Formula for the acceleration of a body during uniformly accelerated motion. Uniformly accelerated linear motion
Mechanics
Kinematics formulas:
Kinematics
Mechanical movement
Mechanical movement is called a change in the position of a body (in space) relative to other bodies (over time).
Relativity of motion. Reference system
To describe the mechanical movement of a body (point), you need to know its coordinates at any moment in time. To determine coordinates, select reference body and connect with him coordinate system. Often the reference body is the Earth, which is associated with a rectangular Cartesian coordinate system. To determine the position of a point at any time, you must also set the beginning of the time count.
The coordinate system, the reference body with which it is associated, and the device for measuring time form reference system, relative to which the movement of the body is considered.
Material point
A body whose dimensions can be neglected under given motion conditions is called material point.
A body can be considered a material point if its dimensions are small compared to the distance it travels, or compared to the distances from it to other bodies.
Trajectory, path, movement
Trajectory of movement called the line along which the body moves. The path length is called the path traveled. Path– scalar physical quantity, can only be positive.
By moving is the vector connecting the starting and ending points of the trajectory.
The movement of a body in which all its points at a given moment in time move equally is called forward movement. To describe the translational motion of a body, it is enough to select one point and describe its movement.
A movement in which the trajectories of all points of the body are circles with centers on the same line and all planes of the circles are perpendicular to this line is called rotational movement.
Meter and second
To determine the coordinates of a body, you must be able to measure the distance on a straight line between two points. Any process of measuring a physical quantity consists of comparing the measured quantity with the unit of measurement of this quantity.
The unit of length in the International System of Units (SI) is meter. A meter is equal to approximately 1/40,000,000 of the earth's meridian. According to modern understanding, a meter is the distance that light travels in emptiness in 1/299,792,458 of a second.
To measure time, some periodically repeating process is selected. The SI unit of measurement of time is second. A second is equal to 9,192,631,770 periods of radiation from a cesium atom during the transition between two levels of the hyperfine structure of the ground state.
In SI, length and time are taken to be independent of other quantities. Such quantities are called main.
Instantaneous speed
To quantitatively characterize the process of body movement, the concept of movement speed is introduced.
Instant speed translational motion of a body at time t is the ratio of a very small displacement Ds to a small period of time Dt during which this displacement occurred:
Instantaneous speed is a vector quantity. The instantaneous speed of movement is always directed tangentially to the trajectory in the direction of body movement.
The unit of speed is 1 m/s. A meter per second is equal to the speed of a rectilinearly and uniformly moving point, at which the point moves a distance of 1 m in 1 s.
Acceleration
Acceleration is called a vector physical quantity equal to the ratio of a very small change in the velocity vector to the short period of time during which this change occurred, i.e. This is a measure of the rate of change of speed:
A meter per second per second is an acceleration at which the speed of a body moving rectilinearly and uniformly accelerates changes by 1 m/s in a time of 1 s.
The direction of the acceleration vector coincides with the direction of the speed change vector () for very small values of the time interval during which the speed change occurs.
If a body moves in a straight line and its speed increases, then the direction of the acceleration vector coincides with the direction of the velocity vector; when the speed decreases, it is opposite to the direction of the speed vector.
When moving along a curved path, the direction of the velocity vector changes during the movement, and the acceleration vector can be directed at any angle to the velocity vector.
Uniform, uniformly accelerated linear motion
Motion at constant speed is called uniform rectilinear movement. With uniform rectilinear motion, the body moves in a straight line and covers the same paths in any equal intervals of time.
A movement in which a body makes unequal movements at equal intervals of time is called uneven movement. With such movement, the speed of the body changes over time.
Equally variable is a movement in which the speed of a body changes by the same amount over any equal periods of time, i.e. movement with constant acceleration.
Uniformly accelerated is called uniformly alternating motion in which the magnitude of the speed increases. Equally slow– uniformly alternating motion, in which the speed decreases.
This is a movement in which the speed of a body changes equally over any equal periods of time, i.e. acceleration is constant.
Examples of such motion are the free fall of bodies near the surface of the Earth and motion under the influence of a constant force.
With uniformly accelerated linear motion, the coordinate of the body changes over time in accordance with the law of motion:
Where x 0 – initial coordinate of the material point, 0 x– projection of initial speed and a x– projection of point acceleration onto axis 0 X.
Projection of the velocity of a material point onto axis 0 X in this case it changes according to the following law:
In this case, the projections of velocity and acceleration can take on different values, including negative ones.
Dependency graphs x (t) And x(t) represent a straight line and a parabola, respectively, and, as in algebra, the coefficients in the equations of a straight line and a parabola can be used to judge the location of the graph of a function relative to the coordinate axes.
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Figure 6 shows graphs for x(t),x (t),s(t) when x 0 > 0, 0 x > 0,a x < 0. Соответственно прямая(t) has a negative slope (tg =a x < 0).
3. Rotational motion and its kinematic parameters. Relationship between angular and linear speeds.
Uniform movement around a circle occurs at a constant absolute speed, i.e. = const (Fig. 7). However, the direction of velocity during such motion continuously changes, therefore the uniform motion of a body in a circle is motion with acceleration.
To describe the uniform motion of a body in a circle, the following physical quantities are introduced: period,circulation frequency,linear speed,angular velocity And centripetal acceleration.
Circulation periodT– the time it takes to complete one full revolution.
Frequency is the number of revolutions made by the body in 1 s. The SI unit of frequency of circulation is c –1.
Frequency and period of revolution are related by the relation.
When a point moves around a circle, the velocity vector constantly changes its direction (Fig. 8).
With uniform motion of a body in a circle, the path segment s, traveled during a period of time t, is the length of the arc of a circle. The relationship is constant over time and is called linear speed module. For a time equal to the circulation period T, the point travels a distance equal to the circumference of the circle 2 R, That's why
The speed of rotation of solid bodies is usually characterized by a physical quantity called angular velocity , the module of which is equal to the ratio of the angle of rotation of the body to the period of time during which this rotation is completed (Fig. 8):
The SI unit of angular velocity is c –1.
Since the orientation of a rigid body is the same in all reference systems moving translationally relative to each other, the angular velocity of rotation of the rigid body will be the same in all reference systems moving translationally relative to each other.
With uniform rotation of a rigid body about a certain axis, any point of this body moves around the same axis in a circle of radius R with linear speed, which is equal to
If the initial coordinates of the point are equal ( R; 0), then its coordinates change according to the law x(t) =R cos t And y(t) =R sin t.
We consider the highway to be straight. Let's write down the equation of motion of a cyclist. Since the cyclist moved uniformly, his equation of motion is:
(we place the origin of coordinates at the starting point, so the initial coordinate of the cyclist is zero).
The motorcyclist was moving at uniform acceleration. He also started moving from the starting point, so his initial coordinate is zero, the initial speed of the motorcyclist is also zero (the motorcyclist began to move from a state of rest).
Considering that the motorcyclist started moving later, the equation of motion for the motorcyclist is:
In this case, the speed of the motorcyclist changed according to the law:
At the moment when the motorcyclist caught up with the cyclist, their coordinates are equal, i.e. or:
Solving this equation for , we find the meeting time:
This is a quadratic equation. We define the discriminant:
Determining the roots:
Let's substitute numerical values into the formulas and calculate:
We discard the second root as not corresponding to the physical conditions of the problem: the motorcyclist could not catch up with the cyclist 0.37 s after the cyclist started moving, since he himself left the starting point only 2 s after the cyclist started.
Thus, the time when the motorcyclist caught up with the cyclist:
Let's substitute this time value into the formula for the law of change in speed of a motorcyclist and find the value of his speed at this moment:
2) Graphic method.
On the same coordinate plane we build graphs of changes over time in the coordinates of the cyclist and motorcyclist (the graph for the cyclist’s coordinates is in red, for the motorcyclist – in green). It can be seen that the dependence of the coordinate on time for a cyclist is a linear function, and the graph of this function is a straight line (the case of uniform rectilinear motion). The motorcyclist was moving with uniform acceleration, so the dependence of the motorcyclist’s coordinates on time is a quadratic function, the graph of which is a parabola.
The part of mechanics in which motion is studied without considering the reasons causing this or that character of motion is called kinematics.Mechanical movement called a change in the position of a body relative to other bodies
Reference system called the body of reference, the coordinate system associated with it and the clock.
Body of reference name the body relative to which the position of other bodies is considered.
Material point is a body whose dimensions can be neglected in this problem.
Trajectory called a mental line that a material point describes during its movement.
According to the shape of the trajectory, the movement is divided into:
A) rectilinear- the trajectory is a straight line segment;
b) curvilinear- the trajectory is a segment of a curve.
Path is the length of the trajectory that a material point describes over a given period of time. This is a scalar quantity.
Moving is a vector connecting the initial position of a material point with its final position (see figure).
It is very important to understand how a path differs from a movement. The most important difference is that movement is a vector with a beginning at the point of departure and an end at the destination (it does not matter at all what route this movement took). And the path is, on the contrary, a scalar quantity that reflects the length of the trajectory traveled.
Uniform linear movement called a movement in which a material point makes the same movements over any equal periods of time
Speed of uniform linear motion is called the ratio of movement to the time during which this movement occurred:
For uneven motion they use the concept average speed. Average speed is often introduced as a scalar quantity. This is the speed of such uniform motion in which the body travels the same path in the same time as during uneven motion:
Instant speed call the speed of a body at a given point in the trajectory or at a given moment in time.
Uniformly accelerated linear motion- this is a rectilinear movement in which the instantaneous speed for any equal periods of time changes by the same amount
The dependence of the body coordinates on time in uniform rectilinear motion has the form: x = x 0 + V x t, where x 0 is the initial coordinate of the body, V x is the speed of movement.
Free fall called uniformly accelerated motion with constant acceleration g = 9.8 m/s 2, independent of the mass of the falling body. It occurs only under the influence of gravity.
Free fall speed is calculated using the formula:
Vertical movement is calculated using the formula:
One type of motion of a material point is motion in a circle. With such movement, the speed of the body is directed along a tangent drawn to the circle at the point where the body is located (linear speed). You can describe the position of a body on a circle using a radius drawn from the center of the circle to the body. The displacement of a body when moving in a circle is described by turning the radius of the circle connecting the center of the circle with the body. The ratio of the angle of rotation of the radius to the period of time during which this rotation occurred characterizes the speed of movement of the body in a circle and is called angular velocity ω:
Angular velocity is related to linear velocity by the relation
where r is the radius of the circle.
The time it takes a body to complete a complete revolution is called circulation period. The reciprocal of the period is the circulation frequency - ν
Since during uniform motion in a circle the velocity module does not change, but the direction of the velocity changes, with such motion there is acceleration. He is called centripetal acceleration, it is directed radially to the center of the circle:
Basic concepts and laws of dynamics
The part of mechanics that studies the reasons that caused the acceleration of bodies is called dynamics
Newton's first law:
There are reference systems relative to which a body maintains its speed constant or is at rest if other bodies do not act on it or the action of other bodies is compensated.
The property of a body to maintain a state of rest or uniform linear motion with balanced external forces acting on it is called inertia. The phenomenon of maintaining the speed of a body under balanced external forces is called inertia. Inertial reference systems are systems in which Newton's first law is satisfied.
Galileo's principle of relativity:
in all inertial reference systems under the same initial conditions, all mechanical phenomena proceed in the same way, i.e. subject to the same laws
Weight is a measure of body inertia
Force is a quantitative measure of the interaction of bodies.
Newton's second law:
The force acting on a body is equal to the product of the mass of the body and the acceleration imparted by this force:
$F↖(→) = m⋅a↖(→)$
The addition of forces consists of finding the resultant of several forces, which produces the same effect as several simultaneously acting forces.
Newton's third law:
The forces with which two bodies act on each other are located on the same straight line, equal in magnitude and opposite in direction:
$F_1↖(→) = -F_2↖(→) $
Newton's III law emphasizes that the action of bodies on each other is in the nature of interaction. If body A acts on body B, then body B acts on body A (see figure).
Or in short, the force of action is equal to the force of reaction. The question often arises: why does a horse pull a sled if these bodies interact with equal forces? This is possible only through interaction with the third body - the Earth. The force with which the hooves press into the ground must be greater than the frictional force of the sled on the ground. Otherwise, the hooves will slip and the horse will not move.
If a body is subjected to deformation, forces arise that prevent this deformation. Such forces are called elastic forces.
Hooke's law written in the form
where k is the spring stiffness, x is the deformation of the body. The “−” sign indicates that the force and deformation are directed in different directions.
When bodies move relative to each other, forces arise that impede the movement. These forces are called friction forces. A distinction is made between static friction and sliding friction. Sliding friction force calculated by the formula
where N is the support reaction force, µ is the friction coefficient.
This force does not depend on the area of the rubbing bodies. The friction coefficient depends on the material from which the bodies are made and the quality of their surface treatment.
Static friction occurs if the bodies do not move relative to each other. The static friction force can vary from zero to a certain maximum value
By gravitational forces are the forces with which any two bodies are attracted to each other.
Law of universal gravitation:any two bodies are attracted to each other with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
Here R is the distance between the bodies. The law of universal gravitation in this form is valid either for material points or for spherical bodies.
Body weight called the force with which the body presses on a horizontal support or stretches the suspension.
Gravity- this is the force with which all bodies are attracted to the Earth:
With a stationary support, the weight of the body is equal in magnitude to the force of gravity:
If a body moves vertically with acceleration, its weight will change.
When a body moves with upward acceleration, its weight
It can be seen that the weight of the body is greater than the weight of the body at rest.
When a body moves with downward acceleration, its weight
In this case, the weight of the body is less than the weight of the body at rest.
Weightlessness is the movement of a body in which its acceleration is equal to the acceleration of gravity, i.e. a = g. This is possible if only one force acts on the body - gravity.
Artificial Earth satellite- this is a body that has a speed V1 sufficient to move in a circle around the Earth
There is only one force acting on the Earth's satellite - the force of gravity directed towards the center of the Earth
First escape velocity- this is the speed that must be imparted to the body so that it revolves around the planet in a circular orbit.
where R is the distance from the center of the planet to the satellite.
For the Earth, near its surface, the first escape velocity is equal to
1.3. Basic concepts and laws of statics and hydrostatics
A body (material point) is in a state of equilibrium if the vector sum of the forces acting on it is equal to zero. There are 3 types of equilibrium: stable, unstable and indifferent. If, when a body is removed from an equilibrium position, forces arise that tend to bring this body back, this stable balance. If forces arise that tend to move the body further from the equilibrium position, this unstable position; if no forces arise - indifferent(see Fig. 3).
When we are not talking about a material point, but about a body that can have an axis of rotation, then in order to achieve an equilibrium position, in addition to the equality of the sum of forces acting on the body to zero, it is necessary that the algebraic sum of the moments of all forces acting on the body be equal to zero.
Here d is the force arm. Shoulder of strength d is the distance from the axis of rotation to the line of action of the force.
Lever equilibrium condition:
the algebraic sum of the moments of all forces rotating the body is equal to zero.
Pressure is a physical quantity equal to the ratio of the force acting on a platform perpendicular to this force to the area of the platform:
Valid for liquids and gases Pascal's law:
pressure spreads in all directions without changes.
If a liquid or gas is in a gravity field, then each layer above presses on the layers below, and as the liquid or gas is immersed inside, the pressure increases. For liquids
where ρ is the density of the liquid, h is the depth of penetration into the liquid.
A homogeneous liquid in communicating vessels is established at the same level. If liquid with different densities is poured into the elbows of communicating vessels, then the liquid with a higher density is installed at a lower height. In this case
The heights of liquid columns are inversely proportional to densities:
Hydraulic Press is a vessel filled with oil or other liquid, in which two holes are cut, closed by pistons. The pistons have different areas. If a certain force is applied to one piston, then the force applied to the second piston turns out to be different.
Thus, the hydraulic press serves to convert the magnitude of the force. Since the pressure under the pistons must be the same, then 
Then A1 = A2.
A body immersed in a liquid or gas is acted upon by an upward buoyant force from the side of this liquid or gas, which is called by the power of Archimedes
The magnitude of the buoyancy force is determined by Archimedes' law: a body immersed in a liquid or gas is acted upon by a buoyant force directed vertically upward and equal to the weight of the liquid or gas displaced by the body:
where ρ liquid is the density of the liquid in which the body is immersed; V submergence is the volume of the submerged part of the body.
Body floating condition- a body floats in a liquid or gas when the buoyant force acting on the body is equal to the force of gravity acting on the body.
1.4. Conservation laws
Body impulse is a physical quantity equal to the product of a body’s mass and its speed:Momentum is a vector quantity. [p] = kg m/s. Along with body impulse, they often use impulse of power. This is the product of force and the duration of its action
The change in the momentum of a body is equal to the momentum of the force acting on this body. For an isolated system of bodies (a system whose bodies interact only with each other) law of conservation of momentum: the sum of the impulses of the bodies of an isolated system before interaction is equal to the sum of the impulses of the same bodies after the interaction.
Mechanical work called a physical quantity that is equal to the product of the force acting on the body, the displacement of the body and the cosine of the angle between the direction of the force and the displacement:
Power is the work done per unit of time:
The ability of a body to do work is characterized by a quantity called energy. Mechanical energy is divided into kinetic and potential. If a body can do work due to its motion, it is said to have kinetic energy. The kinetic energy of the translational motion of a material point is calculated by the formula
If a body can do work by changing its position relative to other bodies or by changing the position of parts of the body, it has potential energy. An example of potential energy: a body raised above the ground, its energy is calculated using the formula
where h is the lift height
Compressed spring energy:
where k is the spring stiffness coefficient, x is the absolute deformation of the spring.
The sum of potential and kinetic energy is mechanical energy. For an isolated system of bodies in mechanics, law of conservation of mechanical energy: if there are no frictional forces between the bodies of an isolated system (or other forces leading to energy dissipation), then the sum of the mechanical energies of the bodies of this system does not change (the law of conservation of energy in mechanics). If there are friction forces between the bodies of an isolated system, then during interaction part of the mechanical energy of the bodies turns into internal energy.
1.5. Mechanical vibrations and waves
Oscillations movements that have varying degrees of repeatability over time are called. Oscillations are called periodic if the values of physical quantities that change during the oscillation process are repeated at regular intervals.Harmonic vibrations are called such oscillations in which the oscillating physical quantity x changes according to the law of sine or cosine, i.e.
The quantity A equal to the largest absolute value of the fluctuating physical quantity x is called amplitude of oscillations. The expression α = ωt + ϕ determines the value of x at a given time and is called the oscillation phase. Period T is the time it takes for an oscillating body to complete one complete oscillation. Frequency of periodic oscillations The number of complete oscillations completed per unit of time is called:
Frequency is measured in s -1. This unit is called hertz (Hz).
Mathematical pendulum is a material point of mass m suspended on a weightless inextensible thread and oscillating in a vertical plane.
If one end of the spring is fixed motionless, and a body of mass m is attached to its other end, then when the body is removed from the equilibrium position, the spring will stretch and oscillations of the body on the spring will occur in the horizontal or vertical plane. Such a pendulum is called a spring pendulum.
Period of oscillation of a mathematical pendulum determined by the formula 
where l is the length of the pendulum.
Period of oscillation of a load on a spring determined by the formula 
where k is the spring stiffness, m is the mass of the load.
Propagation of vibrations in elastic media.
A medium is called elastic if there are interaction forces between its particles. Waves are the process of propagation of vibrations in elastic media.
The wave is called transverse, if the particles of the medium oscillate in directions perpendicular to the direction of propagation of the wave. The wave is called longitudinal, if the vibrations of the particles of the medium occur in the direction of wave propagation.
Wavelength is the distance between two closest points oscillating in the same phase:
where v is the speed of wave propagation.
Sound waves are called waves in which oscillations occur with frequencies from 20 to 20,000 Hz.
The speed of sound varies in different environments. The speed of sound in air is 340 m/s.
Ultrasonic waves are called waves whose oscillation frequency exceeds 20,000 Hz. Ultrasonic waves are not perceived by the human ear.
In rectilinear uniformly accelerated motion the body
- moves along a conventional straight line,
- its speed gradually increases or decreases,
- over equal periods of time, the speed changes by an equal amount.
For example, a car starts moving from a state of rest along a straight road, and up to a speed of, say, 72 km/h it moves uniformly accelerated. When the set speed is reached, the car moves without changing speed, i.e. uniformly. With uniformly accelerated motion, its speed increased from 0 to 72 km/h. And let the speed increase by 3.6 km/h for every second of movement. Then the time of uniformly accelerated movement of the car will be equal to 20 seconds. Since acceleration in SI is measured in meters per second squared, acceleration of 3.6 km/h per second must be converted into the appropriate units. It will be equal to (3.6 * 1000 m) / (3600 s * 1 s) = 1 m/s 2.
Let's say that after some time of driving at a constant speed, the car began to slow down to stop. The movement during braking was also uniformly accelerated (over equal periods of time, the speed decreased by the same amount). In this case, the acceleration vector will be opposite to the velocity vector. We can say that the acceleration is negative.
So, if the initial speed of a body is zero, then its speed after a time of t seconds will be equal to the product of acceleration and this time:
When a body falls, the acceleration of gravity “works”, and the speed of the body at the very surface of the earth will be determined by the formula:
If the current speed of the body and the time it took to develop such a speed from a state of rest are known, then the acceleration (i.e. how quickly the speed changed) can be determined by dividing the speed by the time:
However, the body could begin uniformly accelerated motion not from a state of rest, but already possessing some speed (or it was given an initial speed). Let's say you throw a stone vertically down from a tower using force. Such a body is subject to a gravitational acceleration equal to 9.8 m/s 2 . However, your strength gave the stone even more speed. Thus, the final speed (at the moment of touching the ground) will be the sum of the speed developed as a result of acceleration and the initial speed. Thus, the final speed will be found according to the formula:
However, if the stone was thrown upward. Then its initial speed is directed upward, and the acceleration of free fall is directed downward. That is, the velocity vectors are directed in opposite directions. In this case (as well as during braking), the product of acceleration and time must be subtracted from the initial speed:
From these formulas we obtain the acceleration formulas. In case of acceleration:
at = v – v 0
a = (v – v 0)/t
In case of braking:
at = v 0 – v
a = (v 0 – v)/t
In the case when a body stops with uniform acceleration, then at the moment of stopping its speed is 0. Then the formula is reduced to this form:
Knowing the initial speed of the body and the braking acceleration, the time after which the body will stop is determined:
Now let's print formulas for the path that a body travels during rectilinear uniformly accelerated motion. The graph of speed versus time for rectilinear uniform motion is a segment parallel to the time axis (usually the x axis is taken). The path is calculated as the area of the rectangle under the segment. That is, by multiplying speed by time (s = vt). With rectilinear uniformly accelerated motion, the graph is a straight line, but not parallel to the time axis. This straight line either increases in the case of acceleration or decreases in the case of braking. However, path is also defined as the area of the figure under the graph.
In rectilinear uniformly accelerated motion, this figure is a trapezoid. Its bases are a segment on the y-axis (speed) and a segment connecting the end point of the graph with its projection on the x-axis. The sides are the graph of speed versus time itself and its projection onto the x-axis (time axis). The projection onto the x-axis is not only the side side, but also the height of the trapezoid, since it is perpendicular to its bases.
As you know, the area of a trapezoid is equal to half the sum of the bases and the height. The length of the first base is equal to the initial speed (v 0), the length of the second base is equal to the final speed (v), the height is equal to time. Thus we get:
s = ½ * (v 0 + v) * t
Above was given the formula for the dependence of the final speed on the initial and acceleration (v = v 0 + at). Therefore, in the path formula we can replace v:
s = ½ * (v 0 + v 0 + at) * t = ½ * (2v 0 + at) * t = ½ * t * 2v 0 + ½ * t * at = v 0 t + 1/2at 2
So, the distance traveled is determined by the formula:
s = v 0 t + at 2 /2
(This formula can be arrived at by considering not the area of the trapezoid, but by summing up the areas of the rectangle and right triangle into which the trapezoid is divided.)
If the body begins to move uniformly accelerated from a state of rest (v 0 = 0), then the path formula simplifies to s = at 2 /2.
If the acceleration vector was opposite to the speed, then the product at 2 /2 must be subtracted. It is clear that in this case the difference between v 0 t and at 2 /2 should not become negative. When it becomes zero, the body will stop. A braking path will be found. Above was the formula for the time until a complete stop (t = v 0 /a). If we substitute the value t into the path formula, then the braking path is reduced to the following formula.
