas like such:

There is another designation derivative, very common both in mathematics and physics:

(Reads ¾DE XE for DE TE¿).

Let us dwell in more detail on the sense of designation (29). Mathematician understands his bicon or like a limit:

either as a fraction, in the denominator which is the increment of time DT, and in the numerator the so-called DX differential function x (t). The concept of differential is not difficult, but we will not discuss it now; It is waiting for you in the first year.

The physicist, which is not complained by the requirements of mathematical rigor, understands the designation (29) more informally. Let DX be a change in coordinates during DT. Take the DT interval as small that the DX \u003d DT ratio is close to its limit (30) with accuracy.

And then, the physicist will say, the derivative coordinate in time is simply a fraction, in the numerator of which it costs a sufficiently small change in the coordinate of the DX, and in the denominator there is a sufficiently small period of time DT, during which this change of coordinate occurred. Such a nestor understanding of the derivative is characteristic of reasoning in physics. Next, we will adhere to this particular physical level of rigor.

Let's return to the original example (26) and consider the derivative of the coordinates, and at the same time we will look at the joint use of the designation (28) and (29):

x (t) \u003d 1 + 12t 3T2) x (t) \u003d dt d (1 + 12t 3T2) \u003d 12 6T:

(DT D Differentiation Symbol in front of the bracket is all the same as the barcode from the bracket in the former designations.)

Please note that the calculated derivative coordinate turned out to be equal to the body velocity (27). This is not a random coincidence, and we need to discuss it in more detail.

2.1 Coordinate derivative

First of all, we note that the speed in (27) can be both positive and negative. Namely, the speed is positive at T< 2, обращается в нуль при t = 2 и становится отрицательной при t > 2.

What does it mean? Very simple: We are dealing not with an absolute value of speed, but with the projection of the VX velocity vector on the X axis. Therefore, instead (27) it would be more correct to write:

If you forgot what the vector projection is on the axis, then read the appropriate section of the article ¾ Vectors in physics¿. Here we remark only that the projection mark VX reflects the connection of the direction of speed and the direction of the X axis:

vX\u003e 0, the body moves in the x axis direction; vx.< 0 , тело движется против оси X.

(For example, if Vx \u003d 3 m / s, then this means that the body moves at a speed of 3 m / s to the side opposite to the X axis.)

Therefore, in our example (31) we have the following picture of the movement: with T< 2 тело движется в положительном направлении оси X и постепенно замедляется; при t = 0 тело останавливается; при t > 2 The body, accelerating, moves in the negative direction of the X axis.

Suppose that the body's velocity at an absolute value is v. Two cases of movement directions are possible.

1. If the body moves in the positive direction of the X axis, then a small change in the DX coordinate is positive and equal to the path of the body during DT. therefore

x \u003d dx dt \u003d v:

2. If the body moves in the negative direction of the X axis, then DX< 0. Путь за время dt равен dx, поэтому dx=dt = v или

x \u003d dx dt \u003d v:

Note now that in the first case VX \u003d V, and in the second case VX \u003d V. Thus, both cases are combined into one formula:

and we come to the most important fact: the body's derivative coordinate is equal to the projection of the body velocity on this axis.

It is easy to see that the sign of increasing (descending) function works. Namely:

x\u003e 0) vx\u003e 0) the body moves in the direction of the x axis) of the x coordinate increases; X.< 0) vx < 0) тело двигается против оси X) координата x уменьшается:

2.2 Acceleration

The velocity of the body characterizes the speed of changes to its coordinate. But speed can also change slower or faster. The speed of speed changes is the physical quantity called acceleration.

Let, for example, the vehicle speed with uniform acceleration increased from V0 \u003d 2 m / s to V \u003d 14 m / s per time T \u003d 3 s. The acceleration of the car is calculated by the formula:

Thus, in one second, the vehicle speed increases by 4 m / s.

And why is the acceleration, if the speed, on the contrary, decreased from V0 \u003d 14 m / s to v \u003d 2 m / s during the same time t \u003d 3 c? Then by formula (33) we get:

In one second, as we see, the speed is reduced by 4 m / s.

Is it possible to talk about acceleration if the speed changes uneven? Of course, it is possible, but only it will be an instant acceleration, which also depends on time. The scheme of reasoning is already familiar to you: in formula (33), instead of time, we take a small dt interval, instead of the difference V V0 take the increment of DV speed during DT, and as a result we obtain:

Thus, it turns out that acceleration is a speed derivative.

Formula (34), however, does not describe all situations that occur in the mechanics. For example, with a uniform movement around the circle, the body speed does not change according to the module, and in accordance with (34) we would have to get a \u003d v \u003d 0. But you know perfectly well that the acceleration of the body is available, it is directed to the center of the circle and is called Centripetal. Therefore, formula (34) needs some modification.

This modification is associated with the fact that the acceleration is actually a vector. It turns out that the acceleration vector shows the direction of changing the body of the body. What does this mean, we now find out on simple examples.

Let the body move along the X axis. Let's look at two cases of acceleration directions: along the X axis and against the x axis, respectively.

1. The speed of acceleration ~ A is co-directed with the x axis (Fig.eighteen ). The projection of the acceleration on the X axis is positive: AX\u003e 0.

Fig. 18. AX\u003e 0

IN this case, the speed changes in the positive direction of the X axis. Namely:

If the body moves to the right (VX\u003e 0), it is accelerated: the body speed in the module increases. The VX speed projection is also increasing.

If the body moves to the left (VX< 0), то оно тормозит: скорость тела по модулю уменьшается. Но обратите внимание, что проекция скорости vx , будучи отрицательной, при этом увеличивается.

Thus, if Ax\u003e 0, then the projection of the speed VX increases regardless of

in which direction the body moves.

2. The acceleration vector ~ A is directed opposite to the X axis (Fig.nineteen ). Acceleration projection on the X axis is negative: AX< 0.

Fig. 19. AX.< 0

IN this case, the speed changes in the negative direction of the X axis. Namely:

If the body moves to the right (VX\u003e 0), then it slows down: the body speed in the module decreases. The projection of the speed VX is also reduced.

If the body moves to the left (VX< 0), то оно разгоняется: скорость тела по модулю увеличивается. Но проекция скорости vx , будучи отрицательной, при этом уменьшается.

Thus, if AX< 0, то проекция скорости vx убывает, и опять-таки вне зависимости от того, в каком направлении движется тело.

The connection of the AX acceleration projection of AX, detected in these examples, with an increase in (decrease), the projection of the VX speed leads us to the desired modification of formula (34):

Example. Let us come back for example (26):

x \u003d 1 + 12t 3T2

(The coordinate is measured in meters, time in seconds). Sequentially differentiating two times, we get:

vx \u003d x \u003d 12 6t;

aX \u003d VX \u003d 6:

As we see, the acceleration is constantly in the module and equal to 6 m / s2. The acceleration is directed to the side opposite to the X axis.

The resulting example is the case of an equivalent movement, in which the module and the direction of acceleration are unchanged (or, in short, ~ a \u003d const). Equal asked movement is one of the most important and frequently encountered modes of movement in the mechanics.

From this example, it is not difficult to understand that with an equilibrium movement, the speed projection is a linear function of time, and the coordinate with a quadratic function.

Example. Consider a more exotic case:

x \u003d 2 + 3T 4T2 + 5T3.

Until now, the concept of the derivative we associated with the geometric representation of the schedule of the function. However, it would be a rough mistake to limit the role of the concept of a derivative one only the task of determining the tilt of tangent to this curve. An even more important from a scientific point of view of the task is to calculate the rate of change of any value f (T)varying over time t. It is from this side that Newton and approached differential calculus. In particular, Newton sought to analyze the velocity phenomenon, considering the time and position of the moving particle as variable values \u200b\u200b(according to Newton's expression, "Flyuents"). When some particle moves along the x axis, then its movement is quite defined, the function is specified. x \u003d f (t)indicating the position of the particle x at any time t. "Uniform movement" with a constant velocity B along the axis x is determined by a linear function x \u003d a + btwhere is the position of the particle at the initial moment (when t \u003d 0.).

The movement of the particle on the plane is already described by two functions.

x \u003d f (t), y \u003d g (t),

which determine its coordinates as a function of time. In particular, two linear functions correspond to uniform movement.

x \u003d a + bt, y \u003d c + dt,

where b and d are two "components" of a constant speed, and a and c - coordinates of the initial position of the particle (with t \u003d 0.); The trajectory of the particle is a straight line, the equation of which

(x - a) d - (y - c) b \u003d 0

it turns out by excluding T from two ratios above.

If the particle moves in the vertical plane x, y under the action of alone gravity, then its movement (this is proved in elementary physics) is determined by two equations

where a, B, C, D - Permanent values \u200b\u200bdepending on the state of the particle at the initial moment, and G - acceleration of gravity, equal to approximately 9.81, if the time is measured in seconds, and the distance is in meters. The trajectory of movement, obtained by excluding T of two data equations, is parabola

if only b ≠ 0; Otherwise, the trajectory is the segment of the vertical axis.

If the particle is forced to move along some of this curve (just as the train moves along the rails), it can be determined by the function S (T) (Time T), equal to the length of the arc s, calculated along this curve from some starting point P 0 Before the position of the particle at the point p at time t. For example, if we are talking about a single circle x 2 + y 2 \u003d 1, then function s \u003d Ct. Determines in this circle uniform rotational motion at the rate from.

* The exercise. Draw the trajectories of flat movements given by equations: 1) x \u003d sin t, y \u003d cos t; 2) x \u003d sin 2t, y \u003d cos 3t; 3) x \u003d sin 2t, y \u003d 2 sin 3t; 4) In the parabolic movement described above, assume the initial position of the particle (at t \u003d 0) at the beginning of the coordinates and count b\u003e 0, D\u003e 0. Find the coordinates of the highest point of the trajectory. Find Time T and X value corresponding to the secondary crossing of the trajectory with the x axis.

The first goal that Newton put himself was to find the velocity of the particle moving unevenly. Consider for simplicity Move the particle along some straight line specified by the function x \u003d f (t). If the movement was uniform, i.e. performed at a constant speed, then this velocity could be found by taking two points T and T 1 and the corresponding positions of particles f (T) and f (T 1) And reaching relatives

For example, if T is measured in hours, and x in kilometers, then t 1 - T \u003d 1 difference x 1 - x there will be a number of kilometers traveled in 1 hour, and v. - speed (in kilometers per hour). Saying that the speed is the magnitude constant, meaning only that the difference attitude

does not change with any values \u200b\u200bT and T 1. But if the movement is uneven (which has, for example, a place with a free falling of the body, whose speed increases as it increases), then the ratio (3) does not give the speed values \u200b\u200bat the time T, and represents what is called to call an average rate in the time interval from T to T 1. To get speed at the moment T., you need to calculate the limit mid speed With the pursuit of T 1 to T. Thus, following Newton, we will define the speed like this:

In other words, the speed is derived from the path traveled (the coordinates of the particles on a straight line) in time, or the "instantaneous rate of change" of the path with respect to time - as opposed to middle The rate of change defined by formula (3).

Speed \u200b\u200bchange rate called acceleration. Acceleration is simply derived from the derivative; It is usually denoted by the F "(T) symbol and is called the second derivative Function F (T).

Shchedra algebra. Often it gives more than she is asked.

J.Dalamber

Intergovernmental bonds are a didactic condition and a means of deep and comprehensively assimilating the basics of sciences at school.
In addition, they contribute to the increase in the scientific level of knowledge of students, the development of logical thinking and their creative abilities. The implementation of interdisciplinary bonds eliminates duplication in the study of the material, saves time and creates favorable conditions for the formation of general educational skills and students' skills.
The establishment of interdisciplinary relations in the course of physics increases the efficiency of the polytechnic and practical orientation of training.
The motivational side is very important in teaching mathematics. The mathematical task is perceived by students better if it arises as if they have in front of them formulating after consideration of some physical phenomena or technical problems.
No matter how much the teacher tells about the role of practice in the progress of mathematics and the meaning of mathematics for the study of physics, the development of technology, but if he does not show how physics affects the development of mathematics and how mathematics helps practice in solving its problems, the development of materialistic worldview will be applied Serious damage. But in order to show how mathematics helps in solving its problems, we need tasks that are not invented for methodical purposes, but in fact emerging in various fields of human practical activity

Historical information

Differential calculus was created by Newton and Leibnitsa at the end of the 17th century based on two tasks:

• on the search for tangents to an arbitrary line;
• on the location of the speed at an arbitrary law of movement.

Earlier, the concept of the derivative was found in the works of Italian mathematician Nikolo Tartalli (about 1500 - 1557) - here there was a tangential in the course of the study of the angle of the tool, which ensures the largest range of the projectile.

In the 17th century, the kinematic concept of the derivative was actively developed on the basis of the exercise of Galilee on the movement.

Decides a whole treatise on the role of the derivative in mathematics, the famous scientist Galileo Galilee. Different presentations began to meet in the works of Descartes, French Mathematics Roberval, English scientist L.Gregori. Lopal, Bernoulli, Lagrange, Euler, Gauss made a great contribution to the study of differential calculus.

Some applications derivative in physics

Derivative- the basic concept of differential calculus characterizing speed \u200b\u200bchange function.

Determined As the limit of the relationship of the function of the function to the increment of its argument when the argument increases to zero, if there is a limit.

In this way,

It means to calculate the derivative function f (x) At point x 0 By definition, you need:

Consider several physical problems, when solving which this scheme applies.

The task of instantaneous speed. Mechanical sense of the derivative

Recall how the speed of movement was determined. The material point moves along the coordinate direct. Coordinates x of this point there is a well-known function x (t) of time t. Over time from t 0. before t 0. + Move the point is equal x (t 0 +) – x (t 0) - And its average speed is as follows: .
Typically, the nature of the movement is such that at small, the average speed is practically not changed, i.e. Movement with a large degree of accuracy can be considered uniform. In other words, the value of the average speed when she strives for some well-defined value, which is called instantaneous speed v (t 0) material point at time t 0..

So,

But by definition
Therefore, it is believed that instantaneous speed at the time of time t 0.

Similarly, arguing, we obtain that the derivative of the speed of time is acceleration, i.e.

Task on body heat

So that the body temperature of the mass in 1g rose from 0 degrees to t. degrees, body must be reported to a certain amount of heat Q.. It means Q.there is a function of temperature t.To which the body heats up: Q \u003d Q (T). Let the body temperature rose with t 0. before t.The amount of heat spent for this heating is equal to the ratio of the heat that is necessary on average to heat the body by 1 degree when the temperature changes on degree. This ratio is called the average heat capacity of this body and is indicated. with cf..
Because The average heat capacity does not give the appearance of heat capacity for any temperature value of T, the concept of heat capacity is introduced at a temperature of t 0. (at this point t 0.).
Heat capacity at temperature t 0. (at this point) is called the limit

Task on linear density rod

Consider a non-uniform rod.

For such a rod, the question of the rate of change of mass is faced depending on its length.

Medium linear density Mass of the rod is a function of its length h..

Thus, the linear density of the inhomogeneous rod at this point is defined as follows:

Considering such tasks, you can get similar conclusions in many physical processes. Some of them are shown in the table.

Function	Formula	Output
m (T) - the dependence of the mass of consumable fuel from time.		Derivative time masses there is speed Fuel consumption.
T (T) is the dependence of the temperature of the heated body.		Derivative temperatures in time there is speed Heating body.
m (T) - the dependence of the mass during the decay of the radioactive substance from time.		Derivative mass of radioactive substance in timethere is speed Radioactive decay.
q (T) - the dependence of the amount of electricity flowing through the conductor, on time		Derivative time electricity there is tok Power.
A (T) - time dependence on time		Derivative work on time there is power.

Practical tasks:

The shell flying out of the gun moves under the law x (t) \u003d - 4t 2 + 13t (m). Find the speed of the projectile at the end of 3 seconds.

The amount of electricity flowing through the conductor, starting from time to T \u003d 0 C, is set by the formula Q (T) \u003d 2T 2 + 3T + 1 (Cool). Find the current strength at the end of the fifth second.

The amount of heat Q (J), which is necessary for heating 1 kg of water from 0 o to T o C, is determined by the formula Q (T) \u003d T + 0.00002T 2 + 0.0000003T 3. Calculate water heat capacity if T \u003d 100 o.

The body moves straightforward by law x (t) \u003d 3 + 2t + t 2 (m). Determine its speed and acceleration at the time of 1 s and 3 s.

Find the value of the force F, acting on the point weighing M, moving under the law x (t) \u003d T 2 - 4T 4 (M), at T \u003d 3 s.

The body, the mass of which m \u003d 0.5kg, moves straightforward according to the law X (T) \u003d 2T 2 + T - 3 (m). Find the kinetic energy of the body after 7 seconds after the start of the movement.

Conclusion

You can specify many more tasks from the technique, to solve which it is also necessary to search for the rate of change of the corresponding function.
For example, finding the angular velocity of the rotating body, the linear coefficient of expansion of the bodies when heated, the rate of chemical reaction at the moment of time.
Due to the abundance of tasks leading to the calculation of the rate of change of function or, otherwise, to calculate the limit of the relationship of the function of the function to increments the argument, when the latter strives for zero, it turned out to be necessary to allocate such a limit for arbitrary function and explore its main properties. This limit called derived function.

So, on a number of examples, we showed how different physical processes are described using mathematical tasks how the analysis of solutions allows to draw conclusions and predictions about the progress of processes.
Of course, the number of examples of this kind is enormous, and quite most of them are quite accessible to those interested in students.

"Music can elevate or pacify the soul,
Painting - please please
Poetry - awaken the feelings
Philosophy - satisfy the needs of the mind,
Engineering is to improve the material side of people's life,
And mathematics can achieve all these goals. "

So said the American mathematician Maurice Kline.

Bibliography :

1. Abramov A.N., Vilenkin N.Ya. and others. Selected mathematics questions. Grade 10. - M: Education, 1980.
2. Vilenkin N.Ya., Shibasov A.P.Behind the pages of the textbook of mathematics. - M: Education, 1996.
3. Dobrozhotova M.A., Safonov A.N. Function, its limit and derivative. - M: Education, 1969.
4. Kolmogorov A.N., Abramov A.M. and others. Algebra and began mathematical analysis. - M: Enlightenment, 2010.
5. Kolosov A.A. Book for extracurricular reading in mathematics. - M: Uchochegiz, 1963.
6. Fichtenhalts G.M. Fundamentals of mathematical analysis, Part 1 - M: Science, 1955.
7. Yakovlev G.N. Mathematics for technical schools. Algebra and start analysis, Part 1 - M: Science, 1987.