Types of rectilinear motion in physics. Rectilinear movement. With such a movement, a formula cannot be used to calculate the displacement, since the speed changes in time and we are no longer talking about some specific speed, the value of which can be

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

Types of movements:

A) Uniform rectilinear motion of a material point: Initial conditions

. Initial conditions

G) Harmonic oscillatory motion. important case mechanical movement are fluctuations in which the parameters of the motion of a point (coordinates, speed, acceleration) are repeated at certain intervals.

O motion scriptures . There are various ways to describe the motion of bodies. With the coordinate method setting the position of the body in the Cartesian coordinate system, the movement of a material point is determined by three functions that express the dependence of coordinates on time:

x= x(t), y=y(t) and z= z(t) .

This dependence of coordinates on time is called the law of motion (or the equation of motion).

With the vector method the position of a point in space is determined at any time by the radius vector r= r(t) , drawn from the origin to the point.

There is another way to determine the position of a material point in space for a given trajectory of its movement: using a curvilinear coordinate l(t) .

All three ways of describing the motion of a material point are equivalent, the choice of any of them is determined by considerations of the simplicity of the resulting equations of motion and the clarity of the description.

Under reference system understand the body of reference, which is conditionally considered to be motionless, the coordinate system associated with the body of reference, and the clock, also associated with the body of reference. In kinematics, the frame of reference is chosen in accordance with the specific conditions of the problem of describing the motion of a body.

2. Trajectory of movement. Distance traveled. Kinematic law of motion.

The line along which a certain point of the body moves is called trajectorymovements this point.

The length of the section of the trajectory traversed by the point during its movement is called the way we have traveled .

The change in the radius vector over time is called kinematic law :
In this case, the coordinates of the points will be the coordinates in time: x= x(t), y= y(t) andz= z(t).

With curvilinear motion, the path is greater than the displacement modulus, since the length of the arc is always greater than the length of the chord that contracts it

The vector drawn from the initial position of the moving point to its position in this moment time (increment of the radius vector of a point over the considered time interval), is called moving. The resulting displacement is equal to the vector sum of successive displacements.

With rectilinear motion, the displacement vector coincides with the corresponding section of the trajectory, and the displacement modulus is equal to the distance traveled.

3. Speed. Average speed. Velocity projections.

Speed - the speed of change of coordinates. When a body (material point) moves, we are interested not only in its position in the chosen frame of reference, but also in the law of motion, i.e., the dependence of the radius vector on time. Let the moment of time corresponds to the radius vector moving point, but to a close point in time - radius vector . Then in a short period of time
the point will make a small displacement equal to

To characterize the motion of a body, the concept is introduced average speed his movements:
This quantity is vector, coinciding in direction with the vector
. With an unlimited reduction Δt the average speed tends to the limit value, which is called the instantaneous speed :

Velocity projections.

A) Uniform rectilinear motion of a material point:
Initial conditions

B) Uniformly accelerated rectilinear motion of a material point:
. Initial conditions

C) The movement of the body along the arc of a circle with a constant modulo speed:

Types of motion (uniform, uniformly accelerated) and their graphic description

According to the shape of the trajectory, the movement is divided into curvilinear(the trajectory of the body is a curved line) and rectilinear(the trajectory of the body is a straight line).

When a body moves along a rectilinear trajectory, the modulus of the displacement vector always coincides with the path traveled. When a body moves along a curvilinear trajectory, the module of the displacement vector is always less than the distance traveled

Uniform rectilinear motion.

Rectilinear uniform motion called the movement in which the body for any equal intervals of time makes the same movement.

Speed ​​of uniform rectilinear motion - this is a physical vector quantity equal to the ratio of the displacement of the body S for any period of time to the value of this interval t:

v x \u003d S / t

Speed ​​- it is a physical quantity showing the rate of change of the coordinate.

Speed ​​units - meters per second

Equation of uniform motion (body movement with uniform motion) :

S= v x t

Body coordinate equation:

x \u003d x 0 + v x t

Designations:

X- coordinate of the moving body

x 0- initial coordinate of the moving body

vWed -Average speed of uniform rectilinear motion

v X- Speed ​​of uniform rectilinear motion

S - Body displacement (distance the body has moved)

t - Span of travel time (time)

Graphical representation of uniform rectilinear motion

v

Time dependence of acceleration. Since the acceleration is equal to zero during uniform motion, the dependence a(t) is a straight line that lies on the time axis.

Since the body moves in a straight line and uniformly ( v =const), i.e. speed does not change with time, then the graph with the dependence of speed on time v (t) - straight line, axis parallel time.

The projection of the body displacement is numerically equal to the area of ​​the rectangle under the graph, since the magnitude of the displacement vector is equal to the product of the velocity vector and the time during which the displacement was made.

with rectilinear uniform motion, the module of the displacement vector equal to area rectangle below the speed graph.

Dependence of displacement on time. Graph s(t) - sloping line :

Dependence of the coordinate on time. Chart x(t) - sloping line :

It can be seen from the graph that the velocity projection is equal to:

v x \u003d S / t \u003d tga

Considering this formula, we can say the larger the angle a, the faster the body moves and it covers a greater distance in less time.

The rule for determining the speed according to the schedule s(t) and x(t): The tangent of the slope of the graph to the time axis is equal to the speed of movement.

Uneven linear motion.

Uniform motion is motion at a constant speed. If the speed of a body changes, it is said that it is moving unevenly.

A movement in which a body makes unequal movements in equal intervals of time is called uneven or variable motion.

To characterize non-uniform motion, the concept of average speed is introduced.

Average moving speed is equal to the ratio of the total distance traveled material point to the time interval for which this path has been passed.

In physics, the greatest interest is not the average, but instantaneous speed , which is defined as the limit to which the average speed tends over an infinitesimal time interval Δ t:

instantaneous speedvariable motion is called the speed of the body at a given time or at a given point in the trajectory.

The instantaneous velocity of the body at any point of the curvilinear trajectory is directed tangentially to the trajectory at that point.

The difference between average and instantaneous speeds is shown in the figure.

The movement of a body, in which its speed for any equal intervals of time changes in the same way, is calleduniformly accelerated or uniform motion.

Acceleration -it is a vector physical quantity that characterizes the rate of change in speed, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.

If the speed changes the same throughout the entire time of movement, then the acceleration can be calculated by the formula:

Designations:

v x is the final speed of the body at uniformly accelerated motion in a straight line

v 0x - initial speed of the body

a- body acceleration

t - body movement time

Acceleration shows how quickly the speed of a body changes. If the acceleration is positive, then the speed of the body increases, the movement is accelerated. If the acceleration is negative, then the speed is decreasing, the movement is slow.

Unit of measurement of acceleration in SI [ m/s 2].

Acceleration is measured accelerometer

Speed ​​equation for uniformly accelerated motion:

The equation of uniformly accelerated rectilinear motion(displacement with uniformly accelerated motion):

Designations:

Movement of a body with uniformly accelerated motion in a straight line

Initial body speed

The speed of a body in uniformly accelerated motion in a straight line

body acceleration

body movement time

More formulas for finding displacement during uniformly accelerated rectilinear motion, which can be used in solving problems:

- if the initial, final speeds of movement and acceleration are known.

- if the initial, final speeds of movement and the time of the entire movement are known

Graphical representation of non-uniform rectilinear motion

Mechanical movement is represented graphically. Addiction physical quantities expressed using functions. Designate:

v (t) - change in speed with time

S(t) - change in displacement (path) over time

a(t) - change in acceleration with time

Dependence of acceleration on time. Acceleration does not change with time, has a constant value, graph a(t) is a straight line parallel to the time axis.

Speed ​​versus time. With uniform motion, the speed changes according to linear dependence .

The graph is a sloping line.

The rule for determining the path according to the schedule v(t): The path of the body is the area of ​​the triangle (or trapezoid) under the velocity graph.

The rule for determining the acceleration according to the schedule v(t): The acceleration of the body is the tangent of the slope of the graph to the time axis. If the body slows down, the acceleration is negative, the angle of the graph is obtuse, so we find the tangent of the adjacent angle.

Path versus time. With uniformly accelerated motion, the path changes, according to the quadratic dependence

In coordinates, the dependence has the form .

The graph is a branch of a parabola.

To find the coordinates of a moving body at any time, you need to know the projection of the displacement vector on the coordinate axes, and hence the displacement vector itself. What you need to know for this. The answer depends on what kind of movement the body is making.

Consider first the simplest form of movement - rectilinear uniform motion.

The movement in which the body makes the same movements for any equal intervals is called rectilinear uniform motion.

To find the displacement of a body in uniform rectilinear motion over a certain period of time t, you need to know what movement the body makes per unit of time, since for any other unit of time it makes the same movement.

The movement per unit of time is called speed body movements and denoted by the letter υ . If the displacement in this section is denoted by , and the time interval by t, then the speed can be expressed in relation to . Since displacement is a vector quantity and time is a scalar, velocity is also a vector quantity. The velocity vector is directed in the same way as the displacement vector.

The speed of uniform rectilinear motion body is called a value equal to the ratio of the movement of the body to the period of time during which this movement occurred:

Thus, the speed shows what movement the body makes per unit of time. Therefore, in order to find the displacement of a body, one must know its velocity. The displacement of the body is calculated by the formula:

The displacement vector is directed in the same way as the velocity vector, time t is a scalar value.

According to formulas written in vector form, calculations cannot be carried out, since a vector quantity has not only a numerical value, but also a direction. In calculations, formulas are used that do not include vectors, but their projections on the coordinate axes, since algebraic operations can be performed on projections.

Since the vectors are equal, their projections on the axis are also equal X, from here:

Now you can get the formula for calculating the coordinate x points at any point in time. We know that

It can be seen from this formula that in case of rectilinear uniform motion, the coordinate of the body depends linearly on time, which means that it can be used to describe rectilinear uniform motion.

In addition, it follows from the formula that in order to find the position of the body at any time in a rectilinear uniform motion, you need to know the initial coordinate of the body x0 and the projection of the velocity vector on the axis along which the body moves.

It must be remembered that this formula v x- the projection of the velocity vector, therefore, like any projection of the vector, it can be positive and negative.

Rectilinear uniform motion is rare. More often one has to deal with motion in which the displacements of the body can be different for equal time intervals. This means that the speed of the body somehow changes with time. Cars, trains, planes, etc., a body thrown up, bodies falling to the Earth move at a variable speed.

With such a movement, a formula cannot be used to calculate the displacement, since the speed changes in time and we are no longer talking about some specific speed, the value of which can be substituted into the formula. In such cases, the so-called average speed is used, which is expressed by the formula:

average speed shows what is the displacement that the body makes on average per unit of time.

However, with the help of the concept of average speed, the main problem of mechanics - to determine the position of the body at any moment in time - cannot be solved.

1) Analytical method.

We consider the highway to be straight. Let's write down the equation of motion of a cyclist. Since the cyclist was moving uniformly, his equation of motion is:

(the origin of coordinates is placed at the starting point, so the initial coordinate of the cyclist is zero).

The motorcyclist was moving at a uniform speed. He also started moving from the starting point, so his initial coordinate is zero, the initial speed of the motorcyclist is also equal to zero (the motorcyclist began to move from a state of rest).

Considering that the motorcyclist started moving a little later, the motorcyclist's equation of motion 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 with respect to , we find the meeting time:

This is quadratic equation. We define the discriminant:

Define roots:

Substitute the 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 began to move, 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:

Substitute this value of time into the formula for the law of change in the speed of a motorcyclist and find the value of his speed at this moment:

2) Graphical way.

On one coordinate plane we build graphs of changes in the coordinates of the cyclist and motorcyclist over time (the graph for the coordinates of the cyclist is in red, for the motorcyclist - in green). It can be seen that the dependence of the coordinate on time for a cyclist is 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 quadratic function, whose graph is a parabola.

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Rectilinear uniform motion is a motion in which a body travels the same distance in equal intervals of time.

Uniform movement- this is such a movement of the body in which its speed remains constant (), that is, it moves at the same speed all the time, and acceleration or deceleration does not occur ().

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

The speed of uniform rectilinear motion does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the velocity vector coincides with the displacement vector. With all this, the average speed in any period of time is equal to the initial and instantaneous speed:

Speed ​​of uniform rectilinear motion is a physical vector quantity equal to the ratio of the displacement of the body for any period of time to the value of this interval t:

from this formula. we can easily express body movement with uniform motion:

Consider the dependence of speed and displacement on time

Since our body moves in a straight line and uniformly accelerated (), then the graph with the dependence of speed on time will look like a parallel straight line to the time axis.

depending projections of body velocity versus time there is nothing complicated. The projection of the movement of the body is numerically equal to the area of ​​the rectangle AOBC, since the magnitude of the displacement vector is equal to the product of the velocity vector by the time during which the movement was made.

On the chart we see displacement versus time.

It can be seen from the graph that the velocity projection is equal to: