Types of equivalent movement. Speed, acceleration, uniform and equilibrium straight movement

Equal asked movement - Movement in which the acceleration is constantly in the module and direction.

An example of such a movement is the movement of the body thrown at an angle α (\\ displayStyle \\ Alpha) To the horizon in a homogeneous field of gravity - the body moves with constant acceleration A → \u003d G → (\\ displayStyle (\\ VEC (A)) \u003d (\\ VEC (G)))directed vertically down.

With an equalized movement in direct body speed determined by the formula:

V (T) \u003d V 0 + A T (\\ displayStyle V (T) \u003d V_ (0) + AT)

Knowing what V (t) \u003d d d t x (t) (\\ displayStyle V (T) \u003d (\\ FRAC (D) (DT)) x (t)), find a formula for determining the x coordinate:

x (t) \u003d x 0 + v 0 t + a t 2 2 (\\ displaystyle x (t) \u003d x_ (0) + v_ (0) T + (\\ FRAC (AT ^ (2)) (2)))

Note. Equalioned You can call the move at which the speed module is evenly reduced with time (if vector V → (\\ DisplayStyle (\\ VEC (V))) and A → (\\ DisplayStyle (\\ VEC (A))) contamsized). Equalized movement is also equivalent.

Encyclopedic YouTube.

1 / 5
In the case of a one-dimensional equivalent movement along the coordinate x. There is a formula:
Δ x \u003d V x 2 - V 0 x 2 2 A x (\\ displaystyle \\ Delta X \u003d (\\ FRAC (V_ (X) ^ (2) -v_ (0x) ^ (2)) (2a_ (x)))),
Curvolinear equative (Equipment) movement can also be considered as one-dimensional. In this case, the generalized coordinate is used. S., often referred to the way. This coordinate corresponds to the length of the trajectory passed (the length of the arc curve). Thus, the formula acquires the form:
Δ s \u003d V 2 - V 0 2 2 A τ (\\ displaystyle \\ delta s \u003d (\\ FRAC (V ^ (2) -v_ (0) ^ (2)) (2a _ (\\ Tau)))),
where a τ (\\ displaystyle a _ (\\ tau)) - Tangential acceleration that "replies" for changing body velocity module.
From the above formulas, it is possible to obtain expressions to determine the final body velocity, with known initial speed, acceleration and movement:
V x \u003d ± V 0 x 2 + 2 A x Δ x (\\ displaystyle v_ (x) \u003d \\ pm (\\ sqrt (v_ (0x) ^ (2) + 2a_ (x) \\ Delta X)))
When curvilinear equative We have:
v \u003d ± v 0 2 + 2 a τ Δ s (\\ displaystyle v \u003d \\ pm (\\ sqrt (v_ (0) ^ (2) + 2a _ (\\ Tau) \\ Delta S)))
Similar relations can be recorded for expressions:
v y \u003d ± v 0 y 2 + 2 a y Δ y (\\ displaystyle v_ (y) \u003d \\ pm (\\ sqrt (v_ (0y) ^ (2) + 2a_ (y) \\ Delta Y))); V z \u003d ± V 0 z 2 + 2 a z Δ z (\\ displaystyle v_ (z) \u003d \\ pm (\\ sqrt (v_ (0z) ^ (2) + 2a_ (z) \\ Delta Z))).
And find the final speed on the Pythagoreo theorem
| V → | \u003d VX 2 + VY 2 + VZ 2 (\\ DisplayStyle | (\\ VEC (V)) | \u003d (\\ sqrt (v_ (x) ^ (2) + v_ (y) ^ (2) + v_ (z) ^ ( 2)))).
Theorem on the kinetic energy point

The displacement formula with an equilibrium movement is used in the proof of the theorem on kinetic energy. To do this, it is necessary to transfer acceleration into the left side and to multiply both parts for body weight:
Max Δ x \u003d MVX 2 2 - MV 0 x 2 2 (\\ DISPLAYSTYLE MA_ (X) \\ Delta X \u003d (\\ FRAC (MV_ (X) ^ (2)) (2)) - (\\ FRAC (MV_ (0x) ^ (2)) (2))).
Recovering similar ratios for coordinates y. and z. and having aroused all three equalities to get the relation:
F → ⋅ Δ R → \u003d MV 2 2 - MV 0 2 2 (\\ DISPLAYSTYLE (\\ VEC (F)) \\ CDOT (\\ VEC (\\ Delta R)) \u003d (\\ FRAC (MV ^ (2)) (2) ) - (\\ FRAC (MV_ (0) ^ (2)) (2))).
Left stands work permanent Television power F → (\\ DisplayStyle (\\ VEC (F)))And on the right is the difference in kinetic energies in the final and initial moment of movement. The resulting formula is a mathematical expression of the theorem kinetic energy Points for the case of equivalent movement.

In this topic, we will look at a very special kind of uneven movement. Based on the opposition to uniform motion, uneven movement is a movement with a different speed, along any trajectory. What is the feature of the equilibrium movement? This is uneven movement, but which "EXCALLY accelerated". Acceleration is associated with increasing speed. Recall the word "equal", we obtain an equal increase in speed. And how to understand the "equal increase in speed", how to evaluate the speed equals or not? To do this, we will need to retrieve time, evaluate the speed through the same time interval. For example, the car begins to move, in the first two seconds it develops speed up to 10 m / s, over the next two seconds 20 m / s, after another two seconds it is already moving at a speed of 30 m / s. Every two seconds, the speed increases and each time 10 m / s. This is an equilibrium movement.

The physical quantity characterizing how many times the speed increases is called acceleration.

Is it possible to consider the cyclist equivalent if after stopping in the first minute its speed 7km / h, in the second - 9km / h, in the third 12km / h? It is impossible! The cyclist accelerates, but not the same, first accelerated at 7km / h (7-0), then 2 km / h (9-7), then 3 km / h (12-9).

Usually, the movement with an increasing velocity is called an accelerated movement. Movement with a decreasing speed - slow motion. But physics, any movement with a changing rate is called accelerated movement. Does the car begins from the scene (the speed grows!), Or slows down (speed decreases!), In any case, it moves with acceleration.

Equal asked movement - this is a body movement, in which its speed in any equal periods of time changes (may increase or decrease) the same

Acceleration of body

Acceleration characterizes speed change. This is the number that changes the speed per second. If the acceleration of the body in the module is large, it means that the body quickly gains speed (when it is accelerated) or quickly loses it (when braking). Acceleration - This is a physical vector value, numerically equal relation Changes in speed by a period of time during which this change occurred.

We define acceleration in the next task. In the initial moment of time, the height rate was 3 m / s, at the end of the first second the speed of the ship was 5 m / s, at the end of the second - 7m / s, at the end of the third 9 m / s, etc. Obviously. But how did we define? We consider speed difference in one second. In the first second, 5-3 \u003d 2, in the second second 7-5 \u003d 2, in the third 9-7 \u003d 2. And what if there are no speeds for every second? Such a task: the initial height rate of 3 m / s, at the end of the second second - 7 m / s, at the end of the fourth 11 m / s. In this case, it is necessary 11-7 \u003d 4, then 4/2 \u003d 2. The speed difference we divide for a period of time.

This formula is most often used in solving tasks in a modified form:

The formula is not written in a vector form, so the sign "+" write when the body is accelerated, the sign "-" - when slows down.

The direction of the acceleration vector

The direction of the acceleration vector is shown in the drawings.

In this picture, the machine moves in the positive direction along the OX axis, the speed vector always coincides with the direction of movement (directed to the right). When the vector acceleration coincides with the direction of speed, this means that the car is accelerated. Acceleration is positive.
When acceleration, the acceleration direction coincides with the direction of speed. Acceleration is positive.

In this figure, the machine moves in the positive direction along the OX axis, the speed vector coincides with the direction of movement (directed to the right), the acceleration does not coincide with the direction of speed, which means that the machine is slow. Acceleration is negative.
When braking, the direction of acceleration is opposite to the direction of speed. Acceleration is negative.
We'll figure it out why when braking acceleration is negative. For example, the motor ship for the first second dropped the speed from 9m / s to 7m / s, for the second second to 5m / s, for the third to 3m / s. Speed \u200b\u200bchanges to "-2m / s". 3-5 \u003d -2; 5-7 \u003d -2; 7-9 \u003d -2m / s. This is where the negative acceleration value appears.
When solving tasks, If the body slows down, the acceleration into the formula is substituted with the "minus" sign !!!

Move with equivalent movement

An additional formula called strong
Formula in coordinates

Connection with average speed

With an equilibrium movement, the average speed can be calculated as a medium-ray initial and final speed.

This rule follows a formula that is very convenient to use when solving many tasks

Ratio path

If the body moves equally, the initial speed is zero, the paths passing into sequential equal intervals are related as a sequential number of odd numbers.

The main thing is to remember

1) what is an equidalized movement;
2) what characterizes acceleration;
3) Acceleration - vector. If the body accelerates the acceleration is positive, if slows down - the acceleration is negative;
3) the direction of the acceleration vector;
4) formulas, units of measurement in si

Exercises

Two trains go towards each other: one - accelerated to the north, the other is slowed to the south. How are trains accelerate?

Equally north. Because at the first train, the acceleration coincides towards the movement, and the second - the opposite movement (it slows down).

Mechanics

Kinematics formulas:

Kinematics

Mechanical movement

Mechanical movement It is called changing the position of the body (in space) relative to other bodies (over time).

The relativity of motion. Reference system

To describe mechanical movement Body (points), you need to know its coordinates at any time. To determine the coordinates should choose body reference and tie with him coordinate system. Often the body of reference is the land with which the rectangular decartian coordinate system is associated. To determine the position of the point at any time you also need to set the start of the time.

The coordinate system, the reference body with which it is connected, and the device for measuring time form reference systemRegarding which the movement of the body is considered.

Material point

Body, the sizes of which in these conditions of movement can be neglected, called material point.

The body can be viewed as a material point if its dimensions are small compared to the distance it passes, or compared with distances from it to other bodies.

Trajectory, Path, Move

Movement trajectory The line is called, along which the body moves. The length of the trajectory is called traveled path. Way - Scalar physical quantityMaybe only positive.

Movement The vector is called the vector connecting the initial and endpoint of the trajectory.

The movement of the body at which all of its points at the moment is moving the same, called progressive movement. For description transportation Body is enough to choose one point and describe its movement.

Movement in which the trajectories of all points of the body are circles with centers on one straight line and all planes of circles perpendicular to this straight, called rotational motion.

Meter and second

To determine the coordinates of the body, you need to be able to measure the distance on a straight line between the two points. Any process of measuring the physical quantity consists in comparing the measured value with a unit of measurement of this magnitude.

The unit of length in the international system of units (C) is meter. The meter is approximately 1/40,000,000 parts of the earth meridian. According to the modern presentation, the meter is the distance that the light takes place in the emptiness for 1/299 792 458 share of a second.

To measure time, some periodically repeated process is selected. Unit of time measurement in si adopted second. The second is 9,192,631,770 the radiation period of the cesium atom during the transition between the two levels of the hyperfine structure of the main state.

In the SI length and time taken for independent of other values. Similar values \u200b\u200bare called basic.

Instant speed

For the quantitative characteristics of the body movement process, the concept of motion speed is introduced.

Instant speed The progressive movement of the body at time t is called the ratio of the very small displacement DS to a small period of time DT, for which it happened:

Instant speed - vector magnitude. Instant movement speed is always directed by the trajectory towards the body movement.

The speed unit is 1 m / s. Meter per second equal to speed Straight and evenly moving point, at which the point for time 1 s is moved to a distance of 1 m.

Acceleration

Acceleration The vector physical value is called the ratio of a very small change of the velocity vector to a small period of time for which this change occurred, i.e. This is a speed change of speed:

The meter per second per second is such an acceleration at which the body speed moving is straight and equal to 1 C varies by 1 m / s.

The direction of the acceleration vector coincides with the direction of the velocity change vector () at very small values \u200b\u200bof the period of time, for which the speed changes occurs.

If the body moves in a straight line and its speed increases, the direction of the acceleration vector coincides with the direction of the velocity vector; As descending speed, the opposite direction of the velocity vector.

When moving along a curved trajectory, the direction of the velocity vector varies during the movement, the speed of the acceleration can be directed at any angle to the velocity vector.

Uniform, equivalent rectilinear movement

Movement with a constant speed is called uniform straight movement. With a uniform rectilinear movement, the body moves in direct and in any equal periods of time passes the same paths.

Movement in which the body in equal intervals of time makes unequal movements, called uneven movement. With this movement, the body speed varies over time.

Equipment This movement is called, in which the body's speed in any equal intervals varies to the same value, i.e. Movement with constant acceleration.

Equalious An equally referred movement is called, in which the speed increases. Equalioned - Equipment movement at which the speed is reduced.

Tasks in physics are easy!

Do not forgetthat it is always necessary to solve the tasks in the SI system!

And now to the tasks!

Elementary tasks from the course of school physics in kinematics.
Solving tasks for straight equilibrium movement. When solving the problem, we must make the drawing on which we show all the vector about which are in question in the task. In the condition of the problem, unless otherwise specified, the modules are given. In response, the task must also stand the value module.
Task 1.
The car, moving at a speed of 30 m / s, began to slow down. What will be equal to its speed after 1 minute, if the acceleration when braking is 0.3 m / s 2?
Note! The projection of the acceleration vector on the T axis is negative.

Task 2.
Sledges begin to move from the mountain with an acceleration of 2 m / s 2. What distance will they pass in 2 seconds?

Do not forget to answer the projection to the speed of the acceleration vector!
Task 3.
What is the acceleration of the cyclist if its speed in 5 seconds has changed from 7 to 2 m / s?

From the condition of the problem, it can be seen that in the process of movement, the body speed decreases. Based on this, we determine the direction of the acceleration vector in the drawing. As a result of the calculation, the negative value of the acceleration vector should be.
Task 4.
Sledges begin to move from the mountain from the state of rest with an acceleration of 0.1 m / s 2. What speed will they have 5 seconds after the start of movement?

Task 5.
The train moved with an acceleration of 0.4 m / s 2, after 20 seconds, braking stopped. What is the brake path, if the initial train rate is 20 m / s?
Attention! In the task, the train slows down, do not forget about the minus when substituting the numerical value of the projection of the acceleration vector.

Task 6.
The bus, moving away from the stop, is moving with an acceleration of 0.2 m / s 2. At what distance from the beginning of the movement, its speed will be equal to 10 m / s?

The task can be solved in 2 actions.
This solution is similar to the solution of the system of two equations with two unknowns. As in algebra: two equations - formulas for V x and S x, two unknowns - T and S x.
Task 7.
What speed will the boat, passing out of the state of the rest 200 meters with an acceleration of 2 m / s 2?
Do not forget that not always all the data in the task is set by numbers!
Here you have to pay attention to the words "from the state of rest" - it corresponds to the initial speed of 0.
When removing the square root: time can only be more than 0!
Task 8.
In case of emergency braking, a motorcycle, moving at a speed of 15 m / s, left after 5 seconds. Find a brake path.

Continue see

The formulas of the rectilinear movement of the material point are derived for the three ways to set the movement - with a known dependence of the coordinate from time; With a certain dependence of acceleration from time and acceleration from the coordinate. Reviewed straight uniform and straight equative movement.
Content
Basic rectilinear movement formulas

Let the material point move along the axis. Further and denote the coordinate and speed of the point at the initial moment of time.
If the law of changes to its coordinates is set:
,
That differentiating the coordinate in time, we obtain the speed and acceleration of the point:
;
.

Let us known addiction acceleration from time:
.
Then the dependences of the speed and coordinates are determined by the formulas:
(1) ;
(2) ;
(3) ;
(4) .

Let us known addiction acceleration from coordinate:
.
Then the dependence of the speed of the coordinate is:
(5) .
The dependence of the coordinate on time is determined in an implicit form:
(6) .

For straight uniform movement:
;
;
.

For rectilinear equative movement:
;
;
;
.

The formulas given here can be applied not only for straight movement, but also for some cases of curvilinear movement. For example, for a three-dimensional movement in a rectangular coordinate system, if the movement along the axis does not depend on the projections of the values \u200b\u200bto other coordinate axes. Then formulas (1) - (6) give dependencies for the projections of the values \u200b\u200bon the axis.

Also, these formulas are applicable when moving along a given trajectory with a natural way to set the movement. Only here as the coordinate is the length of the trajectory arc, which is counted from the selected field of reference. Then instead of projections and should be substituted and - projections of speed and acceleration on the selected direction of tangent to the trajectory.

Rectilinear movement with a known dependence of the coordinates from time to time

Consider the case when the material point moves in a straight line. Select the coordinate system with the beginning at an arbitrary point. The axis will send along the point of movement of the point. Then the position of the point is uniquely determined by the value of one coordinate.

If the law is set to change the coordinates from time to:
,
That differentiating in time, we will find the law of changing the speed:
.
When the point moves in the positive direction of the axis (in the figure from left to right). When the point moves in the negative axis direction (in the figure on the right left).

Differentiating speed in time, we find the law of changes in acceleration:
.
Since direct is not curvature, the radius of the curvature of the trajectory can be considered infinitely large ,. Then normal acceleration is zero:
.
That is, the acceleration of the point is tangential (tangent):
.
That is quite natural, since the speed and acceleration of the point are aimed at tangent to the trajectory - direct, along which movement takes place.
If one sign (i.e., both positive or both negative), the speed module increases (speed increases in absolute value). If different signs, the speed module decreases (the speed decreases in absolute value).

Rectilinear movement with a well-known acceleration

Acceleration

Let us know the law of changing acceleration from time to:
.
Our task is to find the law of changing the speed and the law of change of coordinates from time to:
;
.

Apply the formula:
.
This is a differential equation of first order with separating variables
;
.
Here is a constant integration. It can be seen that only according to the known dependence of acceleration on time, it is impossible to unambiguously determine the dependence of the speed of time. We received a whole set of laws change, which differ from each other into an arbitrary constant. To find the law of changing the speed you need, we must set another value. As a rule, this value is the value of the speed at the initial moment of time. To do this by moving from an indefinite integral to a specific:
.
Let be the speed of the point at the initial moment of time. Substitute:
;
;
.
Thus, the law of changing the speed of time has the form:
(1) .

Similarly, we determine the law of changes in the coordinates.
.
(2) .
Here is the value of the coordinate at the initial moment of time.

Substitute (1) in (2).

.

Integration area in double integral.

If you change the order of integration in a double integral, we will get:

.

Thus, we received the following formulas:
(3) ;
(4) .

Coordinate acceleration

Now let us know the law of changing acceleration from the coordinate:
.
We need to solve the differential equation:
.
This differential equation does not contain an independent variable explicitly. General method Solutions of such equations are considered on the page "Differential equations of higher orders that do not contain an independent variable explicitly." According to this method, we believe that is a function from:
;
.
We share variables and integrate:
;
;
;
.
Removing the root must be taken into account that the speed can be both positive and negative. On a slight distance from the point, the sign is determined by the sign constant. However, if the acceleration is directed oppositely, the speed of the point will decrease to zero and the direction of movement will change to the opposite. Therefore, the correct sign, plus or minus, is selected when considering a particular movement.
(5) .
At the beginning of the move
.

Now determine the dependence of the coordinates from time to time. Differential equation for the coordinate has the form:
.
This is a differential equation with separating variables. We share variables and integrate:
(6) .
This equation determines the dependence of the coordinate on the implicit form.

Straight uniform traffic

Apply the results obtained above for the case of a straight uniform movement. In this case, acceleration
.
;
. That is, the speed is constant, and the coordinate linearly depends on time. Formulas (5) and (6) give the same result.

Straight equivalent movement

Now consider a straight equestrian movement.
In this case, the acceleration is the value of constant:
.
According to formulas (1) and (2) we find:
;

.

If we apply to formula (5), we obtain the dependence of the speed of the coordinate:
.

Rectilinear motion in vector form

The resulting formulas can be represented in vector form. To do this, it is sufficient to multiply the equations defining and on a single vector (ort), directed along the axis.

Then the radius-vector point, velocity and acceleration vectors are:
;
;
.

Share an article with friends:

Similar articles