Speed ​​is a derivative of distance with respect to time. The time derivative of the coordinate is the speed

The procedure that we have just performed is so common in mathematics that a special notation was invented for the quantities ε and x: ε is denoted as ∆t, and x as ∆s. The ∆t value means "little addition to t", and it is understood that this addition can be done less. The sign ∆ does not in any way mean multiplication by any quantity, just as sin θ does not mean s · i · n · 0. It's just some time addition, and the ∆ sign reminds us of its special character. Well, if ∆ is not a factor, then it cannot be canceled in the ratio ∆s / ∆t. It's like cutting off all the letters in sin θ / sin 2θ to get 1/2. In these new notation, the velocity is equal to the limit of the ratio ∆s / ∆t as ∆t tends to zero, i.e.

This is essentially formula (8.3), but now it is clearer that everything is changing here, and, in addition, it reminds us exactly which quantities are changing.
There is one more law that is fulfilled with good accuracy. It says: the change in distance is equal to the speed multiplied by the time interval during which this change occurred, that is, ∆s = υ∆t. This rule is strictly valid only when the speed does not change during the interval ∆t, and this, generally speaking, happens only when ∆t is small enough. In such cases, they usually write ds = υdt, where dt means the time interval ∆t, provided that it is arbitrarily small. If the interval ∆t is large enough, then the speed during this time can change and the expression ∆s = υ∆t will already be approximate. However, if we write dt, then it means that the time interval is infinitely small and in this sense the expression ds = υdt is exact. In the new notation, expression (8.5) has the form

The quantity ds / dt is called the "derivative of s with respect to t" (this name reminds us of what is changing), and the complex process of finding the derivative is also called; differentiation. If ds and dt appear separately, and not in the form of a ratio ds / dt, then they are called differentials. To better acquaint you with the new terminology, I will also say that in the previous paragraph we found the derivative of the function 5t 2, or simply the derivative of 5t 2. It turned out to be equal to 10t. As you get more used to the new words, the idea itself will become clearer to you. To train, let's find the derivative of a more complex function. Consider the expression s = At ​​3 + Bt + C, which can describe the motion of a point. The letters A, B, C, as in the usual quadratic equation, denote constant numbers. We need to find the speed of movement described by this formula at any moment of time t. For this, consider the moment t + ∆t, and some addition ∆s is added to s, and we find how ∆s is expressed in terms of ∆t. Insofar as

But we do not need the value of ∆s itself, but the ratio ∆s / ∆t. After dividing by ∆t, we obtain the expression

which, after ∆t tends to zero, turns 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 expansions similar to the previous ones, there are terms proportional to (∆t) 2 or (∆t) 3 or even higher powers, then they can be deleted immediately, since they will still vanish when at the end we will ∆t tend to zero. After a short workout, you will immediately see what needs to be left and what is discarded immediately. There are many rules and formulas for differentiating different kinds of functions. You can either remember them or use special tables. A small list of such rules is given in table. 8.3.

Until now, we have associated the concept of a derivative with the geometric representation of the graph of a function. However, it would be a gross mistake to limit the role of the concept of a derivative to only the problem of

determining the slope of the tangent to a given curve. Even more important, from a scientific point of view, the task is to calculate the rate of change of any value that changes over time. It was from this side that Newton approached differential calculus. In particular, Newton sought to analyze the phenomenon of speed, considering the time and position of a moving particle as variable quantities (in Newton's expression, "fluents"). When a certain particle moves along the x-axis, then its motion is quite definite, once a function is given that indicates the position of the particle x at any moment of time t. "Uniform motion" with constant speed along the x axis is determined by a linear function where a is the position of the particle at the initial moment

The motion of a particle on a plane is already described by two functions

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

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

is obtained by excluding from the two above relations.

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

where constants, depending on the state of the particle at the initial moment, the acceleration of gravity, equal to approximately 9.81, if time is measured in seconds, and distance - in meters. The trajectory of motion, obtained by excluding these two equations, is a parabola

unless otherwise, the path is a vertical axis segment.

If a particle is forced to move along some given curve (similar to how a train moves on rails), then its motion can be determined by a function (a function of time equal to the length of the arc calculated along a given curve from a certain initial point to the position of the particle at point P at the moment of time For example, if we are talking about a unit circle, then the function determines on this circle a uniform rotational motion with a speed of c.

The exercise. Draw the trajectories of plane motions given by the equations: in the parabolic motion described above, assume the initial position of the particle (at at the origin and count Find the coordinates of the highest point of the trajectory. Find the time and value of x corresponding to the secondary intersection of the trajectory with the axis

The first goal that Newton set himself was to find the speed of a particle moving unevenly. Let us consider, for simplicity, the motion of a particle along a certain straight line, given by the function If the motion were uniform, that is, it was performed with a constant speed, then this speed could be found by taking two moments of time and the corresponding positions of the particles and composing the ratio

For example, if measured in hours, and; in kilometers, then at the difference will be the number of kilometers traveled in 1 hour, the speed (kilometers per hour). When they say that velocity is a constant value, they only mean that the difference ratio

does not change at any values ​​But if the motion is uneven (which is, for example, a place in the free fall of a body, the speed of which increases as it falls), then the ratio (3) does not give the value of the speed at the moment a is what is usually called the average speed in the time interval from to To get the speed at the moment, you need to calculate the limit of the average

speed at tending Thus, together with Newton, we define the speed as follows:

In other words, the velocity is the derivative of the "traversed path" (the coordinates of the particle on a straight line) in time, or the "instantaneous rate of change" of the path in relation to time - as opposed to the average rate of change determined by formula (3).

The rate of change of the speed itself is called acceleration. Acceleration is simply a derivative of a derivative; it is usually denoted by the symbol and is called the second derivative of the function

Galileo noticed that the vertical distance x traversed by a body in free fall over time is expressed by the formula

Moving on to the physical applications of the derivative, we will use a slightly different designation, those that are accepted in physics.

First, the designation of the functions changes. Indeed, what functions are we going to differentiate? These functions are physical quantities that depend on time. For example, the coordinate of a body x (t) and its velocity v (t) can be given by formulas like this:

There is another notation for the derivative, which is very common in both mathematics and physics:

(read ¾de iks on de te¿).

Let us dwell in more detail on the meaning of notation (29). A mathematician understands it in two ways, either as a limit:

or as a fraction, the denominator of which is the time increment dt, and the numerator is the so-called differential dx of the function x (t). Differential is not difficult, but we will not discuss it now; it is waiting for you in the first year.

A physicist who is not constrained by the requirements of mathematical rigor understands notation (29) more informally. Let dx be the change in coordinate during time dt. Let us take the interval dt so small that the ratio dx = dt is close to its limit (30) with an accuracy that suits us.

And then, the physicist will say, the derivative of the coordinate with respect to time is simply a fraction, in the numerator of which there is a rather small change in the coordinate dx, and in the denominator there is a rather small time interval dt, during which this change in the coordinate occurred. Such a loose understanding of the derivative is characteristic of reasoning in physics. From now on, we will adhere to this particular physical level of rigor.

Let's go back to the original example (26) and calculate the derivative of the coordinate, and at the same time look at the combined use of designations (28) and (29):

x (t) = 1 + 12t 3t2) x (t) = dt d (1 + 12t 3t2) = 12 6t:

(The differentiation symbol dt d in front of the parenthesis is the same as the dash above the parenthesis in the previous notation.)

Note that the calculated derivative of the coordinate turned out to be equal to the body's velocity (27). This is not a coincidence and we need to discuss it in more detail.

2.1 Derived Coordinate

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

What does it mean? It is very simple: we are not dealing with the absolute value of the velocity, but with the projection vx of the velocity vector onto the X-axis.Therefore, instead of (27), it would be more correct to write:

If you have forgotten what the projection of a vector onto an axis is, then read the corresponding section of the article ¾ Vectors in physics¿. Here we only recall that the sign of the projection vx reflects the relationship between the direction of the velocity and the direction of the X axis:

vx> 0, the body moves in the direction of the X axis; vx< 0 , тело движется против оси X.

(For example, if vx = 3 m / s, then this means that the body is moving at a speed of 3 m / s in the direction opposite to the X-axis.)

Therefore, in our example (31), we have the following picture of motion: at 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.

Let us assume that the speed of the body in absolute value is equal to v. There are two possible directions of movement.

1. If the body moves in the positive direction of the X-axis, then a small change in the coordinate dx is positive and equal to the path traversed by the body in time dt. That's why

x = dx dt = 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 = dx dt = v:

Note now that in the first case vx = v, and in the second case vx = v. Thus, both cases are combined into one formula:

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

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

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

2.2 Acceleration

The speed of a body characterizes the rate at which its coordinates change. But the speed can also change slower or faster. The characteristic of the rate of change of speed is a physical quantity called acceleration.

Suppose, for example, that the speed of a car during uniform acceleration has increased from v0 = 2 m / s to v = 14 m / s in a time t = 3 s. Vehicle acceleration is calculated using the formula:

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

And what is the acceleration if the speed, on the contrary, has decreased from v0 = 14 m / s to v = 2 m / s during the same time t = 3 s? Then, using formula (33), we obtain:

In one second, as we can see, the speed decreases by 4 m / s.

Can we talk about acceleration if the speed changes unevenly? Of course it is possible, but only this will be an instant acceleration, which also depends on time. The scheme of reasoning is already well known to you: in formula (33), instead of the time interval t, we take a small interval dt, instead of the difference v v0, we take the increment dv of the velocity during the time dt, and as a result we obtain:

Thus, it turns out that acceleration is the derivative of speed.

Formula (34), however, does not describe all situations that arise in mechanics. For example, with uniform motion along a circle, the speed of the body does not change in absolute value, and in accordance with (34) we should have received a = v = 0. But you know very well that the body has acceleration, it is directed to the center of the circle and is called centripetal. Therefore, formula (34) needs some modification.

This modification is related to the fact that the acceleration is actually a vector. It turns out that the acceleration vector shows the direction of the change in the speed of the body. What this means, we will now find out with simple examples.

Let the body move along the X-axis. Let's consider two cases of the direction of acceleration: along the X-axis and against the X-axis, respectively.

1. The acceleration vector ~ a is codirectional with the X-axis (Fig. eighteen ). The projection of the acceleration onto the X-axis is positive: ax> 0.

Rice. 18.ax> 0

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

If the body moves to the right (vx> 0), then it accelerates: the body's velocity increases in absolute value. The projection of the speed vx is also increased in this case.

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> 0, then the projection of the velocity vx increases regardless of whether

in which direction the body is moving.

2. The acceleration vector ~ a is directed opposite to the X-axis (Fig. 19 ). X-axis acceleration projection is negative: ax< 0.

Rice. 19.ax< 0

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

If the body moves to the right (vx> 0), then it slows down: the body's velocity decreases in absolute value. The projection of the velocity vx is also reduced in this case.

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 between the sign of the acceleration projection ax and the increase (decrease) in the projection of the velocity vx, discovered in these examples, leads us to the required modification of formula (34):

Example. Let's go back to example (26):

x = 1 + 12t 3t2

(coordinate is measured in meters, time in seconds). Differentiating sequentially two times, we get:

vx = x = 12 6t;

ax = vx = 6:

As you can see, the acceleration is constant in absolute value and is equal to 6 m / s2. Acceleration is directed in the direction opposite to the X-axis.

The given example is the case of uniformly accelerated motion, in which the modulus and direction of acceleration are unchanged (or, in short, ~ a = const). Equally accelerated motion is one of the most important and frequently encountered types of motion in mechanics.

From this example, it is easy to understand that with uniformly accelerated motion, the projection of the velocity is a linear function of time, and the coordinate is a quadratic function.

Example. Consider a more exotic case:

x = 2 + 3t 4t2 + 5t3.

Until now, we have associated the concept of a derivative with the geometric representation of the graph of a function. However, it would be a gross mistake to limit the role of the concept of a derivative to only the problem of determining the slope of the tangent to a given curve. An even more important task from a scientific point of view is the calculation of the rate of change of any quantity f (t) changing over time t. It was from this side that Newton approached differential calculus. In particular, Newton strove to analyze the phenomenon of speed, considering the time and position of a moving particle as variable quantities (according to Newton's expression, "fluents"). When some particle moves along the x-axis, then its motion is quite definite, once the function is given x = f (t) indicating the position of particle x at any time t. "Uniform movement" with constant speed b along the x-axis is determined by a linear function x = a + bt, where a is the position of the particle at the initial moment (at t = 0).

The motion of a particle on a plane is already described by two functions

x = f (t), y = g (t),

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

x = a + bt, y = c + dt,

where b and d are two "components" of constant velocity, and a and c are the coordinates of the initial position of the particle (at t = 0); the particle trajectory is a straight line, the equation of which is

(x - a) d - (y - c) b = 0

is obtained by eliminating t from the two above relations.

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

where a, b, c, d are constants depending on the state of the particle at the initial moment, and g is the acceleration of gravity, equal to approximately 9.81 if time is measured in seconds and distance is in meters. The trajectory of motion, obtained by eliminating t from the two given equations, is a parabola

If only b ≠ 0; otherwise, the path is a vertical axis segment.

If a particle is forced to move along some given curve (just like a train moves on rails), then its motion can be determined by the function s (t) (function of time t), equal to the length of the arc s, calculated along this curve from some starting point P 0 to the position of the particle at point P at time t. For example, if we are talking about a unit circle x 2 + y 2 = 1, then the function s = ct defines on this circle a uniform rotational motion with a speed with.

* The exercise. Draw the trajectories of plane motions given by the equations: 1) x = sin t, y = cos t; 2) x = sin 2t, y = cos 3t; 3) x = sin 2t, y = 2 sin 3t; 4) in the parabolic motion described above, assume the initial position of the particle (at t = 0) at the origin and consider b> 0, d> 0... Find the coordinates of the highest point of the trajectory. Find the time t and the value x corresponding to the secondary intersection of the trajectory with the x-axis.

The first goal that Newton set himself was to find the speed of a particle moving unevenly. For simplicity, consider the motion of a particle along some straight line, given by the function x = f (t)... If the motion were uniform, that is, it was performed with a constant speed, then this speed could be found by taking two moments of time t and t 1 and the corresponding positions of the particles f (t) and f (t 1) and making up the attitude

For example, if t is measured in hours and x in kilometers, then at t 1 - t = 1 difference x 1 - x will be the number of kilometers traveled in 1 hour, and v- speed (in kilometers per hour). When they say that velocity is a constant value, they only mean that the difference ratio

does not change for any values ​​of t and t 1. But if the motion is uneven (which takes place, for example, in the free fall of a body, the speed of which increases as it falls), then relation (3) does not give the value of the speed at the moment t, but represents what is usually called the average speed in the time interval from t to t 1. To get speed at time t, you need to calculate the limit average speed as t 1 tends to t. Thus, following Newton, we define speed as follows:

In other words, the velocity is the time derivative of the distance traveled (the coordinates of the particle on a straight line), or the "instantaneous rate of change" of the path in relation to time - as opposed to middle the rate of change determined by the formula (3).

The rate of change of the speed itself called acceleration. Acceleration is simply a derivative of a derivative; it is usually denoted by the symbol f "(t) and is called second derivative on the function f (t).

Algebra is generous. She often gives more than she is asked.

J. D'Alembert

Interdisciplinary connections are a didactic condition and a means of deep and comprehensive assimilation of the foundations of science at school.
In addition, they contribute to an increase in the scientific level of students' knowledge, the development of logical thinking and their creative abilities. The implementation of interdisciplinary connections eliminates duplication in the study of the material, saves time and creates favorable conditions for the formation of general educational skills and abilities of students.
The establishment of interdisciplinary connections in the physics course increases the effectiveness of the polytechnic and practical orientation of education.
The motivational side is very important in teaching mathematics. A mathematical problem is perceived by students better if it appears as if in front of their eyes, is formulated after considering some physical phenomena or technical problems.
No matter how much the teacher speaks about the role of practice in the progress of mathematics and the importance 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, then the development of a materialistic worldview will be affected serious damage. But in order to show how mathematics helps in solving its problems, we need problems that are not invented for methodological purposes, but actually arising in various areas of human practical activity.

Historical background

Differential calculus was created by Newton and Leibniz at the end of the 17th century on the basis of two problems:

• about finding a tangent to an arbitrary line;
• on the search for speed with an arbitrary law of motion.

Even earlier, the concept of a derivative was encountered in the works of the Italian mathematician Nicolo Tartaglia (about 1500-1557) - here a tangent appeared in the course of studying the issue of the angle of inclination of the gun, at which the greatest flight range of the projectile was ensured.

In the 17th century, on the basis of Galileo's doctrine of motion, the kinematic concept of the derivative was actively developed.

The famous scientist Galileo Galilei dedicates a whole treatise on the role of the derivative in mathematics. Various expositions began to be found in the works of Descartes, the French mathematician Roberval, the English scientist L. Gregory. Lopital, Bernoulli, Lagrange, Euler, Gauss made a great contribution to the study of differential calculus.

Some applications of the derivative in physics

Derivative- the basic concept of differential calculus, which characterizes function change rate.

Determined as the limit of the ratio of the increment of a function to the increment of its argument when the increment of the argument tends to zero, if such a limit exists.

Thus,

Hence, to calculate the derivative of the function f (x) at the point x 0 by definition, you need:

Let us consider several physical problems in the solution of which this scheme is used.

Instantaneous speed problem. The mechanical meaning of the derivative

Let us recall how the speed of movement was determined. The material point moves along the coordinate line. The x-coordinate of this point is the known function x (t) time t. For a period of time from t 0 before t 0+ point displacement is x (t 0 +) – x (t 0) - and its average speed is as follows: .
Usually, the nature of the movement is such that at small, the average speed practically does not change, i.e. movement with a high degree of accuracy can be considered uniform. In other words, the value of the average speed at tends to some well-defined value, which is called the instantaneous speed v (t 0) material point at time t 0.

So,

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

Arguing similarly, we find that the derivative of the velocity with respect to time is the acceleration, i.e.

The problem of the heat capacity of a body

So that the body temperature weighing 1 g rises from 0 degrees to t degrees, the body must be given a certain amount of heat Q... Means, Q there is a temperature function t, to which the body is heated: Q = Q (t). Let the body temperature rise from t 0 before t. The amount of heat expended for this heating is equal to Ratio is the amount of heat that is needed on average to heat a body by 1 degree when the temperature changes by degrees. This ratio is called the average heat capacity of a given body and is denoted since wed.
Because the average heat capacity does not give an idea of ​​the heat capacity for any temperature T, then the concept of heat capacity at a given temperature is introduced t 0(at this point t 0).
Heat capacity at temperature t 0(at a given point) is called the limit

The linear density problem for a bar

Consider a non-uniform bar.

For such a rod, the question arises about the rate of change in mass depending on its length.

Average linear density the mass of a rod is a function of its length NS.

Thus, the linear density of a non-uniform bar at a given point is determined as follows:

Considering such problems, one can obtain similar conclusions for many physical processes. Some of them are shown in the table.

Function	Formula	Output
m (t) is the time dependence of the mass of the consumed fuel.		Derivative masses by time there is speed fuel consumption.
T (t) - time dependence of the temperature of the heated body.		Derivative temperature by time there is speed heating the body.
m (t) is the time dependence of the mass during the decay of a radioactive substance.		Derivative mass of radioactive material in time there is speed radioactive decay.
q (t) - the dependence of the amount of electricity flowing through the conductor on time		Derivative the amount of electricity by time there is amperage.
A (t) - dependence of work on time		Derivative work on time there is power.

Practical tasks:

The projectile ejected from the cannon moves according to the law x (t) = - 4t 2 + 13t (m). Find the projectile speed at the end of 3 seconds.

The amount of electricity flowing through the conductor, starting from the time t = 0 s, is given by the formula q (t) = 2t 2 + 3t + 1 (Cool) Find the current at the end of the fifth second.

The amount of heat Q (J) required to heat 1 kg of water from 0 o to t o C is determined by the formula Q (t) = t + 0.00002t 2 + 0.0000003t 3. Calculate the heat capacity of water if t = 100 o.

The body moves in a straight line according to the law x (t) = 3 + 2t + t 2 (m). Determine its speed and acceleration at times 1 s and 3 s.

Find the magnitude of the force F acting on a point of mass m, moving according to the law x (t) = t 2 - 4t 4 (m), at t = 3 s.

A body, the mass of which is m = 0.5 kg, moves in a straight line according to the law x (t) = 2t 2 + t - 3 (m). Find the kinetic energy of the body 7 seconds after the start of the movement.

Conclusion

It is possible to indicate many more problems from technology, for the solution of which it is also necessary to find the rate of change of the corresponding function.
For example, finding the angular velocity of a rotating body, the linear coefficient of expansion of bodies when heated, the rate of a chemical reaction at a given moment in time.
In view of the abundance of problems leading to the calculation of the rate of change of a function or, in other words, to the calculation of the limit of the ratio of the increment of a function to the increment of an argument when the latter tends to zero, it turned out to be necessary to select such a limit for an arbitrary function and study its basic properties. This limit was called derivative of the function.

So, using a number of examples, we have shown how various physical processes are described using mathematical problems, how the analysis of solutions allows us to draw conclusions and predictions about the course of processes.
Of course, the number of examples of this kind is huge, and a fairly large part of them is quite accessible to interested students.

“Music can uplift or placate the soul,
Painting is pleasing to the eye
Poetry - to awaken feelings
Philosophy - to satisfy the needs of the mind,
Engineering is to improve the material side of people's lives,
And mathematics can achieve all these goals. "

This is what the American mathematician said Maurice Kline.

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2. Vilenkin N.Ya., Shibasov A.P. Behind the pages of a mathematics textbook. - M: Education, 1996.
3. Dobrokhotova M.A., Safonov A.N.... Function, its limit and derivative. - M: Education, 1969.
4. Kolmogorov A.N., Abramov A.M. and other Algebra and the beginning of mathematical analysis. - M: Education, 2010.
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