The movement of a body thrown horizontally occurs. Movement of a body thrown at an angle to the horizontal

If the speed is not directed vertically, then the movement of the body will be curvilinear.

Let us consider the motion of a body thrown horizontally from a height h with speed (Fig. 1). We will neglect air resistance. To describe the movement, it is necessary to select two coordinate axes - Ox and Oy. The origin of the coordinates is compatible with the initial position of the body. From Figure 1 it is clear that.

Then the motion of the body will be described by the equations:

Analysis of these formulas shows that in the horizontal direction the speed of the body remains unchanged, i.e. the body moves uniformly. In the vertical direction, the body moves uniformly with acceleration , i.e., the same as a body freely falling without an initial speed. Let's find the trajectory equation. To do this, we find the time from equation (1) and, substituting its value into formula (2), we obtain

This is the equation of a parabola. Consequently, a body thrown horizontally moves along a parabola. The speed of the body at any moment of time is directed tangentially to the parabola (see Fig. 1). The velocity module can be calculated using the Pythagorean theorem:

Knowing the height h from which the body is thrown, one can find the time after which the body will fall to the ground. At this moment the y coordinate is equal to the height: . From equation (2) we find

Here – initial speed of the body, – speed of the body at the moment of time t, s– horizontal flight range, h– the height above the surface of the earth from which a body is thrown horizontally with speed .

1.1.33. Kinematic equations for velocity projection:

1.1.34. Kinematic coordinate equations:

1.1.35. Body speed at a point in time t:

In the moment falling to the ground y = h, x = s(Fig. 1.9).

1.1.36. Maximum horizontal flight range:

1.1.37. Height above ground level, from which the body is thrown

horizontally:

Motion of a body thrown at an angle α to the horizontal
with initial speed

1.1.38. The trajectory is a parabola(Fig. 1.10). Curvilinear motion along a parabola is caused by the addition of two rectilinear motions: uniform motion along the horizontal axis and uniform motion along the vertical axis.

Rice. 1.10

( – initial speed of the body, – projections of velocity on the coordinate axes at the moment of time t, – body flight time, hmax– maximum body lifting height, smax– maximum horizontal flight range of the body).

1.1.39. Kinematic projection equations:

;

1.1.40. Kinematic coordinate equations:

;

1.1.41. Height of lifting the body to the top point of the trajectory:

At time , (Figure 1.11).

1.1.42. Maximum lifting height:

1.1.43. Body flight time:

At a moment in time , (Fig. 1.11).

1.1.44. Maximum horizontal body flight range:

1.2. Basic equations of classical dynamics

Dynamics(from Greek dynamis– force) is a branch of mechanics devoted to the study of the movement of material bodies under the influence of forces applied to them. Classical dynamics are based on Newton's laws . From these we obtain all the equations and theorems necessary for solving dynamics problems.

1.2.1. Inertial reporting system – This is a frame of reference in which the body is at rest or moves uniformly and rectilinearly.

1.2.2. Force- This is the result of the interaction of the body with the environment. One of the simplest definitions of force: the influence of a single body (or field) that causes acceleration. Currently, four types of forces or interactions are distinguished:

· gravitational(manifest in the form of universal gravitational forces);

· electromagnetic(existence of atoms, molecules and macrobodies);

· strong(responsible for the connection of particles in nuclei);

· weak(responsible for particle decay).

1.2.3. Principle of superposition of forces: if several forces act on a material point, then the resulting force can be found using the vector addition rule:

Body mass is a measure of body inertia. Any body exhibits resistance when trying to set it in motion or change the module or direction of its speed. This property is called inertia.

1.2.5. Pulse(momentum) is the product of mass T body by its speed v:

1.2.6. Newton's first law: Any material point (body) maintains a state of rest or uniform rectilinear motion until the influence of other bodies forces it (it) to change this state.

1.2.7. Newton's second law(basic equation of the dynamics of a material point): the rate of change of the momentum of the body is equal to the force acting on it (Fig. 1.11):


Rice. 1.11	Rice. 1.12

The same equation in projections onto the tangent and normal to the trajectory of a point:

And .

1.2.8. Newton's third law: the forces with which two bodies act on each other are equal in magnitude and opposite in direction (Fig. 1.12):

1.2.9. Law of conservation of momentum for a closed system: the impulse of a closed system does not change over time (Fig. 1.13):

Where P– the number of material points (or bodies) included in the system.


Rice. 1.13

The law of conservation of momentum is not a consequence of Newton's laws, but is fundamental law of nature, which knows no exceptions, and is a consequence of the homogeneity of space.

1.2.10. The basic equation for the dynamics of translational motion of a system of bodies:

where is the acceleration of the center of inertia of the system; – total mass of the system from P material points.

1.2.11. Center of mass of the system material points (Fig. 1.14, 1.15):

Law of motion of the center of mass: the center of mass of a system moves like a material point, the mass of which is equal to the mass of the entire system and which is acted upon by a force equal to the vector sum of all forces acting on the system.

1.2.12. Impulse of a system of bodies:

where is the speed of the center of inertia of the system.


Rice. 1.14	Rice. 1.15

1.2.13. Theorem on the motion of the center of mass: if the system is in an external stationary uniform field of forces, then no actions within the system can change the movement of the center of mass of the system:

1.3. Forces in mechanics

1.3.1. Body weight connection with gravity and ground reaction:

Acceleration of free fall (Fig. 1.16).

Rice. 1.16

Weightlessness is a state in which body weight is zero. In a gravitational field, weightlessness occurs when a body moves only under the influence of gravity. If a = g, That P = 0.

1.3.2. Relationship between weight, gravity and acceleration:

1.3.3. Sliding friction force(Fig. 1.17):

where is the sliding friction coefficient; N– normal pressure force.

1.3.5. Basic relations for a body on an inclined plane(Fig. 1.19). :

· friction force: ;

· resultant force: ;

· rolling force: ;

· acceleration:

Rice. 1.19

1.3.6. Hooke's law for a spring: spring extension X proportional to the elastic force or external force:

Where k– spring stiffness.

1.3.7. Potential energy of an elastic spring:

1.3.8. Work done by a spring:

1.3.9. Voltage– a measure of internal forces arising in a deformable body under the influence of external influences (Fig. 1.20):

where is the cross-sectional area of the rod, d– its diameter, – the initial length of the rod, – the increment in the length of the rod.


Rice. 1.20	Rice. 1.21

1.3.10. Strain diagram – graph of normal stress σ = F/S from relative elongation ε = Δ l/l when the body is stretched (Fig. 1.21).

1.3.11. Young's modulus– quantity characterizing the elastic properties of the rod material:

1.3.12. Bar length increment proportional to voltage:

1.3.13. Relative longitudinal tension (compression):

1.3.14. Relative transverse tension (compression):

where is the initial transverse dimension of the rod.

1.3.15. Poisson's ratio– the ratio of the relative transverse tension of the rod to the relative longitudinal tension:

1.3.16. Hooke's law for a rod: the relative increment in the length of the rod is directly proportional to the stress and inversely proportional to Young’s modulus:

1.3.17. Volumetric potential energy density:

1.3.18. Relative shift ( fig1.22, 1.23 ):

where is the absolute shift.


Rice. 1.22	Fig.1.23

1.3.19. Shear modulusG- a value that depends on the properties of the material and is equal to the tangential stress at which (if such huge elastic forces were possible).

1.3.20. Tangential elastic stress:

1.3.21. Hooke's law for shear:

1.3.22. Specific potential energy bodies in shear:

1.4. Non-inertial frames of reference

Non-inertial reference frame– an arbitrary reference system that is not inertial. Examples of non-inertial systems: a system moving in a straight line with constant acceleration, as well as a rotating system.

Inertial forces are caused not by the interaction of bodies, but by the properties of the non-inertial reference systems themselves. Newton's laws do not apply to inertial forces. Inertial forces are non-invariant with respect to the transition from one frame of reference to another.

In a non-inertial system, you can also use Newton's laws if you introduce inertial forces. They are fictitious. They are introduced specifically to take advantage of Newton's equations.

1.4.1. Newton's equation for a non-inertial reference frame

where is the acceleration of the body of mass T relative to a non-inertial system; – inertial force is a fictitious force due to the properties of the reference system.

1.4.2. Centripetal force– inertial force of the second kind, applied to a rotating body and directed radially to the center of rotation (Fig. 1.24):

where is the centripetal acceleration.

1.4.3. Centrifugal force– inertia force of the first kind, applied to the connection and directed radially from the center of rotation (Fig. 1.24, 1.25):

where is the centrifugal acceleration.


Rice. 1.24	Rice. 1.25

1.4.4. Gravity acceleration dependence g depending on the latitude of the area is shown in Fig. 1.25.

Gravity is the result of the addition of two forces: and ; Thus, g(and therefore mg) depends on the latitude of the area:

where ω is the angular velocity of the Earth's rotation.

1.4.5. Coriolis force– one of the forces of inertia that exists in a non-inertial reference system due to rotation and the laws of inertia, manifesting itself when moving in a direction at an angle to the axis of rotation (Fig. 1.26, 1.27).

where is the angular velocity of rotation.


Rice. 1.26	Rice. 1.27

1.4.6. Newton's equation for non-inertial reference systems taking into account all forces will take the form

where is the inertial force due to the translational motion of the non-inertial reference frame; And – two inertia forces caused by the rotational motion of the reference system; – acceleration of the body relative to a non-inertial reference frame.

1.5. Energy. Job. Power.
Conservation laws

1.5.1. Energy– a universal measure of various forms of movement and interaction of all types of matter.

1.5.2. Kinetic energy– function of the state of the system, determined only by the speed of its movement:

Kinetic energy of a body is a scalar physical quantity equal to half the product of mass m body per square of its speed.

1.5.3. Theorem on the change in kinetic energy. The work of the resultant forces applied to the body is equal to the change in the kinetic energy of the body, or, in other words, the change in the kinetic energy of the body is equal to the work A of all forces acting on the body.

1.5.4. Relationship between kinetic energy and momentum:

1.5.5. Work of force– quantitative characteristic of the process of energy exchange between interacting bodies. Mechanical work .

1.5.6. Constant force work:

If a body moves in a straight line and is acted upon by a constant force F, which makes a certain angle α with the direction of movement (Fig. 1.28), then the work of this force is determined by the formula:

Where F– force module, ∆r– module of displacement of the point of application of force, – angle between the direction of force and displacement.

If< /2, то работа силы положительна. Если >/2, then the work done by the force is negative. When = /2 (the force is directed perpendicular to the displacement), then the work done by the force is zero.


Rice. 1.28	Rice. 1.29

Constant force work F when moving along the axis x to a distance (Fig. 1.29) is equal to the projection of force on this axis multiplied by the displacement:

In Fig. Figure 1.27 shows the case when A < 0, т.к. >/2 – obtuse angle.

1.5.7. Elementary work d A strength F on elementary displacement d r is a scalar physical quantity equal to the scalar product of force and displacement:

1.5.8. Variable force work on trajectory section 1 – 2 (Fig. 1.30):

Rice. 1.30

1.5.9. Instantaneous power equal to the work done per unit time:

1.5.10. Average power for a period of time:

1.5.11. Potential energy body at a given point is a scalar physical quantity, equal to the work done by a potential force when moving a body from this point to another, taken as the zero potential energy reference.

Potential energy is determined up to some arbitrary constant. This is not reflected in the physical laws, since they include either the difference in potential energies in two positions of the body or the derivative of potential energy with respect to coordinates.

Therefore, the potential energy at a certain position is considered equal to zero, and the energy of the body is measured relative to this position (zero reference level).

1.5.12. Principle of minimum potential energy. Any closed system tends to transition to a state in which its potential energy is minimal.

1.5.13. The work of conservative forces equal to the change in potential energy

1.5.14. Vector circulation theorem: if the circulation of any force vector is zero, then this force is conservative.

The work of conservative forces along a closed contour L is zero(Fig. 1.31):


Rice. 1.31

1.5.15. Potential energy of gravitational interaction between the masses m And M(Fig. 1.32):

1.5.16. Potential energy of a compressed spring(Fig. 1.33):


Rice. 1.32	Rice. 1.33

1.5.17. Total mechanical energy of the system equal to the sum of kinetic and potential energies:

E = E k + E P.

1.5.18. Body potential energy on high h above the ground

E n = mgh.

1.5.19. Relationship between potential energy and force:

Or or

1.5.20. Law of conservation of mechanical energy(for a closed system): the total mechanical energy of a conservative system of material points remains constant:

1.5.21. Law of conservation of momentum for a closed system of bodies:

1.5.22. Law of conservation of mechanical energy and momentum with an absolutely elastic central impact (Fig. 1.34):

Where m 1 and m 2 – body masses; and – the speed of the bodies before the impact.


Rice. 1.34	Rice. 1.35

1.5.23. Speeds of bodies after an absolutely elastic impact (Fig. 1.35):

1.5.24. Speed of bodies after a completely inelastic central impact (Fig. 1.36):

1.5.25. Law of conservation of momentum when the rocket is moving (Fig. 1.37):

where and are the mass and speed of the rocket; and the mass and speed of the emitted gases.


Rice. 1.36	Rice. 1.37

1.5.26. Meshchersky equation for a rocket.

If the speed \(~\vec \upsilon_0\) is not directed vertically, then the movement of the body will be curvilinear.

Consider the motion of a body thrown horizontally from a height h with speed \(~\vec \upsilon_0\) (Fig. 1). We will neglect air resistance. To describe the movement, it is necessary to select two coordinate axes - Ox And Oy. The origin of the coordinates is compatible with the initial position of the body. From Figure 1 it is clear that υ 0x = υ 0 , υ 0y = 0, g x = 0, g y = g.

Then the motion of the body will be described by the equations:

\(~\upsilon_x = \upsilon_0,\ x = \upsilon_0 t; \qquad (1)\) \(~\upsilon_y = gt,\ y = \frac(gt^2)(2). \qquad (2) \)

Analysis of these formulas shows that in the horizontal direction the speed of the body remains unchanged, i.e. the body moves uniformly. In the vertical direction, the body moves uniformly with acceleration \(~\vec g\), i.e., the same as a body freely falling without an initial speed. Let's find the trajectory equation. To do this, from equation (1) we find the time \(~t = \frac(x)(\upsilon_0)\) and, substituting its value into formula (2), we obtain\[~y = \frac(g)(2 \ upsilon^2_0) x^2\] .

\(~\upsilon = \sqrt(\upsilon^2_x + \upsilon^2_y) = \sqrt(\upsilon^2_0 + (gt)^2).\)

Knowing the altitude h with which the body is thrown, time can be found t 1 through which the body will fall to the ground. At this moment the coordinate y equal to height: y 1 = h. From equation (2) we find\[~h = \frac(gt^2_1)(2)\]. From here

\(~t_1 = \sqrt(\frac(2h)(g)). \qquad (3)\)

Formula (3) determines the flight time of the body. During this time the body will travel a distance in the horizontal direction l, which is called the flight range and which can be found based on formula (1), taking into account that l 1 = x. Therefore, \(~l = \upsilon_0 \sqrt(\frac(2h)(g))\) is the flight range of the body. The modulus of the body's velocity at this moment is \(~\upsilon_1 = \sqrt(\upsilon^2_0 + 2gh).\).

Literature

Aksenovich L. A. Physics in secondary school: Theory. Tasks. Tests: Textbook. allowance for institutions providing general education. environment, education / L. A. Aksenovich, N. N. Rakina, K. S. Farino; Ed. K. S. Farino. - Mn.: Adukatsiya i vyhavanne, 2004. - P. 15-16.

Theory

If a body is thrown at an angle to the horizon, then in flight it is acted upon by the force of gravity and the force of air resistance. If the resistance force is neglected, then the only force left is gravity. Therefore, due to Newton's 2nd law, the body moves with acceleration equal to the acceleration of gravity; acceleration projections on the coordinate axes are equal a x = 0, and y= -g.

Any complex movement of a material point can be represented as a superposition of independent movements along the coordinate axes, and in the direction of different axes the type of movement may differ. In our case, the motion of a flying body can be represented as the superposition of two independent motions: uniform motion along the horizontal axis (X-axis) and uniformly accelerated motion along the vertical axis (Y-axis) (Fig. 1).

The body's velocity projections therefore change with time as follows:

where is the initial speed, α is the throwing angle.

The body coordinates therefore change like this:

With our choice of the origin of coordinates, the initial coordinates (Fig. 1) Then

The second time value at which the height is zero is zero, which corresponds to the moment of throwing, i.e. this value also has a physical meaning.

We obtain the flight range from the first formula (1). Flight range is the coordinate value X at the end of the flight, i.e. at a time equal to t 0. Substituting value (2) into the first formula (1), we get:

(3)

From this formula it can be seen that the greatest flight range is achieved at a throwing angle of 45 degrees.

The maximum lifting height of the thrown body can be obtained from the second formula (1). To do this, you need to substitute a time value equal to half the flight time (2) into this formula, because It is at the midpoint of the trajectory that the flight altitude is maximum. Carrying out calculations, we get

Basic units of measurement of quantities in the SI system are:

unit of measurement of length - meter (1 m),
time - second (1 s),
mass - kilogram (1 kg),
amount of substance - mole (1 mol),
temperatures - kelvin (1 K),
electric current - ampere (1 A),
For reference: luminous intensity - candela (1 cd, actually not used when solving school problems).

When performing calculations in the SI system, angles are measured in radians.

If a physics problem does not indicate in which units the answer must be given, it must be given in SI units or in quantities derived from them corresponding to the physical quantity that is asked about in the problem. For example, if the problem requires finding speed, and it does not say how it should be expressed, then the answer must be given in m/s.

For convenience, in physics problems it is often necessary to use submultiple (decreasing) and multiple (increasing) prefixes. they can be applied to any physical quantity. For example, mm - millimeter, kt - kiloton, ns - nanosecond, Mg - megagram, mmol - millimole, μA - microampere. Remember that there are no double prefixes in physics. For example, mcg is microgram, not millikilogram. Please note that when adding and subtracting quantities, you can only operate with quantities of the same dimension. For example, kilograms can only be added with kilograms, only millimeters can be subtracted from millimeters, and so on. When converting values, use the following table.

Path and movement

Kinematics is a branch of mechanics in which the movement of bodies is considered without identifying the causes of this movement.

Mechanical movement A body is called a change in its position in space relative to other bodies over time.

Every body has certain dimensions. However, in many mechanics problems there is no need to indicate the positions of individual parts of the body. If the dimensions of a body are small compared to the distances to other bodies, then this body can be considered material point. So, when moving a car over long distances, its length can be neglected, since the length of the car is small compared to the distances it travels.

It is intuitively clear that the characteristics of movement (speed, trajectory, etc.) depend on where we look at it from. Therefore, to describe the motion, the concept of a reference system is introduced. Reference system (FR)– a combination of a reference body (it is considered absolutely solid), a coordinate system attached to it, a ruler (a device that measures distances), a clock and a time synchronizer.

Moving over time from one point to another, a body (material point) describes a certain line in a given CO, which is called body movement trajectory.

By moving the body called a directed straight line segment connecting the initial position of a body with its final position. Displacement is a vector quantity. By moving, the movement can increase, decrease and become equal to zero in the process.

Passed path equal to the length of the trajectory traveled by the body over some time. Path is a scalar quantity. The path cannot decrease. The path only increases or remains constant (if the body does not move). When a body moves along a curved path, the module (length) of the displacement vector is always less than the distance traveled.

At uniform(at constant speed) moving path L can be found by the formula:

Where: v– body speed, t- the time during which it moved. When solving problems in kinematics, displacement is usually found from geometric considerations. Often geometric considerations for finding displacement require knowledge of the Pythagorean theorem.

average speed

Speed– a vector quantity characterizing the speed of movement of a body in space. Speed can be medium or instantaneous. Instantaneous speed describes movement at a given specific moment in time at a given specific point in space, and average speed characterizes the entire movement as a whole, in general, without describing the details of movement in each specific area.

Average travel speed is the ratio of the entire path to the entire time of movement:

Where: L full - the entire path that the body has traveled, t full – all the time of movement.

Average moving speed is the ratio of the total movement to the entire movement time:

This quantity is directed in the same way as the complete movement of the body (that is, from the initial point of movement to the end point). However, do not forget that the total displacement is not always equal to the algebraic sum of displacements at certain stages of movement. The vector of total displacement is equal to the vector sum of displacements at individual stages of movement.

When solving kinematics problems, do not make a very common mistake. The average speed, as a rule, is not equal to the arithmetic mean of the body’s speeds at each stage of movement. The arithmetic mean is obtained only in some special cases.
And even more so, the average speed is not equal to one of the speeds with which the body moved during the movement, even if this speed had approximately an intermediate value relative to other speeds with which the body moved.

Uniformly accelerated linear motion

Acceleration– vector physical quantity that determines the rate of change in the speed of a body. The acceleration of a body is the ratio of the change in speed to the period of time during which the change in speed occurred:

Where: v 0 – initial speed of the body, v– final speed of the body (that is, after a period of time t).

Further, unless otherwise specified in the problem statement, we believe that if a body moves with acceleration, then this acceleration remains constant. This body movement is called uniformly accelerated(or equally variable). With uniformly accelerated motion, the speed of a body changes by the same amount over any equal intervals of time.

Uniformly accelerated motion is actually accelerated when the body increases the speed of movement, and slowed down when the speed decreases. To simplify problem solving, it is convenient to take acceleration with a “–” sign for slow motion.

From the previous formula follows another more common formula that describes change in speed over time with uniformly accelerated motion:

Move (but not path) with uniformly accelerated motion is calculated using the formulas:

The last formula uses one feature of uniformly accelerated motion. With uniformly accelerated motion, the average speed can be calculated as the arithmetic mean of the initial and final speeds (this property is very convenient to use when solving some problems):

Calculating the path is getting more complicated. If the body did not change the direction of movement, then with uniformly accelerated rectilinear motion, the path is numerically equal to the displacement. And if it changed, you need to separately count the path to the stop (the moment of reversal) and the path after the stop (the moment of reversal). And simply substituting time into the formulas for movement in this case will lead to a typical error.

Coordinate with uniformly accelerated motion changes according to the law:

Projection of speed during uniformly accelerated motion it changes according to the following law:

Similar formulas are obtained for the remaining coordinate axes.

Free fall vertically

All bodies located in the gravitational field of the Earth are affected by the force of gravity. In the absence of support or suspension, this force causes bodies to fall towards the surface of the Earth. If we neglect air resistance, then the movement of bodies only under the influence of gravity is called free fall. The force of gravity imparts to any body, regardless of its shape, mass and size, the same acceleration, called the acceleration of gravity. Near the Earth's surface acceleration of gravity is:

This means that the free fall of all bodies near the Earth's surface is uniformly accelerated (but not necessarily rectilinear) motion. First, let's consider the simplest case of free fall, when the body moves strictly vertically. Such motion is uniformly accelerated rectilinear motion, therefore all previously studied patterns and focuses of such motion are also suitable for free fall. Only the acceleration is always equal to the acceleration of gravity.

Traditionally, in free fall, the OY axis is directed vertically. There's nothing wrong with that. You just need in all formulas instead of the index " X" write " at" The meaning of this index and the rule for determining the signs are preserved. Where to direct the OY axis is your choice, depending on the convenience of solving the problem. There are 2 options: up or down.

Let us present several formulas that are solutions to some specific problems in kinematics for vertical free fall. For example, the speed with which a body falling from a height will fall h without initial speed:

Time of a body falling from a height h without initial speed:

The maximum height to which a body thrown vertically upward with initial speed will rise v 0, the time it takes for this body to rise to its maximum height, and the total flight time (before returning to the starting point):

Horizontal throw

When thrown horizontally with initial speed v 0 the movement of a body is conveniently considered as two movements: uniform along the OX axis (along the OX axis there are no forces preventing or assisting the movement) and uniformly accelerated movement along the OY axis.

The speed at any moment of time is directed tangentially to the trajectory. It can be decomposed into two components: horizontal and vertical. The horizontal component always remains unchanged and is equal to v x = v 0 . And the vertical increases according to the laws of accelerated motion v y = GT. Wherein full body speed can be found using the formulas:

It is important to understand that the time a body falls to the ground in no way depends on the horizontal speed with which it was thrown, but is determined only by the height from which the body was thrown. The time a body falls to the ground is found by the formula:

While the body is falling, it simultaneously moves along the horizontal axis. Hence, body flight range or the distance that the body can fly along the OX axis will be equal to:

Angle between horizon and the speed of the body can be easily found from the relation:

Also, sometimes in problems they may ask about the moment of time at which the full speed of the body will be inclined at a certain angle to verticals. Then this angle will be found from the relationship:

It is important to understand which angle appears in the problem (vertical or horizontal). This will help you choose the right formula. If we solve this problem using the coordinate method, then the general formula for the law of coordinate change during uniformly accelerated motion is:

Transforms into the following law of motion along the OY axis for a body thrown horizontally:

With its help, we can find the height at which the body will be located at any given time. In this case, at the moment the body falls to the ground, the coordinate of the body along the OY axis will be equal to zero. It is obvious that the body moves uniformly along the OX axis, therefore, within the framework of the coordinate method, the horizontal coordinate will change according to the law:

Throw at an angle to the horizon (from ground to ground)

Maximum lift height when throwing at an angle to the horizontal (relative to the initial level):

Time to rise to maximum height when throwing at an angle to the horizontal:

Flight range and total flight time of a body thrown at an angle to the horizon (provided that the flight ends at the same altitude from which it began, i.e. the body was thrown, for example, from ground to ground):

The minimum speed of a body thrown at an angle to the horizontal is at the highest point of ascent, and is equal to:

The maximum speed of a body thrown at an angle to the horizontal is at the moments of the throw and fall to the ground, and is equal to the initial one. This statement is only true for ground to ground throws. If the body continues to fly below the level from which it was thrown, then it will acquire greater and greater speed there.

Speed addition

The motion of bodies can be described in various reference systems. From the point of view of kinematics, all reference systems are equal. However, the kinematic characteristics of movement, such as trajectory, displacement, speed, turn out to be different in different systems. Quantities that depend on the choice of the reference system in which they are measured are called relative. Thus, rest and movement of a body are relative.

Thus, the absolute speed of a body is equal to the vector sum of its speed relative to the moving frame of reference and the speed of the moving frame of reference itself. Or, in other words, the speed of a body in a stationary frame of reference is equal to the vector sum of the speed of the body in a moving frame of reference and the speed of the moving frame of reference relative to the stationary one.

Uniform movement around a circle

The movement of a body in a circle is a special case of curvilinear movement. This type of movement is also considered in kinematics. In curvilinear motion, the velocity vector of the body is always directed tangentially to the trajectory. The same thing happens when moving in a circle (see figure). The uniform motion of a body in a circle is characterized by a number of quantities.

Period- the time during which a body, moving in a circle, makes one full revolution. The unit of measurement is 1 s. The period is calculated using the formula:

Frequency– the number of revolutions made by a body moving in a circle per unit time. The unit of measurement is 1 rev/s or 1 Hz. The frequency is calculated using the formula:

In both formulas: N– number of revolutions per time t. As can be seen from the above formulas, period and frequency are reciprocal quantities:

At uniform rotation speed body will be defined as follows:

Where: l– circumference or path traveled by a body in a time equal to the period T. When a body moves in a circle, it is convenient to consider the angular displacement φ (or angle of rotation), measured in radians. Angular velocity ω body at a given point is called the ratio of small angular displacement Δ φ to a short period of time Δ t. Obviously, in a time equal to the period T the body will pass an angle equal to 2 π , therefore, with uniform motion in a circle, the formulas are satisfied:

Angular velocity is measured in rad/s. Don't forget to convert angles from degrees to radians. Arc length l is related to the rotation angle by the relation:

Communication between linear speed module v and angular velocity ω :

When a body moves in a circle with a constant absolute speed, only the direction of the velocity vector changes, therefore the movement of a body in a circle with a constant absolute speed is a motion with acceleration (but not uniformly accelerated), since the direction of the speed changes. In this case, the acceleration is directed radially towards the center of the circle. It is called normal, or centripetal acceleration, since the acceleration vector at any point of the circle is directed towards its center (see figure).

on that website. To do this, you need nothing at all, namely: devote three to four hours every day to preparing for the CT in physics and mathematics, studying theory and solving problems. The fact is that the CT is an exam where it is not enough just to know physics or mathematics, you also need to be able to quickly and without failures solve a large number of problems on different topics and of varying complexity. The latter can only be learned by solving thousands of problems.

Learn all the formulas and laws in physics, and formulas and methods in mathematics. In fact, this is also very simple to do; there are only about 200 necessary formulas in physics, and even a little less in mathematics. In each of these subjects there are about a dozen standard methods for solving problems of a basic level of complexity, which can also be learned, and thus, completely automatically and without difficulty solving most of the CT at the right time. After this, you will only have to think about the most difficult tasks.

Attend all three stages of rehearsal testing in physics and mathematics. Each RT can be visited twice to decide on both options. Again, on the CT, in addition to the ability to quickly and efficiently solve problems, and knowledge of formulas and methods, you must also be able to properly plan time, distribute forces, and most importantly, correctly fill out the answer form, without confusing the numbers of answers and problems, or your own last name. Also, during RT, it is important to get used to the style of asking questions in problems, which may seem very unusual to an unprepared person at the DT.

Successful, diligent and responsible implementation of these three points will allow you to show an excellent result at the CT, the maximum of what you are capable of.

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