Formula for rectilinear uniformly accelerated motion without time. Uniformly accelerated motion. Average and instantaneous speeds of rectilinear uneven motion

Mechanics

Kinematics formulas:

Kinematics

Mechanical movement

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

Relativity of motion. Reference system

To describe the mechanical movement of a body (point), you need to know its coordinates at any moment in time. To determine coordinates, select reference body and connect with him coordinate system. Often the reference body is the Earth, which is associated with a rectangular Cartesian coordinate system. To determine the position of a point at any time, you must also set the beginning of the time count.

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

Material point

A body whose dimensions can be neglected under given motion conditions is called material point.

A body can be considered a material point if its dimensions are small compared to the distance it travels, or compared to the distances from it to other bodies.

Trajectory, path, movement

Trajectory of movement called the line along which the body moves. The path length is called the path traveled. Path– scalar physical quantity, can only be positive.

By moving is the vector connecting the starting and ending points of the trajectory.

The movement of a body in which all its points are in this moment move in the same way in time is called forward movement. To describe the translational motion of a body, it is enough to select one point and describe its movement.

A movement in which the trajectories of all points of the body are circles with centers on the same line and all planes of the circles are perpendicular to this line is called rotational movement.

Meter and second

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

The unit of length in the International System of Units (SI) is meter. A meter is equal to approximately 1/40,000,000 of the earth's meridian. According to modern understanding, a meter is the distance that light travels in emptiness in 1/299,792,458 of a second.

To measure time, some periodically repeating process is selected. The SI unit of measurement of time is second. A second is equal to 9,192,631,770 periods of radiation from a cesium atom during the transition between two levels of the hyperfine structure of the ground state.

In SI, length and time are taken to be independent of other quantities. Such quantities are called main.

Instantaneous speed

To quantitatively characterize the process of body movement, the concept of movement speed is introduced.

Instant speed translational motion of a body at time t is the ratio of a very small displacement Ds to a small period of time Dt during which this displacement occurred:

Instantaneous speed is a vector quantity. The instantaneous speed of movement is always directed tangentially to the trajectory in the direction of body movement.

The unit of speed is 1 m/s. A meter per second is equal to the speed of a rectilinearly and uniformly moving point, at which the point moves a distance of 1 m in 1 s.

Acceleration

Acceleration is called a vector physical quantity equal to the ratio of a very small change in the velocity vector to the short period of time during which this change occurred, i.e. This is a measure of the rate of change of speed:

A meter per second per second is an acceleration at which the speed of a body moving rectilinearly and uniformly accelerates changes by 1 m/s in a time of 1 s.

The direction of the acceleration vector coincides with the direction of the speed change vector () for very small values of the time interval during which the speed change occurs.

If a body moves in a straight line and its speed increases, then the direction of the acceleration vector coincides with the direction of the velocity vector; when the speed decreases, it is opposite to the direction of the speed vector.

When moving along a curved path, the direction of the velocity vector changes during the movement, and the acceleration vector can be directed at any angle to the velocity vector.

Uniform, uniformly accelerated linear motion

Motion at constant speed is called uniform rectilinear movement. With uniform rectilinear motion, a body moves in a straight line and travels the same distances in any equal intervals of time.

A movement in which a body makes unequal movements at equal intervals of time is called uneven movement. With such movement, the speed of the body changes over time.

Equally variable is a movement in which the speed of a body changes by the same amount over any equal periods of time, i.e. movement with constant acceleration.

Uniformly accelerated is called uniformly alternating motion in which the magnitude of the speed increases. Equally slow– uniformly alternating motion, in which the speed decreases.

The most important characteristic when moving a body is its speed. Knowing it, as well as some other parameters, we can always determine the time of movement, distance traveled, initial and final speed and acceleration. Uniformly accelerated motion is only one type of motion. It is usually found in physics problems from the kinematics section. In such problems, the body is taken as a material point, which significantly simplifies all calculations.

Speed. Acceleration

First of all, I would like to draw the reader’s attention to the fact that these two physical quantities are not scalar, but vector. This means that when solving certain types of problems, it is necessary to pay attention to what acceleration the body has in terms of sign, as well as what the vector of the body’s velocity itself is. In general, in problems of a purely mathematical nature, such moments are omitted, but in problems in physics this is quite important, since in kinematics, due to one incorrect sign, the answer may turn out to be erroneous.

Examples

An example is uniformly accelerated and uniformly decelerated motion. Uniformly accelerated motion is characterized, as is known, by acceleration of the body. The acceleration remains constant, but the speed continuously increases at each individual moment. And with uniformly slow motion, the acceleration has a negative value, the speed of the body continuously decreases. These two types of acceleration form the basis of many physical problems and are quite often found in problems in the first part of physics tests.

Example of uniformly accelerated motion

We encounter uniformly accelerated motion everywhere every day. No car is moving in real life evenly. Even if the speedometer needle shows exactly 6 kilometers per hour, you should understand that this is actually not entirely true. Firstly, if we analyze this issue from a technical point of view, then the first parameter that will give inaccuracy will be the device. Or rather, its error.

We find them in all control and measuring instruments. The same lines. Take about ten rulers, at least identical (15 centimeters, for example), or different (15, 30, 45, 50 centimeters). Put them next to each other and you will notice that there are slight inaccuracies and their scales do not quite line up. This is an error. IN in this case it will be equal to half the division value, as with other devices that produce certain values.

The second factor that will cause inaccuracy is the scale of the device. The speedometer does not take into account values such as half a kilometer, one-half kilometer, and so on. It is quite difficult to notice this on the device with the eye. Almost impossible. But there is a change in speed. Albeit by such a small amount, but still. Thus, it will be uniformly accelerated motion, not uniform. The same can be said about a regular step. Let’s say we’re walking, and someone says: our speed is 5 kilometers per hour. But this is not entirely true, and why was explained a little higher.

Body acceleration

Acceleration can be positive or negative. This was discussed earlier. Let us add that acceleration is a vector quantity, which is numerically equal to the change in speed over a certain period of time. That is, through the formula it can be denoted as follows: a = dV/dt, where dV is the change in speed, dt is the time interval (change in time).

Nuances

The question may immediately arise as to how acceleration in this situation can be negative. Those people who ask a similar question motivate this by the fact that even speed cannot be negative, let alone time. In fact, time really cannot be negative. But very often they forget that speed is taken negative values quite possibly. This is a vector quantity, we should not forget about it! It's probably all about stereotypes and incorrect thinking.

So, to solve problems, it is enough to understand one thing: the acceleration will be positive if the body accelerates. And it will be negative if the body slows down. That's it, quite simple. The simplest logical thinking or the ability to see between the lines will, in fact, be part of the solution to a physical problem related to speed and acceleration. A special case is the acceleration of gravity, and it cannot be negative.

Formulas. Problem solving

It should be understood that problems related to speed and acceleration are not only practical, but also theoretical. Therefore, we will analyze them and, if possible, try to explain why this or that answer is correct or, conversely, incorrect.

Theoretical problem

Very often in physics exams in grades 9 and 11 you can come across similar questions: “How will a body behave if the sum of all forces acting on it is zero?” In fact, the wording of the question can be very different, but the answer is still the same. Here, the first thing you need to do is to use superficial buildings and ordinary logical thinking.

The student is given 4 answers to choose from. First: “the speed will be zero.” Second: “the speed of the body decreases over a certain period of time.” Third: “the speed of the body is constant, but it is definitely not zero.” Fourth: “the speed can have any value, but at each moment of time it will be constant.”

The correct answer here is, of course, the fourth. Now let's figure out why this is so. Let's try to consider all the options in turn. As is known, the sum of all forces acting on a body is the product of mass and acceleration. But our mass remains a constant value, we will discard it. That is, if the sum of all forces is zero, the acceleration will also be zero.

So, let's assume that the speed will be zero. But this cannot be, since our acceleration is equal to zero. Purely physically this is permissible, but not in this case, since now we are talking about something else. Let the speed of the body decrease over a period of time. But how can it decrease if the acceleration is constant and equal to zero? There are no reasons or prerequisites for a decrease or increase in speed. Therefore, we reject the second option.

Let us assume that the speed of the body is constant, but it is definitely not zero. It will indeed be constant due to the fact that there is simply no acceleration. But it cannot be said unequivocally that the speed will be different from zero. But the fourth option is right on target. The speed can be any, but since there is no acceleration, it will be constant over time.

Practical problem

Determine which path was traveled by the body in a certain period of time t1-t2 (t1 = 0 seconds, t2 = 2 seconds) if the following data are available. The initial speed of the body in the interval from 0 to 1 second is 0 meters per second, the final speed is 2 meters per second. The speed of the body at the time of 2 seconds is also 2 meters per second.

Solving such a problem is quite simple, you just need to grasp its essence. So, we need to find a way. Well, let's start looking for it, having previously identified two areas. As is easy to see, the body passes through the first section of the path (from 0 to 1 second) with uniform acceleration, as evidenced by the increase in its speed. Then we will find this acceleration. It can be expressed as the difference in speed divided by the time of movement. The acceleration will be (2-0)/1 = 2 meters per second squared.

Accordingly, the distance traveled on the first section of the path S will be equal to: S = V0t + at^2/2 = 0*1 + 2*1^2/2 = 0 + 1 = 1 meter. On the second section of the path, in the period from 1 second to 2 seconds, the body moves uniformly. This means that the distance will be equal to V*t = 2*1 = 2 meters. Now we sum up the distances, we get 3 meters. This is the answer.

Topics of the Unified State Examination codifier: types of mechanical motion, speed, acceleration, equations of rectilinear uniformly accelerated motion, free fall.

Uniformly accelerated motion - this is movement with a constant acceleration vector. Thus, with uniformly accelerated motion, the direction and absolute magnitude of the acceleration remain unchanged.

Dependence of speed on time.

When studying uniform rectilinear motion, the question of the dependence of speed on time did not arise: the speed was constant during the movement. However, with uniformly accelerated motion, the speed changes over time, and we have to find out this dependence.

Let's practice some basic integration again. We proceed from the fact that the derivative of the velocity vector is the acceleration vector:

. (1)

In our case we have . What needs to be differentiated to get a constant vector? Of course, the function. But not only that: you can add an arbitrary constant vector to it (after all, the derivative of a constant vector is zero). Thus,

. (2)

What is the meaning of the constant? At the initial moment of time, the speed is equal to its initial value: . Therefore, assuming in formula (2) we get:

So, the constant is the initial speed of the body. Now relation (2) takes its final form:

. (3)

IN specific tasks we choose a coordinate system and move on to projections onto coordinate axes. Often two axes and a rectangular Cartesian coordinate system are enough, and vector formula (3) gives two scalar equalities:

, (4)

. (5)

The formula for the third velocity component, if needed, is similar.)

Law of motion.

Now we can find the law of motion, that is, the dependence of the radius vector on time. We recall that the derivative of the radius vector is the speed of the body:

We substitute here the expression for speed given by formula (3):

(6)

Now we have to integrate equality (6). It is not difficult. To get , you need to differentiate the function. To obtain, you need to differentiate. Let's not forget to add an arbitrary constant:

It is clear that is the initial value of the radius vector at time . As a result, we obtain the desired law of uniformly accelerated motion:

. (7)

Moving on to projections onto coordinate axes, instead of one vector equality (7), we obtain three scalar equalities:

. (8)

. (9)

. (10)

Formulas (8) - (10) give the dependence of the coordinates of the body on time and therefore serve as a solution to the main problem of mechanics for uniformly accelerated motion.

Let's return again to the law of motion (7). Note that - movement of the body. Then
we get the dependence of displacement on time:

Rectilinear uniformly accelerated motion.

If uniformly accelerated motion is rectilinear, then it is convenient to choose a coordinate axis along the straight line along which the body moves. Let, for example, this be the axis. Then to solve problems we will only need three formulas:

where is the projection of displacement onto the axis.

But very often another formula that is a consequence of them helps. Let us express time from the first formula:

and substitute it into the formula for moving:

After algebraic transformations (be sure to do them!) we arrive at the relation:

This formula does not contain time and allows you to quickly come to an answer in those problems where time does not appear.

Free fall.

An important special case of uniformly accelerated motion is free fall. This is the name given to the movement of a body near the surface of the Earth without taking into account air resistance.

The free fall of a body, regardless of its mass, occurs with a constant free fall acceleration directed vertically downward. In almost all problems, m/s is assumed in calculations.

Let's look at several problems and see how the formulas we derived for uniformly accelerated motion work.

Task. Find the landing speed of a raindrop if the height of the cloud is km.

Solution. Let's direct the axis vertically downwards, placing the origin at the point of separation of the drop. Let's use the formula

We have: - the required landing speed, . We get: , from . We calculate: m/s. This is 720 km/h, about the speed of a bullet.

In fact, raindrops fall at speeds of the order of several meters per second. Why is there such a discrepancy? Windage!

Task. A body is thrown vertically upward at a speed of m/s. Find its speed in c.

Here, so. We calculate: m/s. This means the speed will be 20 m/s. The projection sign indicates that the body will fly down.

Task. From a balcony located at a height of m, a stone was thrown vertically upward at a speed of m/s. How long will it take for the stone to fall to the ground?

Solution. Let's direct the axis vertically upward, placing the origin on the surface of the Earth. We use the formula

We have: so , or . Deciding quadratic equation, we get c.

Horizontal throw.

Uniformly accelerated motion is not necessarily linear. Consider the motion of a body thrown horizontally.

Suppose that a body is thrown horizontally with a speed from a height. Let's find the time and flight range, and also find out what trajectory the movement takes.

Let us choose a coordinate system as shown in Fig. 1 .

We use the formulas:

In our case . We get:

. (11)

We find the flight time from the condition that at the moment of fall the coordinate of the body becomes zero:

Flight range is the coordinate value at the moment of time:

We obtain the trajectory equation by excluding time from equations (11). We express from the first equation and substitute it into the second:

We obtained a dependence on , which is the equation of a parabola. Consequently, the body flies in a parabola.

Throw at an angle to the horizontal.

Let's consider a slightly more complex case of uniformly accelerated motion: the flight of a body thrown at an angle to the horizon.

Let us assume that a body is thrown from the surface of the Earth with a speed directed at an angle to the horizon. Let's find the time and flight range, and also find out what trajectory the body is moving along.

Let us choose a coordinate system as shown in Fig. 2.

We start with the equations:

(Be sure to do these calculations yourself!) As you can see, the dependence on is again a parabolic equation. Try also to show that the maximum lift height is given by the formula.

>>Physics: Speed during uniformly accelerated motion

The theory of uniformly accelerated motion was developed by the famous Italian scientist Galileo Galilei. In his book “Conversations and Mathematical Proofs Concerning Two New Branches of Science Relating to Mechanics and Local Motion,” published in 1638, Galileo first defined uniformly accelerated motion and proved a number of theorems that described its laws.

Getting started studying uniformly accelerated linear motion, let us first find out how the speed of a body is found if the acceleration of this body and the time of movement are known.
With an initial speed equal to zero ( V 0 = 0),
V= at (3.1)
This formula shows that To find the speed of a body after time I after the start of movement, the acceleration of the body must be multiplied by the time of movement.
In the opposite case, when the body makes slow motion and eventually stops ( V= 0), the acceleration formula allows us to find the initial speed of the body:
V 0 = at (3.2)

A clear picture of how the speed of a body changes during uniformly accelerated motion can be obtained by constructing speed graph.

Speed charts were first introduced in the mid-14th century. Franciscan scientist-monk Giovanni di Casalis and the archdeacon of Rouen Cathedral Nicolas Oresme, who later became an adviser to the French king Charles V. They proposed to put time on the horizontal axis, and speed along the vertical axis. In such a coordinate system, velocity graphs for uniformly accelerated motion look like straight lines, the slope of which shows how quickly the speed changes over time.

Formula (3.1), which describes movement with increasing speed, corresponds, for example, to the speed graph shown in Figure 5. The graph shown in Figure 6 corresponds to movement with decreasing speed.

During uniformly accelerated motion, the speed of a body changes continuously. Velocity graphs allow you to determine the speed of a body at different times. But sometimes it is not necessary to know the speed at one or another specific moment in time (this speed is called instant), A average speed along the entire route.

The problem of finding the average speed during uniformly accelerated motion was first solved by Galileo. In his research, he used a graphical method to describe movement.

According to Galileo's theory, if the speed of a body during uniformly accelerated motion increases from 0 to a certain value V, then the average speed will be equal to half the achieved speed:

A similar formula is valid for movement with decreasing speed. If it decreases from some initial value V 0 to 0, then the average speed of such movement is equal to

The results obtained can be illustrated using a speed graph. So, for example, to find the average speed of movement, which corresponds to the graph in Figure 5, we must find half of 6 m/s. The result is 3 m/s. This is the average speed of the movement in question.

1. Who is the author of the first theory of uniformly accelerated motion? 2. What is the speed of a body during uniformly accelerated motion from a state of rest? 3. Using the graph shown in Figure 5, determine the speed of the body 2 s after the start of movement. 4. Using the graph shown in Figure 6, determine the average speed of the body.

S.V. Gromov, N.A. Rodina, Physics 8th grade

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With constant acceleration, the speed of a physical body increases uniformly, starting from zero.

The distance traveled by a uniformly accelerated body, starting from zero speed, is proportional to the square of time.

Galileo Galilei is one of the people who became famous for something that was not for which they should have enjoyed the well-deserved fame. Everyone remembers how this Italian naturalist, at the end of his life, was tried by the Inquisition on suspicion of heresy and forced to renounce the belief that the Earth revolves around the Sun. In fact, this trial had practically no influence on the development of science - in contrast to the experiments previously carried out by Galileo and the conclusions he made on the basis of these experiments, which actually predetermined the further development of mechanics as a branch of physical science.

The motion of physical bodies has been studied since time immemorial, and the foundations of kinematics were laid long before the birth of Galileo. Elementary problems of describing motion are studied today already in primary school. For example, everyone knows that if a car moves uniformly at a speed of 20 km/h, then in 1 hour it will travel 20 km, in 2 hours - 40 km, in 3 hours - 60 km, etc. And until the car moves at a constant speed (the speedometer needle does not deviate from the specified division on its scale), it is not difficult to calculate the distance traveled - just multiply the speed of the car by the time it is on the road. This fact has been known for so long that the name of its discoverer was completely lost in the fog of ancient times.

Difficulties arise as soon as the object begins to move at variable speed. You pull away, for example, from a traffic light - and the speedometer needle creeps up from zero until you release the gas pedal and press the brake pedal. In fact, the speedometer needle practically does not stand still - it moves up or down all the time. At the beginning of each individual second, the real speed of the car is one, and at the end of the second it is already different, and the path traveled by it in a second is not so easy to accurately calculate. This problem - the description of motion with acceleration - worried natural scientists long before Galileo.

Galileo Galilei himself approached it innovatively and, in fact, set the direction for all further development of modern natural science methodology. Instead of sitting and speculatively solving the problem of the movement of accelerating bodies, he came up with experiments that were ingenious in their simplicity, allowing him to experimentally observe what actually happens to accelerating bodies. It may seem to us that there is nothing particularly innovative in this approach, but before Galileo the main method of solving problems was “natural philosophy” - as evidenced by the very name of the then natural science— there was a speculative understanding of what was happening, and not its experimental verification. The very idea of conducting physical experiments was truly radical at that time. To understand the idea of Galileo's experiments, imagine a body falling under the influence of gravity. Let go of any object from your hands and it will fall to the floor; in this case, at the first instant, the speed of its movement will be zero, but it will immediately begin to accelerate - and will continue to accelerate until it falls to the ground. If we can describe the fall of an object to the ground, we can then extend this description to the general case of uniformly accelerated motion.

Today, it is not difficult to measure the dynamics of an object falling - it is possible to accurately record the time from the beginning of the fall to any intermediate point. However, in Galileo's time there were no accurate stopwatches, and any mechanical clock was very primitive and inaccurate by modern standards. Therefore, the scientist first developed an experimental apparatus to circumvent this problem. Firstly, he “diluted” the force of gravity, slowing down the time of fall to reasonable limits from the point of view of available measurement instruments, namely, he forced the bodies to roll down an inclined plane, and not just fall vertically. Then he figured out how to get around the inaccuracy of his contemporary mechanical watches by stringing a series of strings along the path of a ball rolling down an inclined surface, so that it touched them along the way and his movement could be timed by the sounds produced. Time after time, lowering the ball down an inclined path under a series of strings, Galileo moved the strings until he achieved that the ball along its entire path, touching the stretched strings, would produce sounds at regular intervals.

Eventually, Galileo managed to accumulate a sufficient amount of experimental information about uniformly accelerated motion. A body starting from a state of rest then moves as described at the very beginning of this article. Translated into the language of mathematical symbols, uniformly accelerated motion is described by the following equations:

Where a— acceleration, v— speed, d— distance traveled by a body in time t. To feel the meaning of these equations, it is enough to closely observe the falling of objects. The speed of the fall visibly increases with the time that has passed since the beginning of the fall. This follows from the first equation. It is also obvious that during the fall, it takes the body more time to complete the first part of the path than the rest of the path. This is exactly what the second formula describes, since it follows from it that the longer a body accelerates, the greater the distance it covers in the same time.

Galileo made another important observation about a body in a state of free fall under the influence of a force gravitational attraction, although I could not confirm it with direct measurements. By extrapolating the results he obtained from observing objects rolling down an inclined plane, he was able to determine the acceleration of a free fall of a body on the surface of the Earth. The acceleration of free fall is usually denoted g, and it equals (approximately):

g= 9.8 m/s 2 (meters per second per second)

That is, if you drop an object from rest, for every second of falling its speed will increase by 9.8 meters per second. At the end of the first second of the fall, the body will move at a speed of 9.8 m/s, at the end of the second - at a speed of 2 × 9.8 = 18.6 m/s, and so on. Magnitude g determines the acceleration coefficient of the fall of a body located in close proximity to earth's surface, in connection with which g usually called acceleration of free fall, or gravitational acceleration.

Here two important remarks should be made regarding the results obtained by Galileo. Firstly, the scientist obtained a purely experimental value of the quantity g, not based on any theoretical predictions. Much later, Isaac Newton, in his famous works, showed that the quantity g can be calculated theoretically based on a combination of Newton's laws of mechanics and Newton's law of universal gravitation that he formulated. It was Galileo's pioneering work that paved the way for Newton's subsequent triumphant discoveries and the formation of classical mechanics in its generally known form.

Second the most important moment is that the acceleration of gravity does not depend on the mass of the falling body. Essentially, the force of gravity is proportional to the mass of the body, but this is completely compensated by the greater inertia inherent in the more massive body (its reluctance to move, if you like), and therefore (if you do not take into account air resistance) all bodies fall with the same acceleration. This practical conclusion came into complete contradiction with the speculative predictions of ancient and medieval natural philosophers, who were confident that every thing tends to strive towards the center of the universe (which, naturally, seemed to them to be the center of the Earth) and that the more massive the object, the faster it moves towards this rushes to the center.

Galileo, of course, supported his vision with experimental data, but he most likely did not conduct the experiment that is traditionally attributed to him. According to pseudo-scientific folklore, he dropped objects of various masses from the “falling” Leaning Tower of Pisa to demonstrate that they reached the surface of the earth at the same time. In this case, however, Galileo would be disappointed, since heavier objects would inevitably fall to the ground before lighter ones due to the difference in air resistivity. If the objects dropped from the tower were the same size, the force of air resistance that slows down their fall would be the same for all objects. Moreover, from Newton’s laws it follows that lighter objects would be decelerated by air more intensely than heavy ones and would fall to the ground later than heavy objects. And this, naturally, would contradict Galileo's prediction.