What do uniform and uneven motion have in common? mechanical movement

« 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

negative sign 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.

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, while driving a car, the driver moves with different speed when driving around the city and almost at the same speed when leaving 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.

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. Way - scalar, and should not be confused with displacement. By the last term, we denote the segment between the starting point of the path and the end point, which, when curvilinear motion certainly does not coincide with the trajectory. Displacement - having a numeric value equal to the length of the vector.

A natural question arises - in what cases we are talking 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 movement- this is a movement along a straight trajectory at a constant speed, at which the lengths of the traveled intervals of the path 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. It is obvious that different objects pass the same path for different time(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. It's perfect different concepts. - one of the options for uneven movement (i.e., one in which the speed is not a constant value), which has an important hallmark- the speed at this 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 has 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, like speed, is a physical vector quantity. The acceleration vector in the direction always coincides with the velocity vector. An example uniformly accelerated motion can serve as a case of an object in which the attraction of an object by the earth's surface) changes per unit time by a certain amount, called acceleration free fall.

Uniform motion can theoretically be considered as special case uniformly accelerated. 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.

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 freight train goes much faster than our passenger train went 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?

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

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

is a movement in which the body makes the same movements for any equal intervals of time. For example, if we divide some time interval into segments of one second, then with uniform motion the body will move the same distance for each of these segments of time.

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 displacement vector coincides in direction with the velocity vector. Wherein average speed for any period of time is equal to the 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:

V(vector) = s(vector) / t

Thus, the speed of uniform rectilinear motion shows what movement a material point makes per unit of time.

moving with uniform rectilinear motion is determined by the formula:

s(vector) = V(vector) t

Distance traveled in rectilinear motion is equal to the displacement modulus. If the positive direction of the OX axis coincides with the direction of movement, then the projection of the velocity on the OX axis is equal to the velocity and is positive:

v x = v, i.e. v > 0

The projection of displacement onto the OX axis is equal to:

s \u003d vt \u003d x - x 0

where x 0 is the initial coordinate of the body, x is the final coordinate of the body (or the coordinate of the body at any time)

Motion equation, that is, the dependence of the body coordinate on time x = x(t), takes the form:

If the positive direction of the OX axis is opposite to the direction of motion of the body, then the projection of the body velocity on the OX axis is negative, the velocity is less than zero (v< 0), и тогда уравнение движения принимает вид:

4. Equal-variable movement.

Uniform rectilinear motion This is a special case of non-uniform motion.

Uneven movement- this is a movement in which a body (material point) makes unequal movements in equal intervals of time. For example, a city bus moves unevenly, since its movement consists mainly of acceleration and deceleration.

Equal-variable motion- this is a movement in which the speed of a body (material point) changes in the same way for any equal time intervals.

Acceleration of a body in uniform motion remains constant in magnitude and direction (a = const).

Uniform motion can be uniformly accelerated or uniformly slowed down.

Uniformly accelerated motion- this is the movement of a body (material point) with a positive acceleration, that is, with such a movement, the body accelerates with a constant acceleration. In the case of uniformly accelerated motion, the modulus of the body's velocity increases with time, the direction of acceleration coincides with the direction of the velocity of motion.

Uniformly slow motion- this is the movement of a body (material point) with negative acceleration, that is, with such a movement, the body slows down uniformly. With uniformly slow motion, the velocity and acceleration vectors are opposite, and the modulus of velocity decreases with time.

In mechanics, any rectilinear motion is accelerated, so slow motion differs from accelerated motion only by the sign of the projection of the acceleration vector onto the selected axis of the coordinate system.

Average speed of variable motion is determined by dividing the movement of the body by the time during which this movement was made. The unit of average speed is m/s.

Instant Speed is the speed of the body (material point) in this moment time or at a given point of the trajectory, that is, the limit to which the average speed tends with an infinite decrease in the time interval Δt:

V=lim(^t-0) ^s/^t

Instantaneous velocity vector uniform motion can be found as the first derivative of the displacement vector with respect to time:

V(vector) = s'(vector)

Velocity vector projection on the OX axis:

this is the derivative of the coordinate with respect to time (the projections of the velocity vector onto other coordinate axes are similarly obtained).

Acceleration- this is the value that determines the rate of change in the speed of the body, that is, the limit to which the change in speed tends with an infinite decrease in the time interval Δt:

a(vector) = lim(t-0) ^v(vector)/^t

Acceleration vector uniform motion can be found as the first derivative of the velocity vector with respect to time or as the second derivative of the displacement vector with respect to time:

a(vector) = v(vector)" = s(vector)"

Considering that 0 is the speed of the body at the initial moment of time (initial speed), is the speed of the body at a given moment of time (final speed), t is the time interval during which the change in speed occurred, acceleration formula will be as follows:

a(vector) = v(vector)-v0(vector)/t

From here uniform velocity formula at any given time:

v(vector) = v 0 (vector) + a(vector)t

If the body moves rectilinearly along the OX axis of a rectilinear Cartesian coordinate system coinciding in direction with the body trajectory, then the projection of the velocity vector onto this axis is determined by the formula:

v x = v 0x ± a x t

The "-" (minus) sign in front of the projection of the acceleration vector refers to uniformly slow motion. Equations of projections of the velocity vector onto other coordinate axes are written similarly.

Since the acceleration is constant (a \u003d const) with uniformly variable motion, the acceleration graph is a straight line parallel to the 0t axis (time axis, Fig. 1.15).

Rice. 1.15. Dependence of body acceleration on time.

Speed ​​versus time is a linear function, the graph of which is a straight line (Fig. 1.16).

Rice. 1.16. Dependence of body speed on time.

Graph of speed versus time(Fig. 1.16) shows that

In this case, the displacement is numerically equal to the area of ​​\u200b\u200bthe figure 0abc (Fig. 1.16).

The area of ​​a trapezoid is half the sum of the lengths of its bases times the height. The bases of the trapezoid 0abc are numerically equal:

The height of the trapezoid is t. Thus, the area of ​​the trapezoid, and hence the projection of displacement onto the OX axis, is equal to:

In the case of uniformly slow motion, the projection of acceleration is negative, and in the formula for the projection of displacement, the sign “–” (minus) is placed in front of the acceleration.

The general formula for determining the displacement projection is:

The graph of the dependence of the speed of the body on time at various accelerations is shown in Fig. 1.17. The graph of the dependence of displacement on time at v0 = 0 is shown in fig. 1.18.

Rice. 1.17. Dependence of body speed on time for different meanings acceleration.

Rice. 1.18. Dependence of body displacement on time.

The speed of the body at a given time t 1 is equal to the tangent of the angle of inclination between the tangent to the graph and the time axis v \u003d tg α, and the movement is determined by the formula:

If the time of motion of the body is unknown, you can use another displacement formula by solving a system of two equations:

The formula for the abbreviated multiplication of the difference of squares will help us to derive the formula for the displacement projection:

Since the coordinate of the body at any moment of time is determined by the sum of the initial coordinate and the displacement projection, then body motion equation will look like this:

The graph of the x(t) coordinate is also a parabola (as is the displacement graph), but the vertex of the parabola generally does not coincide with the origin. For a x< 0 и х 0 = 0 ветви параболы направлены вниз (рис. 1.18).