Uniform rectilinear motion. Rectilinear uniform motion The equation of the projection of the velocity of the body

To perform calculations of velocities and accelerations, it is necessary to switch from writing equations in vector form to writing equations in algebraic form.

The vectors of the initial velocity and acceleration can have different directions, so the transition from the vector notation of equations to the algebraic one can be very laborious.

It is known that the projection of the sum of two vectors on any coordinate axis is equal to the sum of the projections of the terms of the vectors on the same axis.

Speed Graph

From the equation it follows that the graph of the projection of the velocity of uniformly accelerated motion on time is a straight line. If the projection of the initial velocity on the OX axis is equal to zero, then the straight line passes through the origin.

Main types of movement

1. a n = 0, a t = 0– rectilinear uniform motion;

2. a n = 0, a t = const- rectilinear uniform motion;

3. and n = 0, a t ¹ 0 – rectilinear with variable acceleration;

4. and n = const, a t = 0 - uniform around the circumference

5. a n = const, a t = const- uniform around the circumference

6. a n ¹ const, a t ¹ const- curvilinear with variable acceleration.

Rotational motion of a rigid body.

Rotational motion of a rigid body about a fixed axis - a movement in which all points of a rigid body describe circles, the centers of which lie on one straight line, called axis of rotation.

Uniform circular motion

Consider the simplest form rotary motion, and pay special attention to centripetal acceleration.

With uniform motion in a circle, the value of the velocity remains constant, and the direction of the velocity vector changes during the motion.

From the similarity of triangles OAB and BCD it follows

If the time interval ∆t is small, then the angle a is also small. For small values of the angle a, the length of the chord AB is approximately equal to the length of the arc AB, i.e. . Because , , then we get

Since , we get

Period and frequency

The time it takes for a body to complete one revolution in a circle is called circulation periods (T). Because the circumference is 2pR, the period of revolution with a uniform motion of a body with a speed v along a circle with a radius R equals:

The reciprocal of the period of revolution is called frequency. The frequency shows how many revolutions the body makes per unit time in a circle:

(from -1)

Kinematics of rotary motion

To indicate the direction of rotation, small angles of rotation are assigned a direction: directed along the axis of rotation so that the rotation considered from its end occurs counterclockwise (right screw rule). If the body did N turns: . Average angular speed:

Instantaneous angular velocity:

(12)

Uniform movement- this is movement at a constant speed, that is, when the speed does not change (v \u003d const) and there is no acceleration or deceleration (a \u003d 0).

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

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

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: v cp = v Speed of uniform rectilinear motion is a physical vector quantity equal to the ratio of the displacement of the body for any period of time to the value of this interval t:

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

moving with uniform rectilinear motion is determined by the formula:

Distance traveled in rectilinear motion is equal to the displacement modulus. If the positive direction of the OX axis coincides with the direction of movement, then the projection of the velocity on the OX axis is equal to the velocity and is positive:

V x \u003d v, that is, v > 0

Motion equation, that is, the dependence of the body coordinate on time x = x(t), takes the form:

X \u003d x 0 + vt

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. The dependence of the projection of the velocity of the body on time for uniform rectilinear motion.

The projection of movement onto the coordinate axis is numerically equal to the area of the OABS rectangle (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. The dependence of the projection of the movement of the body on time for uniform rectilinear motion.

The plot of displacement versus time is shown in Fig. 1.13. It can be seen from the graph that the velocity projection is equal to

V = s 1 / t 1 = tg 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 body travels in less time). The tangent of the slope of the tangent to the graph of the dependence of the coordinate on time is equal to the speed: tg α = v

Rice. 1.13. The dependence of the projection of the movement of the body on time for uniform rectilinear motion.

The dependence of the coordinate on time is shown in fig. 1.14. It can be seen from the figure that

Tg α 1 > tg α 2 therefore, the speed of body 1 is higher than the speed of body 2 (v 1 > v 2). tg α 3 \u003d v 3 If the body is at rest, then the coordinate graph is a straight line parallel to the time axis, that is, x \u003d x 0

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

Definition

Uniform rectilinear motion is motion at a constant speed, in which there is no acceleration, and the trajectory of motion is a straight line.

Definition

The speed of uniform rectilinear motion is a physical vector quantity equal to the ratio of the displacement of the body $\overrightarrow(S)$ for any period of time to the value of this interval t:

$$\overrightarrow(v)=\frac(\overrightarrow(S))(t)$$

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

Displacement with uniform rectilinear motion is determined by the formula:

$$ \overrightarrow(S) = \overrightarrow(v) \cdot t $$

The distance traveled in rectilinear motion is equal to the displacement modulus. If the positive direction of the OX axis coincides with the direction of motion, then the projection of the velocity onto the OX axis is equal to the magnitude of the velocity and is positive: $v_x = v$, i.e. $v $>$ 0$

The displacement projection onto the OX axis is: $s = v_t = x - x0$

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)

The equation of motion, that is, the dependence of the body coordinate on time $x = x(t)$, takes the form: $x = x_0 + v_t$

If the positive direction of the OX axis is opposite to the direction of motion of the body, then the projection of the body velocity on the OX axis is negative, the velocity is less than zero ($v $

The dependence of the projection of the body velocity on time is shown in fig. 1. Since the speed is constant ($v = const$), the speed graph is a straight line parallel to the time axis Ot.

Rice. 1. Dependence of the projection of the velocity of the body on time for uniform rectilinear motion.

The displacement projection onto the coordinate axis is numerically equal to the area of the OABS rectangle (Fig. 2), since the magnitude of the displacement vector is equal to the product of the velocity vector and the time during which the displacement was made.

Rice. 2. Dependence of the projection of the displacement of the body on time for uniform rectilinear motion.

The plot of displacement versus time is shown in Fig. 3. It can be seen from the graph that the velocity projection on the Ot axis is numerically equal to the tangent of the slope of the graph to the time axis:

Rice. 3. Dependence of the projection of the displacement of the body on time for uniform rectilinear motion.

The dependence of the coordinate on time is shown in fig. 4. It can be seen from the figure that

tg $\alpha $1 $>$ tg $\alpha $2, therefore, the speed of body 1 is higher than the speed of body 2 (v1 $>$ v2).

tg $\alpha $3 = v3 $

Rice. 4. Dependence of the body coordinate on time for uniform rectilinear motion.

If the body is at rest, then the coordinate graph is a straight line parallel to the time axis, that is, x = x0

Task 1

Two trains are moving towards each other on parallel rails. The speed of the first train is 10 meters per second, the length of the first train is 500 meters. The speed of the second train is 30 meters per second, the length of the second train is 300 meters. Determine how long the second train will pass the first.

Given: $v_1$=10 m/s; $v_2$=30 m/s; $L_1$=500 m; $L_2$=300 m

Find: t --- ?

The time it takes for trains to pass each other can be determined by dividing the total length of the trains by their relative speed. The speed of the first train relative to the second is determined by the formula v= v1+v2 Then the formula for determining the time becomes: $t=\frac(L_1+L_2)(v_1+v_2)=\frac(500+300)(10+30)= 20\c$

Answer: The second train will pass the first one within 20 seconds.

Task 2

Determine the speed of the river current and the speed of the boat in still water, if it is known that the boat travels a distance of 300 kilometers downstream in 4 hours, and against the current in 6 hours.

Given: $L$=300000 m; $t_1$=14400 s; $t_2$=21600 s

Find: $v_p$ - ?; $v_k$ - ?

The speed of the boat downstream relative to the bank is $v_1=v_k+v_p$, and against the current $v_2=v_k-v_p$ . We write the law of motion for both cases:

Having solved the equations for vp and vk, we obtain formulas for calculating the speed of the river and the speed of the boat.

River speed: $v_p=\frac(L\left(t_2-t_1\right))(2t_1t_2)=\frac(300000\left(21600-14400\right))(2\times 14400\times 21600)=3 .47\ m/s$

Boat speed: $v_k=\frac(L\left(t_2+t_1\right))(2t_1t_2)=\frac(300000\left(21600+14400\right))(2\times 14400\times 21600)=17, 36\m/s$

Answer: the speed of the river is 3.47 meters per second, the speed of the boat is 17.36 meters per second.

In the drawings, images of geometric bodies are built using the projection method. But for this one image is not enough, at least two projections are needed. With the help of them, points in space are determined. Therefore, you need to know how to find the projection of a point.

Point projection

To do this, you need to consider the space of a dihedral angle, with a point (A) located inside. Here horizontal P1 and vertical P2 projection planes are used. Point (A) is projected onto the projection planes orthogonally. As for the perpendicular projecting rays, they are combined into a projecting plane, perpendicular to the planes projections. Thus, when combining the horizontal P1 and frontal P2 planes by rotating along the axis P2 / P1, we get a flat drawing.

Then a line with projection points located on it is shown perpendicular to the axis. This results in a complex drawing. Thanks to the built segments on it and the vertical line of communication, it is easy to determine the position of a point relative to the projection planes.

To make it easier to understand how to find a projection, you need to consider right triangle. Its short side is the leg, and the long side is the hypotenuse. If you perform a projection of the leg on the hypotenuse, then it will be divided into two segments. To determine their value, you need to calculate a set of initial data. Consider on this triangle, methods for calculating the main projections.

As a rule, in this problem, the length of the leg N and the length of the hypotenuse D are indicated, whose projection is to be found. To do this, we learn how to find the projection of the leg.

Consider a method for finding the length of the leg (A). Considering that the geometric mean of the projection of the leg and the length of the hypotenuse is equal to the value of the leg we are looking for: N = √(D*Nd).

How to find the projection length

The root of the product can be found by squaring the length of the desired leg (N), and then dividing by the length of the hypotenuse: Nd = (N / √ D)² = N² / D. When only D and N are indicated in the source data, the length projections should be found using the Pythagorean theorem.
Find the length of the hypotenuse D. To do this, use the values of the legs √ (N² + T²), and then substitute the resulting value into the following formula for finding the projection: Nd = N² / √ (N² + T²).

When the source data contains data on the length of the projection of the leg RD, as well as data on the value of the hypotenuse D, the length of the projection of the second leg ND should be calculated using a simple subtraction formula: ND = D - RD.

Speed projection

Let's consider how to find the velocity projection. In order for a given vector to represent a description of the movement, it should be placed in the projection onto the coordinate axes. There is one coordinate axis (ray), two coordinate axes (plane) and three coordinate axes (space). When finding the projection, it is necessary to lower the perpendiculars on the axis from the ends of the vector.

In order to understand the meaning of the projection, you need to know how to find the projection of a vector.

Vector projection

When the body moves perpendicular to the axis, the projection will be represented as a point and have a value of zero. If the movement is parallel to the coordinate axis, then the projection will coincide with the module of the vector. In the case when the body moves in such a way that the velocity vector is directed at an angle φ relative to the (x) axis, the projection to this axis will be a segment: V(x) = V cos(φ), where V is the model of the velocity vector. When the directions of the velocity vector and the coordinate axis coincide, then the projection is positive, and vice versa.

Take the following coordinate equation: x = x(t), y = y(t), z = z(t). In this case, the speed function will be projected onto three axes and will look like this: V(x) = dx / dt = x"(t), V(y) = dy / dt = y"(t), V(z) \u003d dz / dt \u003d z "(t). It follows that to find the speed it is necessary to take derivatives. The velocity vector itself is expressed by an equation of this form: V \u003d V (x) i + V (y) j + V (z) k where i, j, k are the unit vectors of the coordinate axes x, y, z, respectively. Thus, the modulus of velocity is calculated by the following formula: V = √ (V(x) ^ 2 + V(y) ^ 2 + V(z ) ^ 2).

Uniform movement- this is movement at a constant speed, that is, when the speed does not change (v \u003d const) and there is no acceleration or deceleration (a \u003d 0).

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

V cp = v

V x = v, i.e. v > 0

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

S \u003d vt \u003d 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)

Motion equation, that is, the dependence of the body coordinate on time x = x(t), takes the form:

X \u003d x 0 + vt

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

X \u003d x 0 - vt

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. The dependence of the projection of the velocity of the body on time for uniform rectilinear motion.

Rice. 1.12. The dependence of the projection of the movement of the body on time for uniform rectilinear motion.

The plot of displacement versus time is shown in Fig. 1.13. It can be seen from the graph that the velocity projection is equal to

V = s 1 / t 1 = tg 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 body travels in less time). The tangent of the slope of the tangent to the graph of the dependence of the coordinate on time is equal to the speed:

Tgα = v

Rice. 1.13. The dependence of the projection of the movement of the body on time for uniform rectilinear motion.

The dependence of the coordinate on time is shown in fig. 1.14. It can be seen from the figure that

Tgα 1 >tgα 2

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

Tg α 3 = v 3< 0

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

X \u003d x 0

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