mechanical movement

mechanical movement is the process of changing the position of a body in space over time relative to another body, which we consider to be motionless.

The body, conventionally taken for motionless, is the body of reference.

Reference body is a body relative to which the position of another body is determined.

Reference system- this is a reference body, a coordinate system rigidly connected with it, and a device for measuring the time of movement.

Trajectory

body trajectory is a continuous line that describes a moving body (considered as a material point) with respect to the selected reference system.

Distance traveled

Distance traveled is a scalar value equal to the length of the arc of the trajectory traversed by the body in some time.

moving

By moving the body called a directed segment of a straight line connecting the initial position of the body with its subsequent position, a vector quantity.

Average and instantaneous speed of movement. Direction and modulus of speed.

Speed - a physical quantity that characterizes the rate of change of coordinates.

Average moving speed- this is a physical quantity equal to the ratio of the point's displacement vector to the time interval during which this displacement occurred. vector direction average speed coincides with the direction of the displacement vector ∆S

Instant Speed is a physical quantity equal to the limit to which the average speed tends with an infinite decrease in the time interval ∆t. Vector instantaneous velocity is directed tangentially to the trajectory. Module is equal to the first derivative of the path with respect to time.

Path formula for uniformly accelerated motion.

Uniformly accelerated motion- this is a movement in which the acceleration is constant in magnitude and direction.

Movement acceleration

Movement acceleration - a vector physical quantity that determines the rate of change in the speed of the body, that is, the first derivative of the speed with respect to time.

Tangential and normal accelerations.

Tangential (tangential) acceleration is the component of the acceleration vector directed along the tangent to the trajectory at a given point in the trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion.

Direction tangential acceleration vectors a lies on the same axis as the tangent circle, which is the trajectory of the body.

Normal acceleration- is a component of the acceleration vector directed along the normal to the trajectory of motion at a given point on the trajectory of the body.

Vector perpendicular to the linear speed of movement, directed along the radius of curvature of the trajectory.

Speed ​​formula for uniformly accelerated motion

Newton's first law (or law of inertia)

There are such frames of reference, relative to which isolated progressively moving bodies keep their speed unchanged in absolute value and direction.

inertial frame of reference is such a reference system, relative to which a material point, free from external influences, either rests or moves in a straight line and uniformly (ie, at a constant speed).

In nature, there are four type of interaction

1. Gravitational (gravitational force) is the interaction between bodies that have mass.

2. Electromagnetic - valid for bodies with an electric charge, responsible for such mechanical forces as the friction force and the elastic force.

3. Strong - the interaction is short-range, that is, it acts at a distance of the order of the size of the nucleus.

4. Weak. Such an interaction is responsible for some types of interaction among elementary particles, for some types of β-decay and for other processes occurring inside an atom, an atomic nucleus.

Weight - is a quantitative characteristic of the inert properties of the body. It shows how the body reacts to external influences.

Force - is a quantitative measure of the action of one body on another.

Newton's second law.

The force acting on the body is equal to the product of the body mass and the acceleration imparted by this force: F=ma

measured in

The physical quantity equal to the product of the mass of the body and the speed of its movement is called body momentum (or amount of movement). The momentum of the body is a vector quantity. The SI unit of momentum is kilogram-meter per second (kg m/s).

Expression of Newton's second law in terms of the change in momentum of the body

Uniform movement - this is movement at a constant speed, that is, when the speed does not change (v \u003d const) and there is no acceleration or deceleration (a \u003d 0).

Rectilinear motion - this is movement in a straight line, that is, the trajectory of rectilinear movement is a straight line.

Uniformly accelerated motion - movement in which the acceleration is constant in magnitude and direction.

Newton's third law. Examples.

Shoulder of strength.

Shoulder of Strength is the length of the perpendicular from some fictitious point O to the force. The fictitious center, point O, will be chosen arbitrarily, the moments of each force are determined relative to this point. It is impossible to choose one point O to determine the moments of some forces, and choose it elsewhere to find the moments of other forces!

We select point O in an arbitrary place, we do not change its location anymore. Then the arm of gravity is the length of the perpendicular (segment d) in the figure

Moment of inertia tel.

Moment of inertia J(kgm 2) - a parameter similar in physical meaning to the mass in translational motion. It characterizes the measure of inertia of bodies rotating about a fixed axis of rotation. The moment of inertia of a material point with a mass m is equal to the product of the mass by the square of the distance from the point to the axis of rotation: .

The moment of inertia of a body is the sum of the moments of inertia of the material points that make up this body. It can be expressed in terms of body weight and dimensions.

Steiner's theorem.

Moment of inertia J body relative to an arbitrary fixed axis is equal to the sum of the moment of inertia of this body Jc relative to an axis parallel to it, passing through the center of mass of the body, and the product of the body mass m per square distance d between axles:

Jc- known moment of inertia about the axis passing through the center of mass of the body,

J- the desired moment of inertia about a parallel axis,

m- body mass,

d- the distance between the indicated axes.

Law of conservation of angular momentum. Examples.

If the sum of the moments of forces acting on a body rotating around a fixed axis is equal to zero, then the angular momentum is conserved (law of conservation of angular momentum):
.

The law of conservation of angular momentum is very clear in experiments with a balanced gyroscope - a rapidly rotating body with three degrees of freedom (Fig. 6.9).

It is the law of conservation of angular momentum that is used by ice dancers to change the speed of rotation. Or another well-known example - Zhukovsky's bench (Fig. 6.11).

Force work.

The work of force -a measure of the action of a force in the transformation of mechanical motion into another form of motion.

Examples of formulas for the work of forces.

the work of gravity; work of gravity on an inclined surface

elastic force work

The work of the friction force

mechanical energy of the body.

mechanical energy is a physical quantity that is a function of the state of the system and characterizes the ability of the system to do work.

Oscillation characteristic

Phase determines the state of the system, namely the coordinate, speed, acceleration, energy, etc.

Cyclic frequency characterizes the rate of change of the oscillation phase.

The initial state of the oscillatory system characterizes initial phase

Oscillation amplitude A is the largest displacement from the equilibrium position

Period T- this is the period of time during which the point performs one complete oscillation.

Oscillation frequency is the number of complete oscillations per unit time t.

The frequency, cyclic frequency and oscillation period are related as

physical pendulum.

physical pendulum - a rigid body capable of oscillating about an axis that does not coincide with the center of mass.

Electric charge.

Electric charge is a physical quantity that characterizes the property of particles or bodies to enter into electromagnetic force interactions.

Electric charge is usually denoted by the letters q or Q.

The totality of all known experimental facts allows us to draw the following conclusions:

There are two types of electric charges, conventionally called positive and negative.

· Charges can be transferred (for example, by direct contact) from one body to another. Unlike body mass, electric charge is not an inherent characteristic of a given body. The same body in different conditions can have a different charge.

Charges of the same name repel, unlike charges attract. This also shows the fundamental difference between electromagnetic forces and gravitational ones. Gravitational forces are always forces of attraction.

Coulomb's law.

The modulus of the force of interaction of two point stationary electric charges in vacuum is directly proportional to the product of the magnitudes of these charges and inversely proportional to the square of the distance between them.

Г is the distance between them, k is the coefficient of proportionality, depending on the choice of the system of units, in SI

The value showing how many times the force of interaction of charges in a vacuum is greater than in a medium is called the permittivity of the medium E. For a medium with permittivity e, Coulomb's law is written as follows:

In SI, the coefficient k is usually written as follows:

Electrical constant, numerically equal to

Using the electric constant, Coulomb's law has the form:

electrostatic field.

electrostatic field - a field created by electric charges that are immobile in space and unchanged in time (in the absence of electric currents). An electric field is a special kind of matter associated with electric charges and transferring the actions of charges to each other.

The main characteristics of the electrostatic field:

tension

potential

Examples of formulas for the field strength of charged bodies.

1. The intensity of the electrostatic field created by a uniformly charged spherical surface.

Let a spherical surface of radius R (Fig. 13.7) bear a uniformly distributed charge q, i.e. the surface charge density at any point on the sphere will be the same.

We enclose our spherical surface in a symmetric surface S with radius r>R. The intensity vector flux through the surface S will be equal to

According to the Gauss theorem

Hence

Comparing this relation with the formula for the field strength of a point charge, we can conclude that the field strength outside the charged sphere is as if the entire charge of the sphere were concentrated in its center.

For points located on the surface of a charged sphere of radius R, by analogy with the above equation, we can write

Let us draw through the point B, located inside the charged spherical surface, the sphere S with radius r

2. Electrostatic field of the ball.

Let we have a ball of radius R, uniformly charged with bulk density.

At any point A, lying outside the ball at a distance r from its center (r>R), its field is similar to the field of a point charge located in the center of the ball.

Then outside the ball

and on its surface (r=R)

At point B, lying inside the ball at distances r from its center (r>R), the field is determined only by the charge enclosed inside the sphere of radius r. The intensity vector flow through this sphere is equal to

on the other hand, according to the Gauss theorem

From a comparison of the last expressions it follows

where is the permittivity inside the sphere.

3. Field strength of a uniformly charged infinite rectilinear filament (or cylinder).

Let us assume that a hollow cylindrical surface of radius R is charged with a constant linear density .

Let us draw a coaxial cylindrical surface of radius The flow of the field strength vector through this surface

According to the Gauss theorem

From the last two expressions, we determine the field strength created by a uniformly charged thread:

Let the plane have an infinite extent and the charge per unit area is equal to σ. From the laws of symmetry it follows that the field is directed everywhere perpendicular to the plane, and if there are no other external charges, then the fields on both sides of the plane should be the same. Let us limit a part of the charged plane to an imaginary cylindrical box, so that the box is cut in half and its generators are perpendicular, and two bases, each having an area S, are parallel to the charged plane (Figure 1.10).

total vector flow; tension is equal to the vector times the area S of the first base, plus the vector flow through the opposite base. The flux of tension through the side surface of the cylinder is equal to zero, since the lines of tension do not cross them.

Thus, on the other hand, according to the Gauss theorem

Hence

But then the field strength of an infinite uniformly charged plane will be equal to

This expression does not include coordinates, therefore the electrostatic field will be uniform, and its strength at any point in the field is the same.

5. The intensity of the field created by two infinite parallel planes, oppositely charged with the same density.

As can be seen from Figure 13.13, the field strength between two infinite parallel planes having surface charge densities and , is equal to the sum of the field strengths created by the plates, i.e.

Thus,

Outside the plate, the vectors from each of them are directed in opposite directions and cancel each other out. Therefore, the field strength in the space surrounding the plates will be equal to zero E=0.

Electricity.

Electricity - directed (ordered) motion of charged particles

Third party forces.

Third party forces- forces of a non-electric nature, causing the movement of electric charges inside a direct current source. All forces other than the Coulomb forces are considered external.

emf Voltage.

Electromotive Force (EMF) - a physical quantity that characterizes the work of external (non-potential) forces in sources of direct or alternating current. In a closed conducting circuit, the EMF is equal to the work of these forces in moving a single positive charge along the circuit.

EMF can be expressed in terms of the electric field strength of external forces

Voltage (U) is equal to the ratio of the work of the electric field on the movement of the charge
to the value of the transferred charge in the circuit section.

Unit of measure for voltage in the SI system:

Current strength.

Current (I)- a scalar value equal to the ratio of the charge q passed through the cross section of the conductor to the time interval t during which the current flowed. The current strength shows how much charge passes through the cross section of the conductor per unit of time.

current density.

Current density j - a vector whose modulus is equal to the ratio of the strength of the current flowing through a certain area, perpendicular to the direction of the current, to the value of this area.

The SI unit for current density is the ampere per square meter (A/m2).

Ohm's law.

Current is directly proportional to voltage and inversely proportional to resistance.

Joule-Lenz law.

When an electric current passes through a conductor, the amount of heat released in the conductor is directly proportional to the square of the current, the resistance of the conductor, and the time during which the electric current flowed through the conductor.

Magnetic interaction.

Magnetic interaction- this interaction is the ordering of moving electric charges.

A magnetic field.

A magnetic field- this is a special kind of matter, through which the interaction between moving electrically charged particles is carried out.

Lorentz force and Ampère force.

Lorentz force is the force acting from the side of the magnetic field on a positive charge moving at a speed (here, is the speed of the ordered motion of positive charge carriers). Lorentz force modulus:

Amp power is the force with which a magnetic field acts on a current-carrying conductor.

The Ampere force module is equal to the product of the current strength in the conductor and the module of the magnetic induction vector, the length of the conductor and the sine of the angle between the magnetic induction vector and the direction of the current in the conductor.

The Ampere force is maximum if the magnetic induction vector is perpendicular to the conductor.

If the magnetic induction vector is parallel to the conductor, then the magnetic field has no effect on the conductor with current, i.e. Ampere's force is zero.

The direction of Ampère's force is determined by the rule of the left hand.

Biot-Savart-Laplace law.

Bio Savart Laplace's Law- The magnetic field of any current can be calculated as the vector sum of the fields created by individual sections of currents.

Wording

Let a direct current flow along the contour γ, which is in vacuum, the point at which the field is sought, then the magnetic field induction at this point is expressed by the integral (in the SI system)

The direction is perpendicular to and, that is, perpendicular to the plane in which they lie, and coincides with the tangent to the line of magnetic induction. This direction can be found by the rule for finding magnetic induction lines (the rule of the right screw): the direction of rotation of the screw head gives the direction if the translational movement of the gimlet corresponds to the direction of the current in the element. The module of the vector is determined by the expression (in the SI system)

The vector potential is given by the integral (in the SI system)

Loop inductance.

Inductance - physical a value numerically equal to the EMF of self-induction that occurs in the circuit when the current strength changes by 1 ampere in 1 second.
Also, the inductance can be calculated by the formula:

where F is the magnetic flux through the circuit, I is the current strength in the circuit.

SI units for inductance:

The energy of the magnetic field.

The magnetic field has energy. Just as a charged capacitor has a store of electrical energy, a coil with current flowing through its turns has a store of magnetic energy.

Electromagnetic induction.

Electromagnetic induction - the phenomenon of the occurrence of an electric current in a closed circuit with a change in the magnetic flux passing through it.

Lenz's rule.

Lenz's rule

The induction current arising in a closed circuit counteracts the change in the magnetic flux with which it is caused by its magnetic field.

Maxwell's first equation

2. Any displaced magnetic field generates a vortex electric field (the basic law of electromagnetic induction).

Maxwell's second equation:

Electromagnetic radiation.

electromagnetic waves, electromagnetic radiation- propagating in space perturbation (change of state) of the electromagnetic field.

3.1. Wave are vibrations propagating in space over time.
Mechanical waves can propagate only in some medium (substance): in a gas, in a liquid, in a solid. Waves are generated by oscillating bodies that create a deformation of the medium in the surrounding space. A necessary condition for the appearance of elastic waves is the occurrence at the moment of perturbation of the medium of forces preventing it, in particular, elasticity. They tend to bring neighboring particles closer together when they move apart, and push them away from each other when they approach each other. Elastic forces, acting on particles far from the source of perturbation, begin to unbalance them. Longitudinal waves characteristic only of gaseous and liquid media, but transverse- also to solids: the reason for this is that the particles that make up these media can move freely, since they are not rigidly fixed, unlike solids. Accordingly, transverse vibrations are fundamentally impossible.

Longitudinal waves arise when the particles of the medium oscillate, orienting themselves along the propagation vector of the perturbation. Transverse waves propagate in a direction perpendicular to the impact vector. In short: if in a medium the deformation caused by a perturbation manifests itself in the form of shear, tension and compression, then we are talking about a solid body, for which both longitudinal and transverse waves are possible. If the appearance of a shift is impossible, then the medium can be any.

Each wave propagates at a certain speed. Under wave speed understand the propagation speed of the disturbance. Since the speed of the wave is a constant value (for a given medium), the distance traveled by the wave is equal to the product of the speed and the time of its propagation. Thus, to find the wavelength, it is necessary to multiply the speed of the wave by the period of oscillations in it:

Wavelength - the distance between two points in space closest to each other at which oscillations occur in the same phase. The wavelength corresponds to the spatial period of the wave, that is, the distance that a point with a constant phase "travels" in a time interval equal to the period of oscillation, therefore

wave number(also called spatial frequency) is the ratio 2 π radian to wavelength: spatial analogue of circular frequency.

Definition: the wave number k is the growth rate of the phase of the wave φ along the spatial coordinate.

3.2. plane wave - a wave whose front has the shape of a plane.

The plane wave front is unlimited in size, the phase velocity vector is perpendicular to the front. A plane wave is a particular solution of the wave equation and a convenient model: such a wave does not exist in nature, since the front of a plane wave begins at and ends at , which, obviously, cannot be.

The equation of any wave is a solution of a differential equation called a wave equation. The wave equation for the function is written as:

where

· - Laplace operator;

· - desired function;

· - radius of the vector of the desired point;

- wave speed;

· - time.

wave surface is the locus of points that are perturbed by the generalized coordinate in the same phase. A special case of a wave surface is a wave front.

BUT) plane wave - this is a wave, the wave surfaces of which are a set of planes parallel to each other.

B) spherical wave is a wave whose wave surfaces are a collection of concentric spheres.

Ray- line, normal and wave surface. Under the direction of propagation of waves understand the direction of the rays. If the propagation medium of the wave is homogeneous and isotropic, the rays are straight lines (moreover, if the wave is plane - parallel straight lines).

The concept of a ray in physics is usually used only in geometric optics and acoustics, since the manifestation of effects that are not studied in these areas, the meaning of the concept of a ray is lost.

3.3. Energy characteristics of the wave

The medium in which the wave propagates has mechanical energy, which is made up of the energies of the oscillatory motion of all its particles. The energy of one particle with mass m 0 is found by the formula: E 0 = m 0 Α 2 w 2/2. The volume unit of the medium contains n = p/m 0 particles (ρ is the density of the medium). Therefore, a unit volume of the medium has the energy w р = nЕ 0 = ρ Α 2 w 2 /2.

Bulk energy density(W p) is the energy of the oscillatory motion of the particles of the medium contained in a unit of its volume:

Energy flow(Ф) - a value equal to the energy carried by the wave through a given surface per unit time:

Wave intensity or energy flux density(I) - a value equal to the energy flux carried by the wave through a single area, perpendicular to the direction of wave propagation:

3.4. electromagnetic wave

electromagnetic wave- the process of electromagnetic field propagation in space.

Occurrence condition electromagnetic waves. Changes in the magnetic field occur when the current strength in the conductor changes, and the current strength in the conductor changes when the speed of electric charges in it changes, that is, when the charges move with acceleration. Therefore, electromagnetic waves should arise during the accelerated movement of electric charges. At a charge rate of zero, there is only an electric field. At a constant charge rate, an electromagnetic field is generated. With the accelerated movement of the charge, an electromagnetic wave is emitted, which propagates in space at a finite speed.

Electromagnetic waves propagate in matter with a finite speed. Here ε and μ are the dielectric and magnetic permeability of the substance, ε 0 and μ 0 are the electrical and magnetic constants: ε 0 \u003d 8.85419 10 -12 F / m, μ 0 \u003d 1.25664 10 -6 Gn / m.

Velocity of electromagnetic waves in vacuum (ε = μ = 1):

Main Features electromagnetic radiation is considered to be the frequency, wavelength and polarization. The wavelength depends on the propagation speed of the radiation. The group velocity of propagation of electromagnetic radiation in vacuum is equal to the speed of light, in other media this speed is less.

Electromagnetic radiation is usually divided into frequency ranges (see table). There are no sharp transitions between the ranges, they sometimes overlap, and the boundaries between them are conditional. Since the speed of propagation of radiation is constant, the frequency of its oscillations is strictly related to the wavelength in vacuum.

Wave interference. coherent waves. Wave coherence conditions.

Optical path length (OPL) of light. Relation between the difference of the r.d.p. waves with a phase difference of oscillations caused by waves.

The amplitude of the resulting oscillation in the interference of two waves. Conditions for maxima and minima of the amplitude during the interference of two waves.

Interference fringes and interference pattern on a flat screen when two narrow long parallel slits are illuminated: a) with red light, b) with white light.

dependency graph V(t) for this case is shown in Fig.1.2.1. Time interval Δt in formula (1.4) one can take any. Attitude ∆V/∆t does not depend on it. Then ΔV=аΔt. Applying this formula to the interval from t about= 0 up to some point t, you can write an expression for the speed:

V(t)=V0 + at. (1.5)

Here V0– speed value at t about= 0. If the directions of velocity and acceleration are opposite, then they speak of uniformly slow motion (Fig. 1.2.2).

For uniformly slow motion, we similarly obtain

V(t) = V0 – at.

Let us analyze the derivation of the formula for the displacement of a body during uniformly accelerated motion. Note that in this case the displacement and the distance traveled are the same number.

Consider a short period of time Δt. From the definition of average speed Vcp = ∆S/∆t you can find the path ∆S = V cp ∆t. The figure shows that the path ∆S numerically equal to the area of ​​a rectangle with width Δt and height Vcp. If the time interval Δt choose small enough, the average speed on the interval Δt coincides with the instantaneous speed at the midpoint. ∆S ≈ V∆t. This ratio is more accurate, the less Δt. Dividing the total travel time into such small intervals and taking into account that the full path S is the sum of the paths traveled during these intervals, you can make sure that on the velocity graph it is numerically equal to the area of ​​the trapezoid:

S= ½ (V 0 + V)t,

substituting (1.5), we obtain for uniformly accelerated motion:

S \u003d V 0 t + (at 2 / 2)(1.6)

For uniformly slow motion L calculated like this:

L= V 0 t–(at 2 /2).

Let's analyze task 1.3.

Let the speed graph have the form shown in Fig. 1.2.4. Draw qualitatively synchronous graphs of the path and acceleration versus time.

Student:- I have never come across the concept of "synchronous graphics", I also do not really understand what it means to "draw with high quality."

– Synchronous graphs have the same scales along the abscissa axis, on which time is plotted. The graphs are arranged one below the other. Synchronous graphs are convenient for comparing several parameters at once at one point in time. In this problem, we will depict the movement qualitatively, that is, without taking into account specific numerical values. For us, it is quite enough to establish whether the function decreases or increases, what form it has, whether it has breaks or breaks, etc. I think we should start reasoning together.

Divide the entire time of movement into three intervals OV, BD, DE. Tell me, what is the nature of the movement on each of them and by what formula will we calculate the distance traveled?

Student:- Location on OV the body was moving uniformly with zero initial speed, so the formula for the path is:

S 1 (t) = at2/2.

The acceleration can be found by dividing the change in speed, i.e. length AB, for a period of time OV.

Student:- Location on BD the body moves uniformly with a speed V 0 acquired by the end of the section OV. Path Formula - S=Vt. There is no acceleration.

S 2 (t) = at 1 2 /2 + V 0 (t–t1).

Given this explanation, write a formula for the path on the site DE.

Student:- In the last section, the movement is uniformly slow. I will argue like this. Until the point in time t 2 the body has already traveled a distance S 2 \u003d at 1 2 / 2 + V (t 2 - t 1).

To it must be added an expression for the equally slow case, given that the time is counted from the value t2 we get the distance traveled, in time t - t 2:

S 3 \u003d V 0 (t–t 2)–/2.

I foresee the question of how to find the acceleration a one . It equals CD/DE. As a result, we get the path traveled in time t>t 2

S (t)= at 1 2 /2+V 0 (t–t 1)– /2.

Student:- In the first section we have a parabola with branches pointing upwards. On the second - a straight line, on the last - also a parabola, but with branches down.

Your drawing is inaccurate. The path graph has no kinks, i.e., parabolas should be smoothly mated with a straight line. We have already said that the speed is determined by the tangent of the slope of the tangent. According to your drawing, it turns out that at the moment t 1 the speed has two values ​​at once. If you build a tangent on the left, then the speed will be numerically equal to tgα, and if you approach the point on the right, then the speed is equal to tgβ. But in our case, the speed is a continuous function. The contradiction is removed if the graph is constructed in this way.

There is another useful relationship between S, a, V and V 0 . We will assume that the movement occurs in one direction. In this case, the movement of the body from the starting point coincides with the path traveled. Using (1.5), express the time t and exclude it from equality (1.6). This is how you get this formula.

Student:– V(t) = V0 + at, means,

t = (V–V 0)/a,

S = V 0 t + at 2 /2 = V 0 (V– V 0)/a + a[(V– V 0)/a] 2 = .

Finally we have:

S= . (1.6a)

Story.

Once, while studying in Göttingen, Niels Bohr was poorly prepared for a colloquium, and his performance turned out to be weak. Bor, however, did not lose heart and concluded with a smile:

“I have heard so many bad speeches here that I ask you to consider mine as revenge.

In this lesson, we will consider an important characteristic of uneven movement - acceleration. In addition, we will consider non-uniform motion with constant acceleration. This movement is also called uniformly accelerated or uniformly slowed down. Finally, we will talk about how to graphically depict the speed of a body as a function of time in uniformly accelerated motion.

Homework

By solving the tasks for this lesson, you will be able to prepare for questions 1 of the GIA and questions A1, A2 of the Unified State Examination.

1. Tasks 48, 50, 52, 54 sb. tasks of A.P. Rymkevich, ed. ten.

2. Write down the dependences of the speed on time and draw graphs of the dependence of the speed of the body on time for the cases shown in fig. 1, cases b) and d). Mark the turning points on the graphs, if any.

3. Consider the following questions and their answers:

Question. Is gravitational acceleration an acceleration as defined above?

Answer. Of course it is. Free fall acceleration is the acceleration of a body that falls freely from a certain height (air resistance must be neglected).

Question. What happens if the acceleration of the body is directed perpendicular to the speed of the body?

Answer. The body will move uniformly in a circle.

Question. Is it possible to calculate the tangent of the angle of inclination using a protractor and a calculator?

Answer. Not! Because the acceleration obtained in this way will be dimensionless, and the dimension of acceleration, as we showed earlier, must have the dimension of m/s 2 .

Question. What can be said about motion if the graph of speed versus time is not a straight line?

Answer. We can say that the acceleration of this body changes with time. Such a movement will not be uniformly accelerated.

Page 8 of 12

§ 7. Movement with uniformly accelerated
rectilinear motion

1. Using a graph of speed versus time, you can get the formula for moving a body with uniform rectilinear motion.

Figure 30 shows a graph of the projection of the speed of uniform movement on the axis X from time. If we set up a perpendicular to the time axis at some point C, then we get a rectangle OABC. The area of ​​this rectangle is equal to the product of the sides OA and OC. But the side length OA is equal to v x, and the side length OC - t, hence S = v x t. The product of the projection of velocity on the axis X and time is equal to the displacement projection, i.e. s x = v x t.

Thus, the displacement projection for uniform rectilinear motion is numerically equal to the area of ​​the rectangle bounded by the coordinate axes, the velocity graph and the perpendicular raised to the time axis.

2. We obtain in a similar way the formula for the projection of displacement in a rectilinear uniformly accelerated motion. To do this, we use the graph of the dependence of the projection of velocity on the axis X from time (Fig. 31). Select a small area on the graph ab and drop the perpendiculars from the points a and b on the time axis. If the time interval D t, corresponding to the section cd on the time axis is small, then we can assume that the speed does not change during this period of time and the body moves uniformly. In this case the figure cabd differs little from a rectangle and its area is numerically equal to the projection of the movement of the body in the time corresponding to the segment cd.

You can break the whole figure into such strips OABC, and its area will be equal to the sum of the areas of all the strips. Therefore, the projection of the movement of the body over time t numerically equal to the area of ​​the trapezoid OABC. From the geometry course, you know that the area of ​​a trapezoid is equal to the product of half the sum of its bases and height: S= (OA + BC)OC.

As can be seen from figure 31, OA = v 0x , BC = v x, OC = t. It follows that the displacement projection is expressed by the formula: s x= (v x + v 0x)t.

With uniformly accelerated rectilinear motion, the speed of the body at any time is equal to v x = v 0x + a x t, hence, s x = (2v 0x + a x t)t.

From here:

To obtain the equation of motion of the body, we substitute into the displacement projection formula its expression through the difference in coordinates s x = x – x 0 .

We get: x – x 0 = v 0x t+ , or

x = x 0 + v 0x t + .

According to the equation of motion, it is possible to determine the coordinate of the body at any time, if the initial coordinate, initial velocity and acceleration of the body are known.

3. In practice, there are often problems in which it is necessary to find the displacement of a body during uniformly accelerated rectilinear motion, but the time of motion is unknown. In these cases, a different displacement projection formula is used. Let's get it.

From the formula for the projection of the speed of uniformly accelerated rectilinear motion v x = v 0x + a x t let's express the time:

t = .

Substituting this expression into the displacement projection formula, we get:

s x = v 0x + .

From here:

s x = , or
–= 2a x s x.

If the initial velocity of the body is zero, then:

2a x s x.

4. Problem solution example

The skier moves down the mountain slope from a state of rest with an acceleration of 0.5 m / s 2 in 20 s and then moves along the horizontal section, having traveled to a stop of 40 m. With what acceleration did the skier move along the horizontal surface? What is the length of the slope of the mountain?

Given:	Decision
v 01 = 0 a 1 = 0.5 m/s 2 t 1 = 20 s s 2 = 40 m v 2 = 0	The movement of the skier consists of two stages: at the first stage, descending from the slope of the mountain, the skier moves with increasing speed in absolute value; at the second stage, when moving along a horizontal surface, its speed decreases. The values ​​related to the first stage of the movement will be written with index 1, and those related to the second stage with index 2.
a 2? s 1?

We will connect the reference system with the Earth, the axis X let's direct in the direction of the skier's speed at each stage of his movement (Fig. 32).

Let's write the equation for the speed of the skier at the end of the descent from the mountain:

v 1 = v 01 + a 1 t 1 .

In projections on the axis X we get: v 1x = a 1x t. Since the projections of velocity and acceleration on the axis X are positive, the modulus of the skier's speed is: v 1 = a 1 t 1 .

Let's write an equation relating the projections of speed, acceleration and movement of the skier at the second stage of movement:

–= 2a 2x s 2x .

Considering that the initial speed of the skier at this stage of the movement is equal to his final speed at the first stage

v 02 = v 1 , v 2x= 0 we get

– = –2a 2 s 2 ; (a 1 t 1) 2 = 2a 2 s 2 .

From here a 2 = ;

a 2 == 0.125 m / s 2.

The module of movement of the skier at the first stage of movement is equal to the length of the mountain slope. Let's write the equation for displacement:

s 1x = v 01x t + .

Hence the length of the mountain slope is s 1 = ;

s 1 == 100 m.

Answer: a 2 \u003d 0.125 m / s 2; s 1 = 100 m.

Questions for self-examination

1. As according to the plot of the projection of the speed of uniform rectilinear motion on the axis X

2. As according to the graph of the projection of the speed of uniformly accelerated rectilinear motion on the axis X from time to determine the projection of the displacement of the body?

3. What formula is used to calculate the projection of the displacement of a body during uniformly accelerated rectilinear motion?

4. What formula is used to calculate the projection of the displacement of a body moving uniformly accelerated and rectilinearly if the initial speed of the body is zero?

Task 7

1. What is the displacement modulus of a car in 2 minutes if during this time its speed has changed from 0 to 72 km/h? What is the coordinate of the car at the time t= 2 min? The initial coordinate is assumed to be zero.

2. The train moves with an initial speed of 36 km/h and an acceleration of 0.5 m/s 2 . What is the displacement of the train in 20 s and its coordinate at the moment of time t= 20 s if the starting coordinate of the train is 20 m?

3. What is the movement of the cyclist for 5 s after the start of braking, if his initial speed during braking is 10 m/s, and the acceleration is 1.2 m/s 2? What is the coordinate of the cyclist at time t= 5 s, if at the initial moment of time it was at the origin?

4. A car moving at a speed of 54 km/h stops when braking for 15 seconds. What is the displacement modulus of the car when braking?

5. Two cars are moving towards each other from two settlements located at a distance of 2 km from each other. The initial speed of one car is 10 m/s and the acceleration is 0.2 m/s 2 , the initial speed of the other is 15 m/s and the acceleration is 0.2 m/s 2 . Determine the time and coordinate of the meeting point of the cars.

Lab #1

Study of uniformly accelerated
rectilinear motion

Objective:

learn how to measure acceleration in uniformly accelerated rectilinear motion; experimentally establish the ratio of the paths traversed by the body during uniformly accelerated rectilinear motion in successive equal time intervals.

Devices and materials:

chute, tripod, metal ball, stopwatch, measuring tape, metal cylinder.

Work order

1. Fix one end of the chute in the foot of the tripod so that it makes a small angle with the surface of the table. At the other end of the chute, put a metal cylinder into it.

2. Measure the paths traveled by the ball in 3 consecutive time intervals equal to 1 s each. This can be done in different ways. You can put marks on the chute with chalk, fixing the position of the ball at time points equal to 1 s, 2 s, 3 s, and measure the distances s_ between these marks. It is possible, releasing the ball from the same height each time, to measure the path s, passed by him first in 1 s, then in 2 s and in 3 s, and then calculate the path traveled by the ball in the second and third seconds. Record the measurement results in table 1.

3. Find the ratio of the path traveled in the second second to the path traveled in the first second, and the path traveled in the third second to the path traveled in the first second. Make a conclusion.

4. Measure the time the ball traveled along the chute and the distance traveled by it. Calculate its acceleration using the formula s = .

5. Using the experimentally obtained value of acceleration, calculate the paths that the ball must travel in the first, second and third seconds of its movement. Make a conclusion.

Table 1

experience number	Experimental data					Theoretical results
	Time t , with	Path s , cm	Time t , with	Way s, cm	Acceleration a, cm/s2	Timet, with	Path s , cm
1	1		1

Now we must find out the most important thing - how the coordinate of the body changes during its rectilinear uniformly accelerated motion. To do this, as we know, you need to know the displacement of the body, because the projection of the displacement vector is exactly equal to the change in coordinates.

The formula for calculating the displacement is easiest to obtain by a graphical method.

With uniformly accelerated motion of the body along the X axis, the speed changes with time according to the formula v x \u003d v 0x + a x t Since time is included in this formula to the first power, the graph for the projection of speed versus time is a straight line, as shown in Figure 39. Line 1 in this figure corresponds to movement with a positive projection of acceleration (speed increases), a straight line 2 - movement with a negative acceleration projection (speed decreases). Both graphs refer to the case when at the moment of time t = O the body has some initial speed v 0 .

The displacement is expressed as an area. Let's select on the graph of the speed of uniformly accelerated movement (Fig. 40) a small area ab and drop from the points a and b perpendiculars to the axis t. Cut length cd on axle t in the chosen scale is equal to that small period of time during which the speed changed from its value at the point a to its value at point b. Under plot ab the graphics turned out to be a narrow strip absd.

If the time interval corresponding to the segment cd, is small enough, then during this short time the speed cannot noticeably change - the movement during this short period of time can be considered uniform. Strip absd therefore, it differs little from a rectangle, and its area is numerically equal to the projection of the displacement in the time corresponding to the segment cd(see § 7).

But it is possible to divide the entire area of ​​the figure located under the velocity graph into such narrow strips. Therefore, the displacement for all time t numerically equal to the area of ​​the trapezoid OABS. The area of ​​a trapezoid, as is known from geometry, is equal to the product of half the sum of its bases and the height. In our case, the length of one of the bases is numerically equal to v ox, the other is v x (see Fig. 40). The height of the trapezoid is numerically equal to t. It follows that the projection s x displacement is expressed by the formula

3s 15.09

If the projection v ox of the initial velocity is equal to zero (at the initial moment of time the body was at rest!), then formula (1) takes the form:

The graph of the speed of such movement is shown in Figure 41.

When using formulas (1) and(2) REMEMBER THAT Sx, Vox and v x can be both positive" and negative - after all, these are projections of vectors s, vo and v to the x-axis.

Thus, we see that with uniformly accelerated motion, the displacement grows with time differently than with uniform motion: now the square of time enters the formula. This means that the displacement increases faster with time than with uniform motion.

How does the coordinate of the body depend on time? Now it is easy to get the formula for calculating the coordinate X at any time for a body moving with uniform acceleration.

projection s x of the displacement vector is equal to the change in the coordinate x-x 0 . Therefore, one can write

From formula (3) it can be seen that, in order to calculate the x coordinate at any time t, you need to know the initial coordinate, initial velocity and acceleration.

Formula (3) describes rectilinear uniformly accelerated motion, just as formula (2) § 6 describes rectilinear uniform motion.

Another formula for moving. To calculate the displacement, you can get another useful formula that does not include time.

From expression vx = v0x + axt. we get the expression for time

t= (v x - v 0x): a x and substitute it into the formula for moving s x , above. Then we get:

These formulas allow you to find the displacement of the body if the acceleration is known, as well as the initial and final speeds of movement. If the initial speed v o is equal to zero, formulas (4) take the form: