Determination of kinematic characteristics of movement using graphs. Lesson topic: “Graphical representation of movement

Lesson on the topic: "The speed of a straight line uniformly accelerated

movements. Speed ​​graphs."

Learning Objective : introduce a formula for determining the instantaneous speed of a body at any time, continue to develop the ability to build graphs of the dependence of the projection of speed on time, calculate the instantaneous speed of a body at any time, improve students’ ability to solve problems using analytical and graphical methods.

Developmental goal : development of theoretical, creative thinking in schoolchildren, formation of operational thinking aimed at choosing optimal solutions

Motivational goal : awakening interest in the study of physics and computer science

During the classes.

1.Organizational moment .

Teacher: - Hello, guys. Today in the lesson we will study the topic “Speed”, we will repeat the topic “Acceleration”, in the lesson we will learn the formula for determining the instantaneous speed of a body at any moment in time, we will continue to develop the ability to build graphs of the dependence of the projection of velocity on time , calculate the instantaneous speed of a body at any moment in time, we will improve the ability to solve problems using analytical and graphical methods. I am glad to see you healthy in class. Don’t be surprised that I started our lesson with this: the health of each of you is the most important thing for me and other teachers. What do you think can be common between our health and the topic “Speed”?( slide)

Students express their opinions on this issue.

Teacher: - Knowledge on this topic can help predict the occurrence of situations that are dangerous to human life, for example, those that arise during road traffic, etc.

2. Updating knowledge.

The topic “Acceleration” is repeated in the form of students’ answers to the following questions:

1.what is acceleration (slide);

2.formula and units of acceleration (slide);

3. uniformly alternating movement (slide);

4.acceleration graphs (slide);

5. Compose a problem using the material you have studied.

6. The laws or definitions given below have a number of inaccuracies. Give the correct wording.

The movement of the body is calledline segment , connecting the initial and final position of the body.

Speed ​​of uniform rectilinear motion -this is the way traversed by the body per unit time.

Mechanical movement of a body is a change in its position in space.

Rectilinear uniform motion is a motion in which a body travels equal distances in equal intervals of time.

Acceleration is a quantity numerically equal to the ratio of speed to time.

A body that has small dimensions is called a material point.

The main task of mechanics is to know the position of the body

Short-term independent work on cards - 7 minutes.

Red card – score “5”; blue card – score “4”; green card – score “3”

.TO 1

1.what motion is called uniformly accelerated?

2. Write down the formula to determine the projection of the acceleration vector.

3. The acceleration of the body is 5 m/s 2, what does this mean?

4. The parachutist’s descent speed after opening the parachute decreased from 60 m/s to 5 m/s in 1.1 s. Find the skydiver's acceleration.

1.What is acceleration called?

3. The acceleration of the body is 3 m/s 2. What does this mean?

4. With what acceleration is the car moving if in 10 s its speed increased from 5 m/s to 10 m/s

1.What is acceleration called?

2. What are the units of measurement for acceleration?

3.Write down the formula to determine the projection of the acceleration vector.

4. 3. The acceleration of the body is 2 m/s 2, what does this mean?

3.Learning new material .

1. Derivation of the speed formula from the acceleration formula. At the blackboard, under the guidance of the teacher, the student writes the derivation of the formula

2.Graphic representation of movement.

The presentation slide looks at speed graphs

.

4. Solving problems on this topic using GI materials A

Presentation slides.

1. Using a graph of the speed of a body’s movement versus time, determine the speed of the body at the end of the 5th second, assuming that the nature of the body’s movement does not change.

9 m/s

10 m/s

12 m/s

14 m/s

2.According to the graph of the dependence of the speed of movement of the body on time. Find the speed of the body at the moment of timet = 4 s.

3. The figure shows a graph of the speed of movement of a material point versus time. Determine the speed of the body at the moment of timet = 12 s, assuming that the nature of the body’s movement does not change.

4. The figure shows a graph of the speed of a certain body. Determine the speed of the body at the moment of timet = 2 s.

5. The figure shows a graph of the projection of the truck’s speed onto the axleXfrom timemehneither. The projection of the truck's acceleration onto this axis at the momentt =3 sequal to

6.The body begins linear motion from a state of rest, and its acceleration changes with time as shown in the graph. 6 s after the start of movement, the modulus of the body’s velocity will be equal to

7. The motorcyclist and cyclist simultaneously begin uniformly accelerated motion. The acceleration of a motorcyclist is 3 times greater than that of a cyclist. At the same moment in time, the speed of the motorcyclist is greater than the speed of the cyclist

1) 1.5 times

2) √3 times

3) 3 times

5. Lesson summary. (Reflection on this topic.)

What was particularly memorable and impressive from the educational material.

6.Homework.

7. Grades for the lesson.

Graphic representation of uniformly accelerated linear motion.

Moving during uniformly accelerated motion.

Ilevel.

Many physical quantities that describe the movements of bodies change over time. Therefore, for greater clarity of description, movement is often depicted graphically.

Let us show how the time dependences of kinematic quantities describing rectilinear uniformly accelerated motion are graphically depicted.

Uniformly accelerated linear motion- this is a movement in which the speed of a body changes equally over any equal periods of time, i.e. it is a movement with acceleration constant in magnitude and direction.

a=const - acceleration equation. That is, a has a numerical value that does not change over time.

By definition of acceleration

From here we have already found equations for the dependence of speed on time: v = v0 + at.

Let's see how this equation can be used to graphically represent uniformly accelerated motion.

Let us graphically depict the dependences of kinematic quantities on time for three bodies

.

1, the body moves along the 0X axis, while increasing its speed (acceleration vector a is codirectional with the velocity vector v). vx >0, akh > 0

2, the body moves along the 0X axis, while reducing its speed (the acceleration vector a is not codirectional with the velocity vector v). vx >0, ah< 0

2, the body moves against the 0X axis, while reducing its speed (the acceleration vector is not codirectional with the velocity vector v). vx< 0, ах > 0

Acceleration graph

Acceleration, by definition, is a constant value. Then, for the presented situation, the graph of acceleration versus time a(t) will look like:

From the acceleration graph, you can determine how the speed changed - increased or decreased and by what numerical value the speed changed and which body the speed changed more.

Speed ​​graph

If we compare the dependence of the coordinate on time during uniform motion and the dependence of the velocity projection on time during uniformly accelerated motion, we can see that these dependencies are the same:

x= x0 + vx t vx = v 0 x + a X t

This means that the dependency graphs have the same appearance.

To construct this graph, the time of movement is plotted on the abscissa axis, and the speed (projection of speed) of the body is plotted on the ordinate axis. In uniformly accelerated motion, the speed of a body changes over time.

Moving during uniformly accelerated motion.

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

vx = v 0 x + a X t

In this formula, υ0 is the speed of the body at t = 0 (starting speed ), a= const – acceleration. On the speed graph υ ( t) this dependence looks like a straight line (Fig.).

Acceleration can be determined from the slope of the velocity graph a bodies. The corresponding constructions are shown in Fig. for graph I. Acceleration is numerically equal to the ratio of the sides of the triangle ABC: MsoNormalTable">

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

For graph I: υ0 = –2 m/s, a= 1/2 m/s2.

For graph II: υ0 = 3 m/s, a= –1/3 m/s2.

The velocity graph also allows you to determine the projection of movement s bodies for some time t. Let us select on the time axis a certain small period of time Δ t. If this period of time is small enough, then the change in speed over this period is small, i.e. the movement during this period of time 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, the displacement Δ s in time Δ t will be equal to Δ s = υΔ t. This movement is equal to the area of ​​the shaded strip (Fig.). Breaking down the time period from 0 to some point t for small intervals Δ t, we find that the movement s for a given time t with uniformly accelerated rectilinear motion is equal to the area of ​​the trapezoid ODEF. The corresponding constructions were made for graph II in Fig. 1.4.2. Time t taken equal to 5.5 s.

Since υ – υ0 = at s t will be written in the form:

To find the coordinates y bodies at any time t needed to the starting coordinate y 0 add movement in time t: DIV_ADBLOCK189">

Since υ – υ0 = at, the final formula for moving s body with uniformly accelerated motion over a time interval from 0 to t will be written in the form: https://pandia.ru/text/78/516/images/image009_57.gif" width="146 height=55" height="55">

When analyzing uniformly accelerated motion, sometimes the problem arises of determining the movement of a body based on the given values ​​of the initial υ0 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 in the form

If the initial speed υ0 is zero, these formulas take the form MsoNormalTable">

It should be noted once again that the quantities υ0, υ, included in the formulas for uniformly accelerated rectilinear motion s, a, y 0 are algebraic quantities. Depending on the specific type of movement, each of these quantities can take on both positive and negative values.

An example of solving a problem:

Petya slides down the mountainside from a state of rest with an acceleration of 0.5 m/s2 in 20 s and then moves along a horizontal section. Having traveled 40 m, he crashes into the gaping Vasya and falls into a snowdrift, reducing his speed to 0 m/s. With what acceleration did Petya move along the horizontal surface to the snowdrift? What is the length of the mountain slope from which Petya so unsuccessfully slid down?

Stage 1.

The equation for Petit's speed at the end of the descent from the mountain is:

v 1 = v 01 + a 1t 1.

In projections onto the axis X we get:

v 1x = a 1xt.

Let us write an equation connecting the projections of Petya’s velocity, acceleration and displacement at the first stage of movement:

or because Petya was driving from the very top of the hill with an initial speed of V01=0

(If I were Petya, I would be careful about driving down such high hills)

Considering that Petya’s initial speed at this 2nd stage of movement is equal to his final speed at the first stage:

v 02 x = v 1 x, v 2x = 0, where v1 is the speed with which Petya reached the foot of the hill and began to move towards Vasya. V2x - Petya's speed in a snowdrift.

2. Using this acceleration graph, tell us how the speed of the body changes. Write down the equations for the dependence of speed on time if at the moment of the start of movement (t=0) the speed of the body is v0х =0. Please note that with each subsequent section of movement, the body begins to pass at a certain speed (which was achieved during the previous time!).

3. A metro train, leaving the station, can reach a speed of 72 km/h in 20 s. Determine with what acceleration a bag, forgotten in a subway car, is moving away from you. How far will she travel?

4. A cyclist moving at a speed of 3 m/s begins to descend a mountain with an acceleration of 0.8 m/s2. Find the length of the mountain if the descent took 6 s.

5. Having started braking with an acceleration of 0.5 m/s2, the train traveled 225 m to the stop. What was its speed before braking began?

6. Having started to move, the soccer ball reached a speed of 50 m/s, covered a distance of 50 m and crashed into the window. Determine the time it took the ball to travel this path and the acceleration with which it moved.

7. Reaction time of Uncle Oleg’s neighbor = 1.5 minutes, during which time he will figure out what happened to his window and will have time to run out into the yard. Determine what speed young football players should develop so that the joyful owners of the window do not catch up with them, if they need to run 350 m to their entrance.

8. Two cyclists are riding towards each other. The first one, having a speed of 36 km/h, began to climb up the mountain with an acceleration of 0.2 m/s2, and the second, having a speed of 9 km/h, began to descend the mountain with an acceleration of 0.2 m/s2. After how long and in what place will they collide due to their absent-mindedness, if the length of the mountain is 100 m?

Uniform movement– this is movement at a constant speed, that is, when the speed does not change (v = const) and acceleration or deceleration does not occur (a = 0).

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

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

The speed of uniform rectilinear motion does not depend on time and at each point of the trajectory is directed in the same way as the movement of the body. That is, the displacement vector coincides in direction with the velocity vector. In this case, the average speed for any period of time is equal to the instantaneous speed:

Speed ​​of uniform rectilinear motion is a physical vector quantity equal to the ratio of the movement of a body over any period of time to the value of this interval t:

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

Moving with uniform linear motion is determined by the formula:

Distance traveled in linear motion is equal to the displacement module. If the positive direction of the OX axis coincides with the direction of movement, then the projection of the velocity onto the OX axis is equal to the magnitude of the velocity and is positive:

v x = v, that is v > 0

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

s = vt = x – x 0

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

Equation of motion, that is, the dependence of the body coordinates on time x = x(t), takes the form:

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

Dependence of speed, coordinates and path on time

The dependence of the projection of the body velocity on time is shown in Fig. 1.11. Since the speed is constant (v = const), the speed graph is a straight line parallel to the time axis Ot.

Rice. 1.11. Dependence of the projection of body velocity on time for uniform rectilinear motion.

The projection of movement onto the coordinate axis is numerically equal to the area of ​​the rectangle OABC (Fig. 1.12), since the magnitude of the movement vector is equal to the product of the velocity vector and the time during which the movement was made.

Rice. 1.12. Dependence of the projection of body displacement on time for uniform rectilinear motion.

A graph of displacement versus time is shown in Fig. 1.13. The graph shows that the projection of the velocity is equal to

v = s 1 / t 1 = tan α

where α is the angle of inclination of the graph to the time axis.

The larger the angle α, the faster the body moves, that is, the greater its speed (the longer the distance the body travels in less time). The tangent of the tangent to the graph of the coordinate versus time is equal to the speed:

Rice. 1.13. Dependence of the projection of body displacement on time for uniform rectilinear motion.

The dependence of the coordinate on time is shown in Fig. 1.14. From the figure it is clear that

tan α 1 > tan α 2

therefore, the speed of body 1 is higher than the speed of body 2 (v 1 > v 2).

tan α 3 = v 3< 0

If the body is at rest, then the coordinate graph is a straight line parallel to the time axis, that is

Rice. 1.14. Dependence of body coordinates on time for uniform rectilinear motion.

Relationship between angular and linear quantities

Individual points of a rotating body have different linear velocities. The speed of each point, being directed tangentially to the corresponding circle, continuously changes its direction. The magnitude of the speed is determined by the speed of rotation of the body and the distance R of the point in question from the axis of rotation. Let the body turn through an angle in a short period of time (Figure 2.4). A point located at a distance R from the axis travels a path equal to

Linear speed of a point by definition.

Tangential acceleration

Using the same relation (2.6) we obtain

Thus, both normal and tangential accelerations increase linearly with the distance of the point from the axis of rotation.

Basic concepts.

Periodic oscillation is a process in which a system (for example, a mechanical one) returns to the same state after a certain period of time. This period of time is called the oscillation period.

restoring force- the force under the influence of which the oscillatory process occurs. This force tends to return a body or a material point, deviated from its position of rest, to its original position.

Depending on the nature of the impact on the oscillating body, a distinction is made between free (or natural) vibrations and forced vibrations.

Free vibrations occur when only a restoring force acts on the oscillating body. In the event that no energy dissipation occurs, free oscillations are undamped. However, real oscillatory processes are damped, because the oscillating body is subject to motion resistance forces (mainly friction forces).

Forced vibrations are performed under the influence of an external periodically changing force, which is called forcing. In many cases, systems undergo oscillations that can be considered harmonic.

Harmonic vibrations are called oscillatory movements in which the displacement of a body from the equilibrium position occurs according to the law of sine or cosine:

To illustrate the physical meaning, consider a circle and rotate the radius OK with angular velocity ω counterclockwise (7.1) counterclockwise. If at the initial moment of time the OK lay in the horizontal plane, then after time t it will shift by an angle. If the starting angle is non-zero and equal to φ 0 , then the angle of rotation will be equal to The projection onto the XO 1 axis is equal to . As the radius OK rotates, the magnitude of the projection changes, and the point will oscillate relative to the point - up, down, etc. In this case, the maximum value of x is equal to A and is called the amplitude of oscillations; ω - circular or cyclic frequency; - oscillation phase; – initial phase. For one revolution of point K around the circle, its projection will make one complete oscillation and return to the starting point.

Period T is called the time of one complete oscillation. After time T, the values ​​of all physical quantities characterizing the oscillations are repeated. In one period, the oscillating point travels a path numerically equal to four amplitudes.

Angular velocity is determined from the condition that during the period T the radius OK will make one revolution, i.e. will rotate by an angle of 2π radians:

Oscillation frequency- the number of oscillations of a point per second, i.e. the oscillation frequency is defined as the reciprocal of the oscillation period:

Spring pendulum elastic forces.

A spring pendulum consists of a spring and a massive ball mounted on a horizontal rod along which it can slide. Let a ball with a hole be attached to a spring and slide along a guide axis (rod). In Fig. 7.2a shows the position of the ball at rest; in Fig. 7.2, b - maximum compression and in Fig. 7.2,c - arbitrary position of the ball.

Under the influence of a restoring force equal to the compression force, the ball will oscillate. Compression force F = -kx, where k is the spring stiffness coefficient. The minus sign indicates that the direction of the force F and the displacement x are opposite. Potential energy of a compressed spring

kinetic

To derive the equation of motion of the ball, it is necessary to relate x and t. The conclusion is based on the law of conservation of energy. The total mechanical energy is equal to the sum of the kinetic and potential energy of the system. In this case:

. In position b): .

Since the law of conservation of mechanical energy is satisfied in the movement under consideration, we can write:

. Let's determine the speed from here:

But in turn and therefore . Let's separate the variables . Integrating this expression, we get: ,

where is the integration constant. From the latter it follows that

Thus, under the action of elastic force, the body performs harmonic oscillations. Forces of a different nature than elastic, but in which the condition F = -kx is satisfied, are called quasi-elastic. Under the influence of these forces, bodies also perform harmonic vibrations. Wherein:

Mathematical pendulum.

A mathematical pendulum is a material point suspended on an inextensible weightless thread, performing oscillatory motion in one vertical plane under the influence of gravity.

Such a pendulum can be considered a heavy ball of mass m, suspended on a thin thread, the length l of which is much greater than the size of the ball. If it is deflected by an angle α (Fig. 7.3.) from the vertical line, then under the influence of force F, one of the components of weight P, it will oscillate. The other component, directed along the thread, is not taken into account, because is balanced by the tension of the thread. At small displacement angles, then the x coordinate can be measured in the horizontal direction. From Fig. 7.3 it is clear that the weight component perpendicular to the thread is equal to

The minus sign on the right side means that the force F is directed towards decreasing angle α. Taking into account the smallness of the angle α

To derive the law of motion of mathematical and physical pendulums, we use the basic equation of the dynamics of rotational motion

Moment of force relative to point O: , and moment of inertia: M=FL. Moment of inertia J in this case Angular acceleration:

Taking these values ​​into account, we have:

His decision ,

As we can see, the period of oscillation of a mathematical pendulum depends on its length and the acceleration of gravity and does not depend on the amplitude of the oscillations.

Damped oscillations.

All real oscillatory systems are dissipative. The energy of mechanical vibrations of such a system is gradually spent on work against frictional forces, therefore free vibrations always fade - their amplitude gradually decreases. In many cases, when there is no dry friction, as a first approximation we can assume that at low speeds of movement the forces causing attenuation of mechanical vibrations are proportional to the speed. These forces, regardless of their origin, are called resistance forces.

Let's rewrite this equation as follows:

and denote:

where represents the frequency with which free oscillations of the system would occur in the absence of environmental resistance, i.e. at r = 0. This frequency is called the natural frequency of oscillation of the system; β is the attenuation coefficient. Then

We will look for a solution to equation (7.19) in the form where U is some function of t.

Let us differentiate this expression twice with respect to time t and, substituting the values ​​of the first and second derivatives into equation (7.19), we obtain

The solution to this equation significantly depends on the sign of the coefficient at U. Let us consider the case when this coefficient is positive. Let us introduce the notation then With a real ω, the solution to this equation, as we know, is the function

Thus, in the case of low resistance of the medium, the solution to equation (7.19) will be the function

The graph of this function is shown in Fig. 7.8. The dotted lines show the limits within which the displacement of the oscillating point lies. The quantity is called the natural cyclic frequency of oscillations of the dissipative system. Damped oscillations are non-periodic oscillations, because they never repeat, for example, the maximum values ​​of displacement, speed and acceleration. The quantity is usually called the period of damped oscillations, or more correctly, the conditional period of damped oscillations,

The natural logarithm of the ratio of displacement amplitudes following each other through a time interval equal to the period T is called the logarithmic attenuation decrement.

Let us denote by τ the period of time during which the amplitude of the oscillations decreases by e times. Then

Consequently, the attenuation coefficient is a physical quantity inverse to the time period τ during which the amplitude decreases by a factor of e. The quantity τ is called the relaxation time.

Let N be the number of oscillations after which the amplitude decreases by a factor of e, Then

Consequently, the logarithmic damping decrement δ is a physical quantity reciprocal to the number of oscillations N, after which the amplitude decreases by a factor of e

Forced vibrations.

In the case of forced oscillations, the system oscillates under the influence of an external (forcing) force, and due to the work of this force, the energy losses of the system are periodically compensated. The frequency of forced oscillations (forcing frequency) depends on the frequency of change of the external force. Let us determine the amplitude of forced oscillations of a body of mass m, considering the oscillations undamped due to a constantly acting force.

Let this force change with time according to the law where is the amplitude of the driving force. Restoring force and resistance force Then Newton's second law can be written in the following form.

3.1. Uniform motion in a straight line.

3.1.1. Uniform motion in a straight line- movement in a straight line with acceleration constant in magnitude and direction:

3.1.2. Acceleration()- a physical vector quantity showing how much the speed will change in 1 s.

In vector form:

where is the initial speed of the body, is the speed of the body at the moment of time t.

In projection onto the axis Ox:

where is the projection of the initial velocity onto the axis Ox, - projection of the body velocity onto the axis Ox at a point in time t.

The signs of the projections depend on the direction of the vectors and the axis Ox.

3.1.3. Projection graph of acceleration versus time.

With uniformly alternating motion, the acceleration is constant, therefore it will appear as straight lines parallel to the time axis (see figure):

3.1.4. Speed ​​during uniform motion.

In vector form:

In projection onto the axis Ox:

For uniformly accelerated motion:

For uniform slow motion:

3.1.5. Projection graph of speed versus time.

The graph of the projection of speed versus time is a straight line.

Direction of movement: if the graph (or part of it) is above the time axis, then the body is moving in the positive direction of the axis Ox.

Acceleration value: the greater the tangent of the angle of inclination (the steeper it goes up or down), the greater the acceleration module; where is the change in speed over time

Intersection with the time axis: if the graph intersects the time axis, then before the intersection point the body slowed down (uniformly slow motion), and after the intersection point it began to accelerate in the opposite direction (uniformly accelerated motion).

3.1.6. Geometric meaning of the area under the graph in the axes

Area under the graph when on the axis Oy the speed is delayed, and on the axis Ox- time is the path traveled by the body.

In Fig. 3.5 shows the case of uniformly accelerated motion. The path in this case will be equal to the area of ​​the trapezoid: (3.9)

3.1.7. Formulas for calculating path

All formulas presented in the table work only when the direction of movement is maintained, that is, until the straight line intersects with the time axis on the graph of the velocity projection versus time.

If the intersection has occurred, then the movement is easier to divide into two stages:

before crossing (braking):

After the intersection (acceleration, movement in the opposite direction)

In the formulas above - the time from the beginning of movement to the intersection with the time axis (time before stopping), - the path that the body has traveled from the beginning of movement to the intersection with the time axis, - the time elapsed from the moment of crossing the time axis to this moment t, - the path that the body has traveled in the opposite direction during the time elapsed from the moment of crossing the time axis to this moment t, - the module of the displacement vector for the entire time of movement, L- the path traveled by the body during the entire movement.

3.1.8. Movement in the th second.

During this time the body will travel the following distance:

During this time the body will travel the following distance:

Then during the th interval the body will travel the following distance:

Any period of time can be taken as an interval. Most often with.

Then in 1 second the body travels the following distance:

In 2 seconds:

In 3 seconds:

If we look carefully, we will see that, etc.

Thus, we arrive at the formula:

In words: the paths traversed by a body over successive periods of time are related to each other as a series of odd numbers, and this does not depend on the acceleration with which the body moves. We emphasize that this relation is valid for

3.1.9. Equation of body coordinates for uniform motion

Coordinate equation

The signs of the projections of the initial velocity and acceleration depend on the relative position of the corresponding vectors and the axis Ox.

To solve problems, it is necessary to add to the equation the equation for changing the velocity projection onto the axis:

3.2. Graphs of kinematic quantities for rectilinear motion

3.3. Free fall body

By free fall we mean the following physical model:

1) The fall occurs under the influence of gravity:

2) There is no air resistance (in problems they sometimes write “neglect air resistance”);

3) All bodies, regardless of mass, fall with the same acceleration (sometimes they add “regardless of the shape of the body,” but we are considering the movement of only a material point, so the shape of the body is no longer taken into account);

4) The acceleration of gravity is directed strictly downwards and is equal on the surface of the Earth (in problems we often assume for convenience of calculations);

3.3.1. Equations of motion in projection onto the axis Oy

Unlike movement along a horizontal straight line, when not all tasks involve a change in direction of movement, in free fall it is best to immediately use the equations written in projections onto the axis Oy.

Body coordinate equation:

Velocity projection equation:

As a rule, in problems it is convenient to select the axis Oy in the following way:

Axis Oy directed vertically upward;

The origin coincides with the level of the Earth or the lowest point of the trajectory.

With this choice, the equations and will be rewritten in the following form:

3.4. Movement in a plane Oxy.

We considered the motion of a body with acceleration along a straight line. However, the uniformly variable motion is not limited to this. For example, a body thrown at an angle to the horizontal. In such problems, it is necessary to take into account movement along two axes at once:

Or in vector form:

And changing the projection of speed on both axes:

3.5. Application of the concept of derivative and integral

We will not provide a detailed definition of the derivative and integral here. To solve problems we need only a small set of formulas.

Derivative:

Where A, B and that is, constant values.

Integral:

Now let's see how the concepts of derivative and integral apply to physical quantities. In mathematics, the derivative is denoted by """, in physics, the derivative with respect to time is denoted by "∙" above the function.

Speed:

that is, the speed is a derivative of the radius vector.

For velocity projection:

Acceleration:

that is, acceleration is a derivative of speed.

For acceleration projection:

Thus, if the law of motion is known, then we can easily find both the speed and acceleration of the body.

Now let's use the concept of integral.

Speed:

that is, the speed can be found as the time integral of the acceleration.

Radius vector:

that is, the radius vector can be found by taking the integral of the velocity function.

Thus, if the function is known, we can easily find both the speed and the law of motion of the body.

The constants in the formulas are determined from the initial conditions - values ​​and at the moment of time

3.6. Velocity triangle and displacement triangle

3.6.1. Speed ​​triangle

In vector form with constant acceleration, the law of speed change has the form (3.5):

This formula means that a vector is equal to the vector sum of vectors and the vector sum can always be depicted in a figure (see figure).

In each problem, depending on the conditions, the velocity triangle will have its own form. This representation allows the use of geometric considerations in the solution, which often simplifies the solution of the problem.

3.6.2. Triangle of movements

In vector form, the law of motion with constant acceleration has the form:

When solving a problem, you can choose the reference system in the most convenient way, therefore, without losing generality, we can choose the reference system in such a way that, that is, we place the origin of the coordinate system at the point where the body is located at the initial moment. Then

that is, the vector is equal to the vector sum of the vectors and Let us depict it in the figure (see figure).

As in the previous case, depending on the conditions, the displacement triangle will have its own shape. This representation allows the use of geometric considerations in the solution, which often simplifies the solution of the problem.

Questions.

1. Write down the formula by which you can calculate the projection of the instantaneous velocity vector of rectilinear uniformly accelerated motion if you know: a) the projection of the initial velocity vector and the projection of the acceleration vector; b) projection of the acceleration vector given that the initial speed is zero.

2. What is the projection graph of the velocity vector of uniformly accelerated motion at an initial speed: a) equal to zero; b) not equal to zero?

3. How are the movements, the graphs of which are presented in Figures 11 and 12, similar and different from each other?

In both cases, the movement occurs with acceleration, but in the first case the acceleration is positive, and in the second case it is negative.

Exercises.

1. A hockey player lightly hit the puck with his stick, giving it a speed of 2 m/s. What will be the speed of the puck 4 s after impact if, as a result of friction with ice, it moves with an acceleration of 0.25 m/s 2?

2. A skier slides down a mountain from a state of rest with an acceleration equal to 0.2 m/s 2 . After what period of time will its speed increase to 2 m/s?

3. In the same coordinate axes, construct graphs of the projection of the velocity vector (on the X axis, codirectional with the initial velocity vector) for rectilinear uniformly accelerated motion for the cases: a) v ox = 1 m/s, a x = 0.5 m/s 2 ; b) v ox = 1 m/s, a x = 1 m/s 2; c) v ox = 2 m/s, a x = 1 m/s 2.
The scale is the same in all cases: 1 cm - 1 m/s; 1cm - 1s.

4. In the same coordinate axes, construct graphs of the projection of the velocity vector (on the X axis, codirectional with the initial velocity vector) for rectilinear uniformly accelerated motion for the cases: a) v ox = 4.5 m/s, a x = -1.5 m/s 2 ; b) v ox = 3 m/s, a x = -1 m/s 2
Choose the scale yourself.

5. Figure 13 shows graphs of the velocity vector modulus versus time for the rectilinear motion of two bodies. With what absolute acceleration does body I move? body II?