Find a point symmetrical to a point online. The simplest problems with a straight line on a plane
Mechanics
Kinematic 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 motion of a body (point), you need to know its coordinates at any time. To determine the 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 point in time, it is also necessary to set the origin of the time reference.
The coordinate system, the body of reference with which it is associated, and the device for measuring time form reference system, relative to which the motion of the body is considered.
Material point
A body whose dimensions can be neglected under given conditions of motion is called material point.
A body can be considered as 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 length of the trajectory is called the way we have traveled. Path is a scalar physical quantity that can only be positive.
moving is called a vector connecting the start and end points of the trajectory.
The movement of a body, in which all its points at a given moment in time move in the same way, is called progressive movement. To describe the translational motion of a body, it is sufficient to select one point and describe its motion.
A movement in which the trajectories of all points of the body are circles with centers on one straight line and all the planes of the circles are perpendicular to this straight line is called rotational movement.
Meter and second
To determine the coordinates of a body, it is necessary to be able to measure the distance on a straight line between two points. Any process of measuring a physical quantity consists in 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 approximately 1/40,000,000 of the earth's meridian. By modern idea A meter is the distance that light travels in empty space in 1/299,792,458 of a second.
To measure time, some periodically repeating process is selected. The unit of time in SI is accepted second. A second is equal to 9,192,631,770 periods of radiation of 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.
Instant Speed
To quantitatively characterize the process of body movement, the concept of speed of movement is introduced.
instantaneous speed translational motion of the 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 velocity of movement is always directed tangentially to the trajectory in the direction of body motion.
The unit of speed is 1 m/s. Meter per second equal to speed A point moving in a straight line and uniformly at which the point moves a distance of 1 m in a time of 1 s.
Acceleration
acceleration is called a vector physical quantity equal to the ratio of a very small change in the velocity vector to a small period of time during which this change occurred, i.e. is a measure of the rate of change of speed:
A meter per second per second is such an acceleration at which the speed of a body moving in a straight line and uniformly accelerated changes by 1 m / s in a time of 1 s.
The direction of the acceleration vector coincides with the direction of the velocity change vector () for very small values of the time interval during which the velocity changes.
If the 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 curvilinear trajectory, the direction of the velocity vector changes in the process of movement, and the acceleration vector can be directed at any angle to the velocity vector.
Uniform, uniformly accelerated rectilinear motion
Moving at a constant speed is called uniform rectilinear motion. With uniform rectilinear motion a body moves in a straight line and travels the same distance in equal intervals of time.
A movement in which a body makes unequal movements in equal intervals of time is called uneven movement. With such a movement, the speed of the body changes with time.
equivariable is called such a movement in which the speed of the body for any equal time intervals changes by the same amount, i.e. movement with constant acceleration.
uniformly accelerated called uniformly variable motion, in which the magnitude of the speed increases. equally slow- uniformly variable motion, in which the magnitude of the speed decreases.
The most important characteristic in the movement of the body is its speed. Knowing it, as well as some other parameters, we can always determine the time of movement, the distance traveled, the initial, final speed and acceleration. Uniformly accelerated motion is only one of the types of motion. Usually it is found in physics problems from the kinematics section. In such problems, the body is taken as a material point, which greatly 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 s are not scalar, but vector. And this means that when solving a certain kind of problems, it is necessary to pay attention to what kind of acceleration the body has in terms of sign, and also what is the vector of the body's velocity itself. In general, in problems of an exclusively mathematical plan, 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 slow motion. Uniformly accelerated motion is characterized, as is known, by acceleration of the body. The acceleration remains constant, but the speed is continuously increasing at every single instant of time. 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 often encountered in the problems of the first part of physics tests.
An example of uniformly accelerated motion
We meet uniformly accelerated movement every day everywhere. No car moves uniformly in real life. Even if the speedometer needle shows exactly 6 kilometers per hour, it should be understood that this is actually not entirely true. Firstly, if we analyze this issue from a technical point of view, then the device will become the first parameter that will give an inaccuracy. Or rather, his error.
We meet them in all instrumentation. The same lines. Take ten pieces of at least the same (15 centimeters each, for example) rulers, at least different (15, 30, 45, 50 centimeters). Put them next to each other and you will notice that there are small inaccuracies, and their scales do not quite match. This is the error. In this case, it will be equal to half the division price, as with other devices that give certain values.
The second factor that will give inaccuracy is the scale of the instrument. The speedometer does not take into account such values as half a kilometer, one second of a 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 on such a small scale, but still. Thus, it will be uniformly accelerated motion, not uniform. The same can be said about the normal step. We are walking, let's say we are on foot, and someone says: our speed is 5 kilometers per hour. But this is not entirely true, and why, it was told a little higher.
body acceleration
Acceleration can be positive or negative. This was discussed earlier. We 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 the acceleration in this scenario 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 cannot really be negative. But very often they forget that the speed can take negative values. It's a vector quantity, don't forget about it! It's 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, simple enough. Protozoa logical thinking or the ability to see between the lines will already be, in fact, part of the solution to the physics problem associated with speed and acceleration. special case is the free fall acceleration and cannot be negative.
Formulas. Problem solving
It should be understood that the tasks associated with 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 task
Very often in physics exams in grades 9 and 11 you can meet 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 may be very different, but the answer is still the same. Here, the first thing to do is to use superficial buildings and ordinary logical thinking.
There are 4 answers to choose from. First: “the speed will be equal to 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 equal to zero.” Fourth: "speed can have any value, but at each moment of time it will be constant."
The correct answer here is, of course, the fourth one. Now let's see why this is so. Let's try to consider all the options in turn. As you know, the sum of all forces acting on a body is the product of mass and acceleration. But the mass remains a constant value for us, 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 other. Let the speed of the body decrease over a certain period of time. But how can it decrease if the acceleration is constant and it is equal to zero? There are no reasons and prerequisites for a decrease or increase in speed. Therefore, we reject the second option.
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 is impossible to say unequivocally that the speed will be different from zero. But the fourth option - right in the bull's-eye. The speed can be anything, but since there is no acceleration, it will be constant in time.
Practical task
Determine which path was traveled by the body in certain period time t1-t2 (t1 = 0 seconds, t2 = 2 seconds) if the following data is 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 catch its essence. So, you need to find a way. Well, let's start looking for it, after selecting two areas. As it is easy to see, the first part of the path (from 0 to 1 second) the body passes uniformly accelerated, as evidenced by the increase in its speed. Then we 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: 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. So the distance will be equal to V*t = 2*1 = 2 meters. Now summing the distances, we get 3 meters. This is the answer.
Formulas for the rectilinear motion of a material point are derived for three ways of specifying the motion - with a known dependence of the coordinate on time; with a known dependence of acceleration on time and acceleration on coordinates. Rectilinear uniform and rectilinear uniformly accelerated motions are considered.
ContentBasic formulas for rectilinear motion
Let the material point move along the axis. Further, and denote the coordinate and velocity of the point at the initial moment of time .
If the law of change of its coordinates from time is given:
,
then differentiating the coordinate with respect to time, we obtain the speed and acceleration of the point:
;
.
Let us the dependence of acceleration on time is known:
.
Then the dependences of the speed and coordinates on time are determined by the formulas:
(1)
;
(2)
;
(3)
;
(4)
.
Let us the dependence of the acceleration on the coordinate is known:
.
Then the dependence of the velocity on the coordinate has the form:
(5)
.
The dependence of the coordinate on time is defined implicitly:
(6)
.
For rectilinear uniform motion:
;
;
.
For rectilinear uniformly accelerated motion:
;
;
;
.
The formulas given here can be applied not only to rectilinear motion, but also for some cases curvilinear motion . For example, for three-dimensional movement in a rectangular coordinate system, if the movement along the axis does not depend on the projections of quantities on other coordinate axes. Then formulas (1) - (6) give dependencies for the projections of quantities onto the axis .
Also, these formulas are applicable when moving along a given trajectory with a natural way of setting motion. Only here, the length of the arc of the trajectory, measured from the selected reference point, acts as a coordinate. Then instead of the projections and one should substitute and - the projections of the velocity and acceleration on the chosen direction of the tangent to the trajectory.
Rectilinear motion with a known dependence of the coordinate on time
Consider the case when a material point moves in a straight line. We choose a coordinate system with the origin at an arbitrary point . Let's direct the axis along the line of movement of the point. Then the position of the point is uniquely determined by the value of one coordinate .
If the law of coordinate change from time is given:
,
then differentiating with respect to time , we find the law of speed change:
.
When the point moves in the positive direction of the axis (left to right in the figure). When the point moves in the negative direction of the axis (right to left in the figure).
Differentiating the speed with respect to time, we find the law of change of acceleration:
.
Since the straight line has no curvature, the radius of curvature of the trajectory can be considered infinitely large, . Then the normal acceleration is zero:
.
That is, the acceleration of the point is tangential (tangential):
.
Which is quite natural, since both the speed and acceleration of the point are directed tangentially to the trajectory - the straight line along which the movement occurs.
If and of the same sign (that is, both are positive or both are negative), then the speed modulus increases (the speed increases in absolute value). If and of different signs, then the speed modulus decreases (the speed decreases in absolute value).
Rectilinear motion with known acceleration
Time Dependent Acceleration
Let us know the law of change of acceleration with time:
.
Our task is to find the law of change of speed and the law of change of coordinates from time:
;
.
Let's apply the formula:
.
This is a first-order differential equation with separable variables
;
.
Here is the constant of integration. This shows that only by the known dependence of acceleration on time, it is impossible to unambiguously determine the dependence of speed on time. We have obtained a whole set of speed change laws that differ from each other by an arbitrary constant . To find the law of change of speed we need, we must specify one more value. As a rule, this value is the value of the speed at the initial moment of time . To do this, let's go from indefinite integral to a particular one:
.
Let be the speed of the point at the initial moment of time . Substitute :
;
;
.
Thus, the law of change of speed with time has the form:
(1)
.
Similarly, we define the law of change of coordinates from time.
.
(2)
.
Here - the value of the coordinate at the initial moment of time .
We substitute (1) into (2).
.
Domain of integration in the double integral.
If we change the order of integration in the double integral, we get:
.
Thus, we have received the following formulas:
(3)
;
(4)
.
Coordinate dependent acceleration
Let now we know the law of change of acceleration from the coordinate:
.
We need to decide differential equation:
.
This differential equation does not explicitly contain the independent variable. General Method solutions of such equations are considered on the page “Differential equations of higher orders that do not contain an independent variable in an explicit form”. According to this method, we consider that is a function of :
;
.
We separate the variables and integrate:
;
;
;
.
When extracting the root, one must take into account that the speed can be both positive and negative. At a small distance from the point , the sign is determined by the sign of the constant . However, if the acceleration is directed opposite to the speed, then the speed of the point will decrease to zero and the direction of motion will change to the opposite. Therefore, the correct sign, plus or minus, is chosen when considering a particular movement.
(5)
.
At the beginning of the movement
.
Now we determine the dependence of the coordinate on time. The differential equation for the coordinate is:
.
This is a separable differential equation. We separate the variables and integrate:
(6)
.
This equation defines the dependence of the coordinate on time in an implicit form.
Rectilinear uniform motion
Let us apply the results obtained above for the case of rectilinear uniform motion. In this case, the acceleration
.
;
. That is, the speed is constant, and the coordinate depends linearly on time. Formulas (5) and (6) give the same result.
Rectilinear uniformly accelerated motion
Now consider rectilinear uniformly accelerated motion.
In this case, the acceleration is a constant value:
.
According to formulas (1) and (2) we find:
;
.
If we apply formula (5), then we obtain the dependence of the velocity on the coordinate:
.
Rectilinear motion in vector form
The resulting formulas can be represented in vector form. To do this, it suffices to multiply the equations that determine and by the unit vector (ort) directed along the axis.
Then the radius vector of the point, the velocity and acceleration vectors have the form:
;
;
.
Equivalent movement. Velocity and displacement equations for uniform motion. Graphical representation uniform movement.
Short answer
uniformly accelerated or uniform motion.
Designations:
Initial body speed
body acceleration
body movement time
S(t) - change in displacement (path) over time
a(t) - change in acceleration with time
Dependence of acceleration on time. Acceleration does not change with time, has a constant value, graph a(t) is a straight line parallel to the time axis.
Speed versus time. With uniform motion, the speed changes, according to a linear relationship. The graph is a sloping line.
The rule for determining the path according to the schedule v(t): The path of the body is the area of the triangle (or trapezoid) under the velocity graph.
The rule for determining the acceleration according to the schedule v(t): The acceleration of the body is the tangent of the slope of the graph to the time axis. If the body slows down, the acceleration is negative, the angle of the graph is obtuse, so we find the tangent of the adjacent angle.
Path versus time. With uniformly accelerated motion, the path changes, according to a square dependence. In coordinates, the dependence has the form . The graph is a branch of a parabola.
A movement in which a body makes unequal movements in equal intervals of time is called uneven or variable motion.
To characterize uneven motion, the concept of average speed is introduced:
The average speed of movement is equal to the ratio of the entire path traveled by a material point to the time interval for which this path has been traveled.
In physics, the greatest interest is not the average, but instantaneous speed , which is defined as the limit to which the average speed tends over an infinitesimal time interval Δ t:
instantaneous speedvariable motion is called the speed of the body at a given time or at a given point in the trajectory.
The instantaneous velocity of the body at any point of the curvilinear trajectory is directed tangentially to the trajectory at that point.
The movement of a body, in which its speed for any equal intervals of time changes in the same way, is calleduniformly accelerated or uniform motion.
Speed at uniformly accelerated motion in a straight line - is the initial speed of the body plus the acceleration of the given body multiplied by the travel time
Moving with uniformly accelerated motion in a straight line- this is the distance traveled by the body in a straight line (the distance between the start and end points of movement)
Designations:
Movement of a body with uniformly accelerated motion in a straight line
Initial body speed
The speed of a body in uniformly accelerated motion in a straight line
body acceleration
body movement time
More formulas for finding displacement during uniformly accelerated rectilinear motion, which can be used in solving problems:
- if the initial, final speeds of movement and acceleration are known.
- if the initial, final speeds of movement and the time of the entire movement are known
Graphical representation of non-uniform rectilinear motion
Mechanical movement represent graphically. The dependence of physical quantities is expressed using functions. Designate:
(t) - change in speed with time