Uniform movement. What is acceleration

Uniformly accelerated motion is a motion with acceleration, the vector of which does not change in magnitude and direction. Examples of such movement: a bicycle that rolls down a hill; a stone thrown at an angle to the horizon.

Let's consider the last case in more detail. At any point of the trajectory, the stone is subjected to an acceleration free fall g → , which does not change in magnitude and is always directed in the same direction.

The motion of a body thrown at an angle to the horizon can be represented as the sum of motions about the vertical and horizontal axes.

Along the X axis the motion is uniform and rectilinear, and along the Y axis it is uniformly accelerated and rectilinear. We will consider the projections of the velocity and acceleration vectors on the axis.

Formula for speed at uniformly accelerated motion:

Here v 0 is the initial speed of the body, a = c o n s t is the acceleration.

Let us show on the graph that with uniformly accelerated motion, the dependence v (t) has the form of a straight line.

Acceleration can be determined from the slope of the velocity graph. In the figure above, the acceleration modulus is equal to the ratio of the sides of the triangle ABC.

a = v - v 0 t = B C A C

How more angleβ , the greater the slope (steepness) of the graph with respect to the time axis. Accordingly, the greater the acceleration of the body.

For the first graph: v 0 = - 2 m s; a \u003d 0, 5 m s 2.

For the second graph: v 0 = 3 m s; a = - 1 3 m s 2 .

From this graph, you can also calculate the movement of the body in time t. How to do it?

Let's single out a small time interval ∆ t on the graph. We will assume that it is so small that the movement during the time ∆ t can be considered uniform movement with a speed, equal speed body in the middle of the interval ∆ t . Then, the displacement ∆ s during the time ∆ t will be equal to ∆ s = v ∆ t .

Let's divide all time t into infinitely small intervals ∆ t . The displacement s in time t is equal to the area of ​​the trapezoid O D E F .

s = O D + E F 2 O F = v 0 + v 2 t = 2 v 0 + (v - v 0) 2 t .

We know that v - v 0 = a t , so the final formula for moving the body will be:

s = v 0 t + a t 2 2

In order to find the coordinate of the body at a given time, you need to add displacement to the initial coordinate of the body. A change in coordinates during uniformly accelerated motion expresses the law of uniformly accelerated motion.

Law of uniformly accelerated motion

y = y 0 + v 0 t + a t 2 2 .

Another common problem that arises in the analysis of uniformly accelerated motion is finding the displacement for given values ​​of the initial and final velocities and acceleration.

Eliminating t from the above equations and solving them, we obtain:

s \u003d v 2 - v 0 2 2 a.

From the known initial speed, acceleration and displacement, you can find the final speed of the body:

v = v 0 2 + 2 a s .

For v 0 = 0 s = v 2 2 a and v = 2 a s

Important!

The values ​​v , v 0 , a , y 0 , s included in the expressions are algebraic quantities. Depending on the nature of the movement and the direction of the coordinate axes in a particular task, they can take both positive and negative values.

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Acceleration of a point in rectilinear motion

mechanical movement. Basic concepts of mechanics.

mechanical movement- change in the position of bodies (or their parts) in space over time relative to other bodies.

From this definition it follows that mechanical movement- movement relative.

The body in relation to which the given mechanical movement is considered is called reference body.

reference system- this is a set of a reference body, a coordinate system and a time reference system associated with this body, in relation to which the movement (or equilibrium) of any other material points or bodies is being studied(Fig. 1).

inertial frame of reference(a.s.o.)– a reference system in which the law of inertia is valid: a material point, when no forces act on it (or mutually balanced forces act), is at rest or uniform rectilinear motion.

Any frame of reference moving with respect to And. With. O. progressively, uniformly and rectilinearly, there is also And. With. O. Therefore, theoretically, there can be any number of equal And. With. O., which have the important property that the laws of physics are the same in all such systems (the so-called principle of relativity).

If the frame of reference is moving relative to the I.S.O. unevenly and rectilinearly, then it is non-inertial and the law of inertia is not fulfilled in it. This is explained by the fact that with respect to a non-inertial frame of reference, a material point will have an acceleration even in the absence of active forces, due to the accelerated translational or rotational motion of the frame of reference itself.

The concept of and. With. O. is a scientific abstraction. The real reference system is always associated with some specific body (the Earth, the hull of a ship or aircraft, etc.), in relation to which the motion of certain objects is studied. Since there are no motionless bodies in nature (a body that is motionless relative to the Earth will move with it accelerated in relation to the Sun and stars, etc.), then any real frame of reference is non-inertial and can be considered as And. With. O. with some degree of approximation.

With a very high degree of accuracy And. With. O. we can consider the so-called heliocentric (stellar) system with the beginning at the center of the Sun (more precisely, at the center of mass solar system) and with axes directed to three stars. To solve most technical problems And. With. O. In practice, a system rigidly connected to the Earth can serve, and in cases requiring greater accuracy (for example, in gyroscopy), with the beginning at the center of the Earth and the axes directed to the stars.

When moving from one And. With. O. to the other, in classical Newtonian mechanics Galilean transformations are valid for spatial coordinates and time, and in relativistic mechanics (that is, at speeds close to the speed of light), Lorentz transformations are valid.

Material point- a body whose dimensions, shape and internal structure can be neglected under the conditions of this problem.

A material point is an abstract object.

Absolutely solid (ATT) - a body, the distance between any two points of which remains unchanged (the deformation of the body can be neglected).

ATT is an abstract object.

finite movement - movement in a limited area of ​​​​space, infinite motion is a movement that is unlimited in space.

Point position A in space, the radius is set - by a vector or its three projections on the coordinate axes (Fig. 2).

Fig.2.

Kinematics

Kinematics- a section of mechanics devoted to the study of the laws of motion of bodies without taking into account their masses and acting forces.

Basic concepts of kinematics

Path - scalar physical quantity, equal to the length of the trajectory section, passed by the material point for the considered period of time; in SI: = m(meter).

In classical physics, it was implicitly assumed that the linear dimensions of a body are absolute, i.e. are the same in all inertial frames of reference. However, in special theory relativity proves length relativity(reduction of the linear dimensions of the body in the direction of its movement).

Linear dimensions the largest bodies in the frame of reference relative to which the body is at rest:Δ l =Δ i.e. > , where is the proper length of the body, i.e. body length measured in ISO, relative to which the body is at rest, where .

moving –vector,connecting the position of a moving point at the beginning and end of a certain period of time(Fig. 6); in SI: .

Fig.6.

- movement, ABCD- path. Fig.7.

Example. The movement of the point is given by the equations:

Write the equation for the trajectory of the point and determine its coordinates after the start of the movement.

Fig.8.

The initial position of the point (at ) is determined by the coordinates , cm. In 1 sec. the point will be at the position with coordinates:

Time(t) – one of the categories(along with space) denoting the form of existence of matter; form of flow of physical and mental processes; expresses the order of change of phenomena; a condition for the possibility of change, as well as one of the coordinates of the space–time along which the world lines are stretched physical bodies ; in SI: - second.

In classical physics, it was implicitly assumed that time is an absolute value, i.e. the same in all inertial frames of reference. However, in the special theory of relativity, the dependence of time on the choice of inertial system reference: , where is the time measured by the clock of an observer moving along with the frame of reference. This led to the conclusion that relativity of simultaneity, namely: in contrast to classical physics, where it was assumed that simultaneous events in one inertial frame of reference are simultaneous in another inertial frame of reference, in the relativistic case spatially separated events that are simultaneous in one inertial frame of reference may be non-simultaneous in another frame of reference.

Z.2. Speed

Speed(often denoted, or from English. velocity or fr. vitesse)– a vector physical quantity that characterizes the speed of movement and the direction of movement of a material point in space relative to the selected reference system.

Instant Speed is a vector quantity equal to the first derivative of the radius vector moving point in time(the speed of a body at a given point in time or at a given point in the trajectory):

The instantaneous velocity vector is directed tangentially to the trajectory in the direction of the point movement (Fig. 9).

Rice. 9.

In the same time , That's why

Thus, the coordinates of the velocity vector are the rates of change of the corresponding coordinate of the material point:

or in notation:

Then the speed modulus can be represented as: In general, the path is different from the displacement modulus. However, if we consider the path passable point for a short period of time , That . Therefore, the modulus of the velocity vector is equal to the first derivative of the path length with respect to time: .

If the point velocity modulus does not change over time , that movement is called uniform.

For uniform motion, the relation is true: .

If the modulus of velocity changes with time, then the movement is called uneven.

Uneven movement is characterized by average speed and acceleration.

The average ground speed of the non-uniform movement of a point in a given section of its trajectory is a scalar value , equal to the ratio of the length of this section, the trajectory to the duration of time passing it through(Fig. 10): , where is the path traveled by the point in time .

Rice. 10. Vectors of instantaneous and average speed.

Rice. eleven.

In classical mechanics, speed is a relative quantity, i.e. is transformed upon transition from one inertial frame of reference to another according to Galileo's transformations.

When considering a complex movement (that is, when a point or a body moves in one frame of reference, and the frame of reference itself moves relative to another), the question arises about the relationship of velocities in 2 frames of reference, which establishes the classical law of addition of velocities:

the speed of the body relative to the fixed frame of reference is equal to the vector sum of the speed of the body relative to the moving frame and the speed of the moving frame itself relative to the fixed one:

where is the speed of a point relative to a fixed frame of reference, is the speed of a moving frame of reference relative to a fixed frame, is the speed of a point relative to a moving frame of reference.

Example:

1. The absolute speed of a fly crawling along the radius of a rotating gramophone record is equal to the sum of the speed of its movement relative to the record and the speed that the point of the record under the fly has relative to the ground (that is, from which the record carries it due to its rotation).

2. If a person walks along the corridor of the car at a speed of 5 kilometers per hour relative to the car, and the car moves at a speed of 50 kilometers per hour relative to the Earth, then the person moves relative to the Earth at a speed of 50 + 5 = 55 kilometers per hour when walking in the direction of travel train, and at a speed of 50 - 5 = 45 kilometers per hour when he goes in the opposite direction. If a person in the carriage corridor moves relative to the Earth at a speed of 55 kilometers per hour, and a train at a speed of 50 kilometers per hour, then the speed of a person relative to the train is 55–50 = 5 kilometers per hour.

3. If the waves move relative to the coast at a speed of 30 kilometers per hour, and the ship also at a speed of 30 kilometers per hour, then the waves move relative to the ship at a speed of 30–30 = 0 kilometers per hour, that is, they become motionless relative to the ship.

In the relativistic case, the relativistic law of addition of velocities is applied: .

It follows from the last formula that the speed of light is the maximum speed of transmission of interactions in nature.

Acceleration

Acceleration is a value that characterizes the rate of change of speed.

Acceleration(usually denoted ) – derivative of velocity with respect to time, a vector quantity showing how much the velocity vector of a point (body) changes when it moves per unit of time(i.e., acceleration takes into account not only the change in the magnitude of the speed, but also its direction).

For example, near the Earth, a body falling to the Earth, in the case where air resistance can be neglected, increases its speed by about 9.81 m / s every second, that is, its acceleration, called free fall acceleration .

Derivative of acceleration with respect to time, i.e. the quantity characterizing the rate of change of acceleration is called jerk.

The acceleration vector of a material point at any time is found by differentiating the velocity vector of a material point with respect to time:

.

Acceleration modulus algebraic value:

- movement accelerated(speed increases in magnitude);

- movement delayed(speed decreases in magnitude);

- the movement is uniform.

If – movement equally variable(uniformly accelerated or equally retarded).

Average acceleration

Average acceleration - this is the ratio of the change in speed to the period of time during which this change occurred:

Where - mean acceleration vector.

The direction of the acceleration vector coincides with the direction of the change in speed (here, this is the initial speed, that is, the speed at which the body began to accelerate).

At a moment in time, the body has a speed . At the moment of time, the body has a speed (Fig. 12). According to the rule of subtracting vectors, we find the vector of change in speed. Then the acceleration can be defined as follows:

Rice. 12.

Instant acceleration.

Instantaneous acceleration of a body (material point) at a given moment of time is a physical quantity equal to the limit to which the average acceleration tends when the time interval tends to zero. In other words, this is the acceleration that the body develops in a very short period of time:

.

The direction of acceleration also coincides with the direction of change in speed for very small values ​​of the time interval during which the change in speed occurs.

The acceleration vector can be set by projections onto the corresponding coordinate axes in a given frame of reference:

those. the projection of the acceleration of a point on the coordinate axes are equal to the first derivatives of the projections of the velocity or the second derivatives of the corresponding coordinates of the point in time. The module and direction of acceleration can be found from the formulas:

,

where are the angles formed by the acceleration vector with the coordinate axes.

Acceleration of a point in rectilinear motion

If the vector , i.e. does not change with time, the movement is called uniformly accelerated. For uniformly accelerated motion, the formulas are valid:

With accelerated rectilinear motion, the speed of the body increases in absolute value, that is, and the direction of the acceleration vector coincides with the velocity vector, (i.e.).

Rice. 13.

Acceleration of a point during curvilinear motion

When moving along a curvilinear trajectory, not only the modulus of speed changes, but also its direction. In this case, the acceleration vector is represented as two components.

Indeed, when a body moves along a curvilinear trajectory, its velocity changes in magnitude and direction. The change in the velocity vector over a certain small period of time can be set using a vector (Fig. 14).

The vector of velocity change in a short time can be decomposed into two components: directed along the vector (tangential component), and directed perpendicular to the vector (normal component).

Then the instantaneous acceleration is: .

Tangential (tangential) acceleration – is the component of the acceleration vector directed along the tangent to the trajectory at a given point in the trajectory. Tangential acceleration characterizes the change in speed modulo during curvilinear motion:

Normal(centripetal) acceleration

Normal acceleration is a component of the acceleration vector directed along the normal to the motion trajectory at a given point on the body motion trajectory. That is, the normal acceleration vector is perpendicular to the linear speed of movement (Fig. 15). Normal acceleration characterizes the change in speed in direction and is denoted by the symbol . The normal acceleration vector is directed along the radius of curvature of the trajectory. From fig. 15 shows that

Rice. 17. Movement along arcs of circles.

Normal acceleration depends on the modulus of velocity and on the radius of the circle along the arc of which the body is moving at the moment.

With rectilinear motion, the vectors and are directed along one straight line, which is at the same time the trajectory of motion. Along the same straight line in the direction of movement of the bodies, it was agreed to direct the coordinate axis (X axis). In this case, the difference vector, and hence the acceleration vector a, lies on the same straight line (see § 6). But where is it directed - in the direction of movement (just like the X axis) or against it?

In § 6 we saw that the projection of the difference of two vectors on some axis is equal to the difference of their projections on the same axis. Therefore, for the projections of the vectors and onto the X-axis, we can write

Here a is the projection of the vector a onto the axis of the projection of the vectors and onto the same axis.

Since all three vectors lie on the same straight line (X-axis), the absolute values ​​of their projections are equal to the absolute values ​​of the vectors themselves.

Consider 2 cases of accelerated motion of the body.

First case. The speed of the body increases in absolute value (the body “accelerates”). This means that Then from formula (1) it can be seen that the acceleration projection a is positive and equal to Vector a, therefore, it is directed in the same way as the X axis, i.e., in the direction of motion. When, for example, an armor-piercing projectile moves when fired in the gun barrel, its speed increases and the acceleration is directed in the same direction as the speed (Fig. 39).

Second case. The body slows down, i.e., the absolute value of its speed decreases. From formula (1), it can be seen that the acceleration projection a in this case is negative:

From formula (1) you can get the expression for the speed:

In this formula, we repeat, the projections of the vectors on the X axis, which can be both positive and negative.

When solving problems, it is convenient to write the expression for velocity (2) in such a way that it can immediately be seen from it how the acceleration vector is directed.

If the speed of the body increases (acceleration), then

When the speed of the body decreases (braking),

It is clear that a body that is being decelerated must stop at some point. This will happen, as can be seen from formula (26), when it becomes equal, i.e., at the moment of time But if the acceleration remains constant (in modulus and direction) after this moment, then the body, having stopped, will begin to move in the opposite direction . This can be seen from the fact that when it becomes greater than the speed will change its sign to the opposite. So

moves, for example, a body thrown vertically upwards: having reached highest point trajectory, the body starts moving down.

If the acceleration vector is directed in the same way as the coordinate axis, then it follows from formula (2a) that

If the coordinate axis is chosen so that the direction of the acceleration vector is opposite to the direction of the coordinate axis, then from formula (26) it follows that

The sign in this formula means that the velocity vector, as well as the acceleration vector, is directed opposite to the direction of the coordinate axis. The velocity modulus, of course, also in this case increases with time.

Usually we call movement with increasing absolute value speed accelerated movement, and movement with decreasing speed slow movement. But in mechanics, any uneven movement is accelerated movement. Whether the car starts off or brakes, in both cases it moves with acceleration. Accelerated rectilinear motion differs from the slow one only by the sign of the projection of the acceleration vector.

We know that both displacement, and speed, and the trajectory of motion are different with respect to different reference bodies moving relative to each other.

What about acceleration? Is it relative?

The acceleration of a body, as we now know, is determined by the vector difference of two values ​​of its speed at different times. When moving from one coordinate system to another, moving uniformly and rectilinearly relative to the first, both speed values ​​will change. But they will change by the same amount. Their difference will remain unchanged. Therefore, the acceleration will remain unchanged.

In all frames of reference, moving relative to each other in a straight line and uniformly, the acceleration of the body is the same.

But the accelerations of the body will be different in frames of reference moving with acceleration relative to each other. In this case, the accelerations add up in the same way as the velocities (see § 10).

Task. A car passes by an observer at a speed of 10 m/s. At this point, the driver applies the brakes and the car begins to accelerate. How long does it take from the moment the driver applies the brakes to the car stopping?

Solution. Let us choose the place where the observer is located as the origin, and direct the coordinate axis in the direction of the vehicle movement. Then the projection of the vehicle speed on this axis will be positive. Since the speed of the car

decreases, then the acceleration projection is negative and we must use formula (26):

Substituting the numerical values ​​of the given values ​​into this formula, we obtain:

For the positive direction of the coordinate axis, one can also take the direction opposite to the movement. Then the projection of the initial speed of the car will be negative and the projection of acceleration will be positive, and then formula (2a) should be applied:

The result is the same. Yes, it cannot depend on how the direction of the coordinate axis is chosen!

Exercise 9

1. What is acceleration and why do you need to know it?

2. With any uneven movement, the speed changes. How does acceleration characterize this change?

3. What is the difference between slow rectilinear motion and accelerated?

4. What is uniformly accelerated motion?

5. A trolleybus, starting off, moves with constant acceleration. How long will it take for it to acquire a speed of 54 km/h?

6. A car moving at a speed of 36 km / h stops when braking for 4 seconds. How fast is the car moving when braking?

7. The truck, moving with constant acceleration, increased its speed from 15 to 25 m/s on a certain section of the road. How long did this increase in speed take if the acceleration of the truck is

8. What speed of movement would be achieved if the body moved in a straight line with acceleration for 0.5 hours at an initial speed equal to zero?

In general uniformly accelerated motion called such a movement in which the acceleration vector remains unchanged in magnitude and direction. An example of such a movement is the movement of a stone thrown at a certain angle to the horizon (ignoring air resistance). At any point in the trajectory, the acceleration of the stone is equal to the acceleration of free fall. For a kinematic description of the movement of a stone, it is convenient to choose a coordinate system so that one of the axes, for example, the axis OY, was directed parallel to the acceleration vector. Then the curvilinear motion of the stone can be represented as the sum of two motions - rectilinear uniformly accelerated motion along the axis OY And uniform rectilinear motion in the perpendicular direction, i.e. along the axis OX(Fig. 1.4.1).

Thus, the study of uniformly accelerated motion is reduced to the study of rectilinear uniformly accelerated motion. In the case of rectilinear motion, the velocity and acceleration vectors are directed along the straight line of motion. Therefore, the speed v and acceleration a in projections on the direction of motion can be considered as algebraic quantities.

With uniformly accelerated rectilinear motion, the speed of the body is determined by the formula

(*)

In this formula, υ 0 is the speed of the body at t = 0 (starting speed ), a= const - acceleration. On the velocity graph υ ( t), this dependence looks like a straight line (Fig. 1.4.2).

The slope of the velocity graph can be used to determine the acceleration a body. The corresponding constructions are made in Figs. 1.4.2 for graph I. The acceleration is numerically equal to the ratio of the sides of the triangle ABC:

The greater the angle β that forms the velocity graph with the time axis, i.e. the greater the slope of the graph ( steepness), the greater the acceleration of the body.

For graph I: υ 0 \u003d -2 m / s, a\u003d 1/2 m / s 2.

For graph II: υ 0 \u003d 3 m / s, a\u003d -1/3 m / s 2

The velocity graph also allows you to determine the displacement projection s body for a while t. Let us allocate on the time axis some small time interval Δ t. If this time interval is small enough, then the change in speed over this interval is small, i.e., the movement during this time interval can be considered uniform with a certain average speed, which is equal to the instantaneous speed υ of the body in the middle of the interval Δ t. Therefore, displacement Δ s in time Δ t will be equal to Δ s = υΔ t. This displacement is equal to the area of ​​the shaded strip (Fig. 1.4.2). Breaking down the time span from 0 to some point t for small intervals Δ t, we get that the displacement s for a given time t with uniformly accelerated rectilinear motion is equal to the area of ​​the trapezoid ODEF. Corresponding constructions are made for graph II in fig. 1.4.2. Time t taken equal to 5.5 s.

Since υ - υ 0 = at, the final formula for moving s bodies with uniformly accelerated motion over a time interval from 0 to t will be written in the form:

(**)

To find the coordinate y body at any given time. t to the starting coordinate y 0 add displacement over time t:

(***)

This expression is called law of uniformly accelerated motion .

When analyzing a uniformly accelerated motion, sometimes the problem arises of determining the displacement of a body according to the given values ​​​​of the initial υ 0 and final υ velocities and acceleration a. This problem can be solved using the equations written above by eliminating time from them. t. The result is written as

From this formula, you can get an expression for determining the final speed υ of the body, if the initial speed υ 0 is known, acceleration a and moving s:

If the initial speed υ 0 is equal to zero, these formulas take the form

It should again be noted that the quantities υ 0, υ, included in the formulas of uniformly accelerated rectilinear motion, s, a, y 0 are algebraic quantities. Depending on the specific type of movement, each of these quantities can take both positive and negative values.

In this topic, we will consider a very special kind of non-uniform motion. Based on the opposition to uniform movement, uneven movement is movement at an unequal speed, along any trajectory. What is the characteristic of uniformly accelerated motion? This is an uneven movement, but which "equally accelerating". Acceleration is associated with an increase in speed. Remember the word "equal", we get an equal increase in speed. And how to understand "an equal increase in speed", how to evaluate the speed is equally increasing or not? To do this, we need to detect the time, estimate the speed through the same time interval. For example, a car starts moving, in the first two seconds it develops a speed of up to 10 m/s, in 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 by 10 m/s. This is uniformly accelerated motion.

The physical quantity that characterizes how much each time the speed increases is called acceleration.

Can a cyclist's movement be considered uniformly accelerated if, after stopping, his speed is 7 km/h in the first minute, 9 km/h in the second, and 12 km/h in the third? It is forbidden! The cyclist accelerates, but not equally, first accelerating by 7 km/h (7-0), then by 2 km/h (9-7), then by 3 km/h (12-9).

Usually, the movement with increasing speed is called accelerated movement. Movement with decreasing speed - slow motion. But physicists call any motion with a changing speed accelerated motion. Whether the car starts off (speed increases!), or slows down (speed decreases!), in any case, it moves with acceleration.

Uniformly accelerated motion- this is the movement of the body, in which its speed for any equal intervals of time changes(may increase or decrease) equally

body acceleration

Acceleration characterizes the rate of change of speed. This is the number by which the speed changes every second. If the modulo acceleration of the body is large, this means that the body quickly picks up speed (when it accelerates) or quickly loses it (when decelerating). Acceleration- this is a physical vector quantity, numerically equal to the ratio of the change in speed to the period of time during which this change occurred.

Let's determine the acceleration in the following problem. At the initial moment of time, the speed of the ship was 3 m/s, at the end of the first second the speed of the ship became 5 m/s, at the end of the second - 7 m/s, at the end of the third - 9 m/s, etc. Obviously, . But how do we determine? We consider the speed difference in one second. In the first second 5-3=2, in the second second 7-5=2, in the third 9-7=2. But what if the speeds are not given for every second? Such a task: the initial speed of the ship is 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, 11-7= 4, then 4/2=2. We divide the speed difference by the time interval.

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

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

Direction of the acceleration vector

The direction of the acceleration vector is shown in the figures

In this figure, the car is moving in a positive direction along the Ox axis, the velocity vector always coincides with the direction of movement (directed to the right). When the acceleration vector coincides with the direction of speed, this means that the car is accelerating. The acceleration is positive.

During acceleration, the direction of acceleration coincides with the direction of speed. The acceleration is positive.

In this picture, the car is moving in the positive direction along the Ox axis, the velocity vector is the same as the direction of motion (rightward), the acceleration is NOT the same as the direction of the speed, which means that the car is decelerating. The acceleration is negative.

When braking, the direction of acceleration is opposite to the direction of speed. The acceleration is negative.

Let's figure out why the acceleration is negative when braking. For example, in the first second, the ship dropped speed from 9m/s to 7m/s, in the second second to 5m/s, in the third to 3m/s. The speed changes to "-2m/s". 3-5=-2; 5-7=-2; 7-9=-2m/s. That's where it comes from negative meaning acceleration.

When solving problems, if the body slows down, the acceleration in the formulas is substituted with a minus sign!!!

Moving with uniformly accelerated motion

An additional formula called untimely

Formula in coordinates

Communication with medium speed

With uniformly accelerated movement, the average speed can be calculated as the arithmetic mean of the initial and final speed

From this rule follows a formula that is very convenient to use when solving many problems

Path ratio

If the body moves uniformly accelerated, the initial speed is zero, then the paths traveled in successive equal time intervals are related as a series of odd numbers.

The main thing to remember

1) What is uniformly accelerated motion;
2) What characterizes acceleration;
3) Acceleration is a vector. If the body accelerates, the acceleration is positive; if it slows down, the acceleration is negative;
3) 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 - slowly to the south. How are train accelerations directed?

Same to the north. Because the first train has the same acceleration in the direction of movement, and the second has the opposite movement (it slows down).