Uniform and uneven movement in nature. mechanical movement

The simplest form of mechanical motion is the motion of a body along a straight line. with a constant modulo and direction speed. Such a movement is called uniform . With uniform motion, the body covers equal distances in any equal intervals of time. For a kinematic description of uniform rectilinear motion, the coordinate axis OX convenient to place along the line of movement. The position of the body during uniform motion is determined by setting one coordinate x. The displacement vector and the velocity vector are always directed parallel to the coordinate axis OX.

Therefore, the displacement and speed during rectilinear motion can be projected onto the axis OX and consider their projections as algebraic quantities.

If at some point in time t 1 body was at the point with coordinate x 1 , and at a later moment t 2 - at the point with coordinate x 2 , then the displacement projection Δ s per axle OX in time Δ t = t 2 - t 1 equals

This value can be both positive and negative depending on the direction in which the body moved. With uniform motion along a straight line, the displacement modulus coincides with the distance traveled. The speed of uniform rectilinear motion is the ratio

If υ > 0, then the body moves towards the positive direction of the axis OX; at υ< 0 тело движется в противоположном направлении.

Coordinate dependency x from time t (law of motion) is expressed for uniform rectilinear motion linear mathematical equation :

In this equation, υ = const is the speed of the body, x 0 - the coordinate of the point where the body was at the moment of time t= 0. Graph of the law of motion x(t) is a straight line. Examples of such graphs are shown in fig. 1.3.1.

For the law of motion depicted in graph I (Fig. 1.3.1), with t= 0 the body was at the point with coordinate x 0 = -3. Between moments in time t 1 = 4 s and t 2 = 6 s the body has moved from the point x 1 = 3 m to the point x 2 = 6 m. Thus, for Δ t = t 2 - t 1 = 2 s the body moved by Δ s = x 2 - x 1 \u003d 3 m. Therefore, the speed of the body is

The value of the speed turned out to be positive. This means that the body was moving in the positive direction of the axis OX. Note that on the motion graph, the speed of the body can be geometrically defined as the ratio of the sides BC and AC triangle ABC(see fig. 1.3.1)

The greater the angle α, which forms a straight line with the time axis, i.e., the greater the slope of the graph ( steepness), the greater the speed of the body. Sometimes they say that the speed of the body is equal to the tangent of the angle α of the slope of the straight line x (t). From the point of view of mathematics, this statement is not quite correct, since the sides BC and AC triangle ABC have different dimensions: side BC measured in meters, and the side AC- in seconds.

Similarly, for the movement shown in Fig. 1.3.1 line II, we find x 0 = 4 m, υ = -1 m/s.

On fig. 1.3.2 law of motion x (t) of the body is depicted using straight line segments. In mathematics, such graphs are called piecewise linear. This movement of the body along a straight line is not uniform. In different parts of this graph, the body moves at different speeds, which can also be determined by the slope of the corresponding segment to the time axis. At the break points of the graph, the body instantly changes its speed. On the graph (Fig. 1.3.2), this happens at the time points t 1 = -3 s, t 2 = 4 s, t 3 = 7 s and t 4 = 9 s. According to the motion schedule, it is easy to find that on the interval ( t 2 ; t 1) the body moved at a speed υ 12 = 1 m/s, on the interval ( t 3 ; t 2) - at a speed υ 23 = -4/3 m/s and on the interval ( t 4 ; t 3) - with a speed υ 34 = 4 m/s.

It should be noted that under the piecewise linear law of rectilinear motion of the body, the distance traveled l does not match movement s. For example, for the law of motion depicted in Fig. 1.3.2, the movement of the body in the time interval from 0 s to 7 s is zero ( s= 0). During this time, the body has traveled a path l= 8 m.

As kinematics, there is one in which the body for any arbitrarily taken equal lengths of time passes the same length of the path segments. This is uniform motion. An example is the movement of a skater in the middle of a distance or a train on a flat stretch.

Theoretically, the body can move along any trajectory, including curvilinear. At the same time, there is the concept of a path - this is the name of the distance traveled by a body along its trajectory. A path is a scalar quantity and should not be confused with a move. By the last term, we denote the segment between the starting point of the path and the end point, which, during curvilinear motion, obviously does not coincide with the trajectory. Displacement is a vector quantity that has a numeric value equal to the length of the vector.

A natural question arises - in what cases is it about uniform motion? Will the movement of, for example, a carousel in a circle at the same speed be considered uniform? No, because with such a movement, the velocity vector changes its direction every second.

Another example is a car traveling in a straight line at the same speed. Such a movement will be considered uniform as long as the car does not turn anywhere and its speedometer has the same number. Obviously, uniform motion always occurs in a straight line, the velocity vector does not change. The path and displacement in this case will be the same.

Uniform motion is motion along a straight path at a constant speed, in which the lengths of the distances traveled for any equal lengths of time are the same. A special case of uniform motion can be considered a state of rest, when the speed and distance traveled are equal to zero.

Speed ​​is a qualitative characteristic of uniform motion. Obviously, different objects cover the same path in different times (pedestrian and car). The ratio of the path traveled by a uniformly moving body to the length of time for which this path has been traveled is called the speed of movement.

Thus, the formula describing uniform motion looks like this:

V = S / t; where V is the speed of movement (is a vector quantity);

S - path or movement;

Knowing the speed of movement, which is unchanged, we can calculate the path traveled by the body for any arbitrary period of time.

Sometimes they mistakenly mix uniform and uniformly accelerated motion. These are completely different concepts. - one of the variants of uneven movement (i.e., one in which the speed is not a constant value), which has an important distinguishing feature - the speed in this case changes over the same time intervals by the same amount. This value, equal to the ratio of the difference in speeds to the length of time during which the speed has changed, is called acceleration. This number, which shows how much the speed increased or decreased per unit of time, can be large (then they say that the body quickly picks up or loses speed) or insignificant when the object accelerates or slows down more smoothly.

Acceleration, as well as speed, is physical. The acceleration vector in the direction always coincides with the velocity vector. An example of uniformly accelerated motion is the case of an object in which the attraction of the object by the earth's surface) changes per unit time by a certain amount, called the acceleration of free fall.

Uniform motion can theoretically be considered as a special case of uniformly accelerated motion. It is obvious that since the speed does not change during such a movement, then acceleration or deceleration does not occur, therefore, the magnitude of the acceleration with uniform movement is always zero.

« Physics - Grade 10 "

When solving problems on this topic, it is necessary first of all to choose a reference body and associate a coordinate system with it. In this case, the movement occurs in a straight line, so one axis is sufficient to describe it, for example, the OX axis. Having chosen the origin, we write down the equations of motion.

Task I.

Determine the module and direction of the speed of a point if, with uniform movement along the OX axis, its coordinate during the time t 1 \u003d 4 s changed from x 1 \u003d 5 m to x 2 \u003d -3 m.

Decision.

The module and direction of a vector can be found from its projections on the coordinate axes. Since the point moves uniformly, we find the projection of its velocity on the OX axis by the formula

The negative sign of the velocity projection means that the speed of the point is directed opposite to the positive direction of the OX axis. Velocity modulus υ = |υ x | = |-2 m/s| = 2 m/s.

Task 2.

From points A and B, the distance between which along a straight highway l 0 = 20 km, simultaneously two cars began to move uniformly towards each other. The speed of the first car υ 1 = 50 km/h, and the speed of the second car υ 2 = 60 km/h. Determine the position of the cars relative to point A after the time t = 0.5 hours after the start of movement and the distance I between the cars at this point in time. Determine the paths s 1 and s 2 traveled by each car in time t.

Decision.

Let's take point A as the origin of coordinates and direct the coordinate axis OX towards point B (Fig. 1.14). The movement of cars will be described by the equations

x 1 = x 01 + υ 1x t, x 2 = x 02 + υ 2x t.

Since the first car moves in the positive direction of the OX axis, and the second in the negative direction, then υ 1x = υ 1, υ 2x = -υ 2. In accordance with the choice of origin x 01 = 0, x 02 = l 0 . Therefore, after a time t

x 1 \u003d υ 1 t \u003d 50 km / h 0.5 h \u003d 25 km;

x 2 \u003d l 0 - υ 2 t \u003d 20 km - 60 km / h 0.5 h \u003d -10 km.

The first car will be at point C at a distance of 25 km from point A on the right, and the second at point D at a distance of 10 km on the left. The distance between the cars will be equal to the modulus of the difference between their coordinates: l = | x 2 - x 1 | = |-10 km - 25 km| = 35 km. The distances traveled are:

s 1 \u003d υ 1 t \u003d 50 km / h 0.5 h \u003d 25 km,

s 2 \u003d υ 2 t \u003d 60 km / h 0.5 h \u003d 30 km.

Task 3.

The first car leaves point A for point B at a speed υ 1 After a time t 0, a second car leaves point B in the same direction at a speed υ 2. The distance between points A and B is equal to l. Determine the coordinate of the meeting point of cars relative to point B and the time from the moment of departure of the first car through which they will meet.

Decision.

Let's take point A as the origin of coordinates and direct the coordinate axis OX towards point B (Fig. 1.15). The movement of cars will be described by the equations

x 1 = υ 1 t, x 2 = l + υ 2 (t - t 0).

At the time of the meeting, the coordinates of the cars are equal: x 1 \u003d x 2 \u003d x in. Then υ 1 t in \u003d l + υ 2 (t in - t 0) and the time until the meeting

Obviously, the solution makes sense for υ 1 > υ 2 and l > υ 2 t 0 or for υ 1< υ 2 и l < υ 2 t 0 . Координата места встречи

Task 4.

Figure 1.16 shows the graphs of the dependence of the coordinates of points on time. Determine from the graphs: 1) the speed of the points; 2) after what time after the start of the movement they will meet; 3) the paths traveled by the points before the meeting. Write the equations of motion of points.

Decision.

For a time equal to 4 s, the change in the coordinates of the first point: Δx 1 \u003d 4 - 2 (m) \u003d 2 m, the second point: Δx 2 \u003d 4 - 0 (m) \u003d 4 m.

1) The speed of the points is determined by the formula υ 1x = 0.5 m/s; υ 2x = 1 m/s. Note that the same values ​​could be obtained from the graphs by determining the tangents of the angles of inclination of the straight lines to the time axis: the speed υ 1x is numerically equal to tgα 1 , and the speed υ 2x is numerically equal to tgα 2 .

2) The meeting time is the moment in time when the coordinates of the points are equal. It is obvious that t in \u003d 4 s.

3) The paths traveled by the points are equal to their movements and are equal to the changes in their coordinates in the time before the meeting: s 1 = Δх 1 = 2 m, s 2 = Δх 2 = 4 m.

The equations of motion for both points have the form x = x 0 + υ x t, where x 0 = x 01 = 2 m, υ 1x = 0.5 m / s - for the first point; x 0 = x 02 = 0, υ 2x = 1 m / s - for the second point.

95. Give examples of uniform motion.
It is very rare, for example, the movement of the Earth around the Sun.

96. Give examples of uneven movement.
The movement of the car, aircraft.

97. A boy slides down a mountain on a sleigh. Can this movement be considered uniform?
No.

98. Sitting in the car of a moving passenger train and watching the movement of an oncoming freight train, it seems to us that the freight train is going much faster than our passenger train was going before the meeting. Why is this happening?
Relative to the passenger train, the freight train moves with the total speed of the passenger and freight trains.

99. The driver of a moving car is in motion or at rest in relation to:
a) roads
b) car seats;
c) gas stations;
d) the sun;
e) trees along the road?
In motion: a, c, d, e
At rest: b

100. Sitting in the car of a moving train, we watch in the window a car that goes forward, then seems to be stationary, and finally moves back. How can we explain what we see?
Initially, the speed of the car is higher than the speed of the train. Then the speed of the car becomes equal to the speed of the train. After that, the speed of the car decreases compared to the speed of the train.

101. The plane performs a "dead loop". What is the trajectory of movement seen by observers from the ground?
ring trajectory.

102. Give examples of the movement of bodies along curved paths relative to the earth.
The movement of the planets around the sun; the movement of the boat on the river; Flight of bird.

103. Give examples of the movement of bodies that have a rectilinear trajectory relative to the earth.
moving train; person walking straight.

104. What types of movement do we observe when writing with a ballpoint pen? Chalk?
Equal and uneven.

105. Which parts of the bicycle, during its rectilinear movement, describe rectilinear trajectories relative to the ground, and which ones are curvilinear?
Rectilinear: handlebar, saddle, frame.
Curvilinear: pedals, wheels.

106. Why is it said that the Sun rises and sets? What is the reference body in this case?
The reference body is the Earth.

107. Two cars are moving along the highway so that some distance between them does not change. Indicate with respect to which bodies each of them is at rest and with respect to which bodies they move during this period of time.
Relative to each other, the cars are at rest. Vehicles move relative to surrounding objects.

108. Sledges roll down the mountain; the ball rolls down the inclined chute; the stone released from the hand falls. Which of these bodies move forward?
The sled is moving forward from the mountain and the stone released from the hands.

109. A book placed on a table in a vertical position (Fig. 11, position I) falls from the shock and takes position II. Two points A and B on the cover of the book described the trajectories AA1 and BB1. Can we say that the book moved forward? Why?

Do you think you are moving or not when you read this text? Almost every one of you will immediately answer: no, I'm not moving. And it will be wrong. Some might say I'm moving. And they are wrong too. Because in physics, some things are not quite what they seem at first glance.

For example, the concept of mechanical motion in physics always depends on the reference point (or body). So a person flying in an airplane moves relative to the relatives left at home, but is at rest relative to a friend sitting next to him. So, bored relatives or a friend sleeping on his shoulder are, in this case, reference bodies for determining whether our aforementioned person is moving or not.

Definition of mechanical movement

In physics, the definition of mechanical motion studied in the seventh grade is as follows: a change in the position of a body relative to other bodies over time is called mechanical motion. Examples of mechanical movement in everyday life would be the movement of cars, people and ships. Comets and cats. Air bubbles in a boiling kettle and textbooks in a schoolboy's heavy backpack. And every time a statement about the movement or rest of one of these objects (bodies) will be meaningless without indicating the body of reference. Therefore, in life we ​​most often, when we talk about movement, we mean movement relative to the Earth or static objects - houses, roads, and so on.

Trajectory of mechanical movement

It is also impossible not to mention such a characteristic of mechanical movement as a trajectory. A trajectory is a line along which a body moves. For example, footprints in the snow, the footprint of an airplane in the sky, and the footprint of a tear on a cheek are all trajectories. They can be straight, curved or broken. But the length of the trajectory, or the sum of the lengths, is the path traveled by the body. The path is marked with the letter s. And it is measured in meters, centimeters and kilometers, or in inches, yards and feet, depending on what units of measurement are accepted in this country.

Types of mechanical movement: uniform and uneven movement

What are the types of mechanical movement? For example, during a trip by car, the driver moves at different speeds when driving around the city and at almost the same speed when entering the highway outside the city. That is, it moves either unevenly or evenly. So the movement, depending on the distance traveled for equal periods of time, is called uniform or uneven.

Examples of uniform and non-uniform motion

There are very few examples of uniform motion in nature. The Earth moves almost evenly around the Sun, raindrops drip, bubbles pop up in soda. Even a bullet fired from a pistol moves in a straight line and evenly only at first glance. From friction against the air and the attraction of the Earth, its flight gradually becomes slower, and the trajectory decreases. Here in space, a bullet can move really straight and evenly until it collides with some other body. And with uneven movement, things are much better - there are many examples. The flight of a football during a football game, the movement of a lion hunting its prey, the travel of a chewing gum in the mouth of a seventh grader, and a butterfly fluttering over a flower are all examples of uneven mechanical movement of bodies.