1. Methods for specifying the motion of a point in a given reference system

The main tasks of point kinematics are:

1. Description of methods for specifying the movement of a point.

2. Determination of the kinematic characteristics of the movement of a point (speed, acceleration) according to a given law of motion.

Mechanical movement − change in position of one body relative to another (reference body) with which the coordinate system called reference system .

The geometric locus of successive positions of a moving point in the reference frame under consideration is called trajectory points.

Set movement − is to provide a method by which one can determine the position of a point at any time in relation to a chosen reference system. The main ways to specify the movement of a point include:

vector, coordinate and natural .

1.Vector method of specifying movement (Fig. 1).

The position of the point is determined by the radius vector drawn from the fixed point associated with the reference body: − vector equation of motion of the point.

2. Coordinate method of specifying movement (Fig. 2).

In this case, the coordinates of the point are specified as a function of time:

- equations of motion of a point in coordinate form.

These are also parametric equations for the trajectory of a moving point, in which time plays the role of a parameter. To write its equation in explicit form, we must exclude . In the case of a spatial trajectory, excluding , we obtain:

In the case of a flat trajectory

excluding , we get:

Or .

3. The natural way to define movement (Fig. 3).

In this case, set:

1) trajectory of a point,

2) the origin of the trajectory,

3) positive direction of reference,

4) law of change of arc coordinate: .

This method is convenient to use when the trajectory of a point is known in advance.

2. Speed ​​and acceleration of a point

Consider the movement of a point over a short period of time(Fig. 4):

Then is the average speed of a point over a period of time.

The speed of a point at a given time is found as the limit of the average speed at :

Point speed − this is the kinematic measure of its movement, equal to time derivative of the radius vector of this point in the reference frame under consideration.

The velocity vector is directed tangentially to the trajectory of the point in the direction of movement.

Average acceleration characterizes the change in the velocity vector over a short period of time(Fig. 5).

The acceleration of a point at a given time is found as the limit of the average acceleration at :

Point acceleration − this is a measure of the change in its speed, equal to the derivative in time from the speed of this point or the second derivative of the radius vector of the point in time .

The acceleration of a point characterizes the change in the velocity vector in magnitude and direction. The acceleration vector is directed towards the concavity of the trajectory.

3. Determination of the speed and acceleration of a point using the coordinate method of specifying motion

The connection between the vector method of specifying motion and the coordinate method is given by the relation

(Fig. 6).

From the definition of speed:

Projections of velocity on the coordinate axes are equal to the derivatives of the corresponding coordinates with respect to time

, , . .

The magnitude and direction of velocity are determined by the expressions:

The dot above here and henceforth denotes differentiation with respect to time

From the definition of acceleration:

Acceleration projections on the coordinate axes are equal to the second derivatives of the corresponding coordinates with respect to time:

, , .

The module and direction of acceleration are determined by the expressions:

, , .

4 Speed ​​and acceleration of a point using the natural method of specifying motion

4.1 Natural axes.

Determining the speed and acceleration of a point using the natural method of specifying motion

Natural axes (tangent, principal normal, binormal) are the axes of a moving rectangular coordinate system with the origin at a moving point. Their position is determined by the trajectory of movement. The tangent (with a unit vector) is directed along the tangent in the positive direction of the arc coordinate reference and is found as the limit position of the secant passing through a given point (Fig. 9). The touching plane passes through the tangent (Fig. 10), which is located as the limiting position of the plane p as point M1 tends to point M. The normal plane is perpendicular to the tangent. The line of intersection of the normal and osculating planes is the main normal. The unit vector of the main normal is directed towards the concavity of the trajectory. The binormal (with the unit vector) is directed perpendicular to the tangent and the main normal so that the vectors , and form a right-handed triple of vectors. The coordinate planes of the introduced moving coordinate system (contiguous, normal and rectifying) form a natural trihedron, which moves together with the moving point, like a rigid body. Its movement in space is determined by the trajectory and the law of change in the arc coordinate.

From the definition of point speed

where , is the unit tangent vector.

Then

, .

Algebraic speed − projection of the velocity vector onto the tangent, equal to the derivative of the arc coordinate with respect to time. If the derivative is positive, then the point moves in the positive direction of the arc coordinate.

From the definition of acceleration

− vector variable in direction and

The derivative is determined only by the type of trajectory in the vicinity of a given point, while introducing into consideration the angle of rotation of the tangent, we have , where is the unit vector of the main normal, is the curvature of the trajectory, and is the radius of curvature of the trajectory at a given point.

Let the function now be known. In Fig. 5.10
And
 velocity vectors of a moving point at moments t and  t. To get the velocity vector increment
move the vector parallel
exactly M:

Average acceleration of a point over a period of time  t is called the velocity vector increment ratio
to a period of time t:

Hence, the acceleration of a point at a given time is equal to the first derivative with respect to time of the point’s velocity vector or the second derivative of the radius vector with respect to time

. (5.11)

Point acceleration this is a vector quantity that characterizes the rate of change of the velocity vector over time.

Let's construct a speed hodograph (Fig. 5.11). By definition, the velocity hodograph is the curve that is drawn by the end of the velocity vector when a point moves, if the velocity vector is plotted from the same point.

Determining the speed of a point using the coordinate method of specifying its movement

Let the motion of a point be specified by the coordinate method in the Cartesian coordinate system

X = x(t), y = y(t), z = z(t)

The radius vector of a point is equal to

.

Since the unit vectors
are constant, then by definition

. (5.12)

Let us denote the projections of the velocity vector on the axis Oh, OU And Oz through V x , V y , V z

(5.13)

Comparing equalities (5.12) and (5.13) we obtain

(5.14)

In what follows, the derivative with respect to time will be denoted by the dot above, i.e.

.

The velocity modulus of a point is determined by the formula

. (5.15)

The direction of the velocity vector is determined by the direction cosines:

Determining the acceleration of a point using the coordinate method of specifying its movement

The velocity vector in the Cartesian coordinate system is equal to

.

A-priory

Let us denote the projections of the acceleration vector on the axis Oh, OU And Oz through A x , A y , A z Accordingly, we expand the velocity vector along the axes:

. (5.17)

Comparing equalities (5.16) and (5.17) we obtain

The module of the point acceleration vector is calculated similarly to the module of the point velocity vector:

, (5.19)

and the direction of the acceleration vector is direction cosines:

Determining the speed and acceleration of a point using the natural method of specifying its movement

This method uses natural axes starting at the current position of the point M on the trajectory (Fig. 5.12) and unit vectors Unit vector directed tangentially to the trajectory towards the positive reference of the arc, unit vector directed along the main normal of the trajectory towards its concavity, unit vector directed along the binormal to the trajectory at the point M.

Orty And lie in osculating plane, unit vectors And V normal plane, unit vectors And - in straightening plane.

The resulting trihedron is called natural.

Let the law of point motion be given s = s(t).

Radius vector points M relative to any fixed point will be a complex function of time
.

From differential geometry, the Serre-Frenet formulas are known, establishing connections between unit vectors of natural axes and the vector function of the curve

where  is the radius of curvature of the trajectory.

Using the definition of speed and the Serre-Frenet formula, we obtain:

. (5.20)

Denoting the projection of velocity onto the tangent and taking into account that the velocity vector is directed tangentially, we have

. (5.21)

Comparing equalities (5.20) and (5.21), we obtain formulas for determining the velocity vector in magnitude and direction

Magnitude positive if the point M moves in the positive direction of the arc reference s and negative in the opposite case.

Using the definition of acceleration and the Serre-Frenet formula, we obtain:

Let us denote the projection of the acceleration of the point on a tangent , main normal and binormal
respectively.

Then the acceleration is

From formulas (5.23) and (5.24) it follows that the acceleration vector always lies in the contacting plane and is expanded in directions And :

(5.25)

Projection of acceleration onto a tangent
called tangent or tangential acceleration. It characterizes the change in speed.

Projection of acceleration onto the main normal
called normal acceleration. It characterizes the change in the velocity vector in direction.

The magnitude of the acceleration vector is equal to
.

If And of the same sign, then the movement of the point will be accelerated.

If And different signs, then the movement of the point will be slow.

Let's find how the speed and acceleration of a point are calculated if the motion is given by equations (3) or (4). The question of determining the trajectory in this case has already been considered in § 37.

Formulas (8) and (10), which determine the values ​​of v and a, contain the time derivatives of the vectors . In equalities containing derivatives of vectors, the transition to dependencies between projections is carried out using the following theorem: the projection of the derivative of a vector onto an axis fixed in a given reference system is equal to the derivative of the projection of the differentiable vector onto the same axis, i.e.

1. Determining the speed of a point. Velocity vector of a point From here, based on formulas (I), taking into account that we find:

where the dot above the letter is a symbol for differentiation with respect to time. Thus, the projections of the point’s velocity onto the coordinate axes are equal to the first derivatives of the corresponding coordinates of the point with respect to time.

Knowing the projections of velocity, we will find its magnitude and direction (i.e., the angles that the vector v forms with the coordinate axes) using the formulas

2. Determination of the acceleration of a point. Acceleration vector of a point From here, based on formulas (11), we obtain:

i.e. the projections of the acceleration of a point onto the coordinate axes are equal to the first derivatives of the velocity projections or the second derivatives of the corresponding coordinates of the point with respect to time. The magnitude and direction of acceleration can be found from the formulas

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

So, if the motion of a point is given in Cartesian rectangular coordinates by equations (3) or (4), then the speed of the point is determined by formulas (12) and (13), and the acceleration by formulas (14) and (15). Moreover, in the case of movement occurring in one plane, the projection onto the axis should be discarded in all formulas

The speed of a point is a vector that determines at any given moment in time the speed and direction of movement of the point.

The speed of uniform motion is determined by the ratio of the path traveled by a point in a certain period of time to the value of this period of time.

Speed; S-path; t- time.

Speed ​​is measured in units of length divided by unit of time: m/s; cm/s; km/h, etc.

In the case of rectilinear motion, the velocity vector is directed along the trajectory in the direction of its movement.

If a point travels unequal paths in equal periods of time, then this movement is called uneven. Speed ​​is a variable quantity and is a function of time.

The average speed of a point over a given period of time is the speed of such uniform rectilinear motion at which the point during this period of time would receive the same displacement as in its movement under consideration.

Let's consider point M, which moves along a curvilinear trajectory specified by the law

Over a period of time?t, point M will move to position M1 along the arc MM 1. If the time period?t is small, then arc MM 1 can be replaced by a chord and, to a first approximation, find the average speed of the point

This speed is directed along the chord from point M to point M 1. We find the true speed by going to the limit at?t> 0

When?t> 0, the direction of the chord in the limit coincides with the direction of the tangent to the trajectory at point M.

Thus, the value of the speed of a point is defined as the limit of the ratio of the increment of the path to the corresponding period of time as the latter tends to zero. The direction of the velocity coincides with the tangent to the trajectory at a given point.

Point acceleration

Note that in the general case, when moving along a curved path, the speed of a point changes both in direction and in magnitude. The change in speed per unit time is determined by acceleration. In other words, the acceleration of a point is a quantity that characterizes the rate of change in speed over time. If during the time interval?t the speed changes by an amount, then the average acceleration

The true acceleration of a point at a given time t is the value to which the average acceleration tends at?t> 0, that is

As the time interval tends to zero, the acceleration vector will change both in magnitude and direction, tending to its limit.

Acceleration dimension

Acceleration can be expressed in m/s 2 ; cm/s 2, etc.

In the general case, when the motion of a point is given in a natural way, the acceleration vector is usually decomposed into two components, directed tangentially and normal to the trajectory of the point.

Then the acceleration of the point at time t can be represented as follows

Let us denote the component limits by and.

The direction of the vector does not depend on the value of the time interval?t.

This acceleration always coincides with the direction of the velocity, that is, it is directed tangentially to the trajectory of the point and is therefore called tangential or tangential acceleration.

The second component of the acceleration of a point is directed perpendicular to the tangent to the trajectory at a given point towards the concavity of the curve and affects the change in the direction of the velocity vector. This component of acceleration is called normal acceleration.

Since the numerical value of the vector is equal to the increment in the speed of the point over the considered period?t of time, then the numerical value of the tangential acceleration

The numerical value of the tangential acceleration of a point is equal to the time derivative of the numerical value of the velocity. The numerical value of the normal acceleration of a point is equal to the square of the point’s speed divided by the radius of curvature of the trajectory at the corresponding point on the curve

The total acceleration during uneven curvilinear motion of a point is composed geometrically of the tangential and normal accelerations.

For example, a car that starts moving moves faster as it increases its speed. At the point where the motion begins, the speed of the car is zero. Having started moving, the car accelerates to a certain speed. If you need to brake, the car will not be able to stop instantly, but over time. That is, the speed of the car will tend to zero - the car will begin to move slowly until it stops completely. But physics does not have the term “slowdown”. If a body moves, decreasing speed, this process is also called acceleration, but with a “-” sign.

Medium acceleration is called the ratio of the change in speed to the period of time during which this change occurred. Calculate the average acceleration using the formula:

where is it . The direction of the acceleration vector is the same as the direction of change in speed Δ = - 0

where 0 is the initial speed. At a moment in time t 1(see figure below) at the body 0. At a moment in time t 2 the body has speed. Based on the rule of vector subtraction, we determine the vector of speed change Δ = - 0. From here we calculate the acceleration:

.

In the SI system unit of acceleration called 1 meter per second per second (or meter per second squared):

.

A meter per second squared is the acceleration of a rectilinearly moving point, at which the speed of this point increases by 1 m/s in 1 second. In other words, acceleration determines the degree of change in the speed of a body in 1 s. For example, if the acceleration is 5 m/s2, then the speed of the body increases by 5 m/s every second.

Instantaneous acceleration of a body (material point) at a given moment in time is a physical quantity that is equal to the limit to which the average acceleration tends as the time interval tends to 0. In other words, this is the acceleration developed by the body in a very short period of time:

.

Acceleration has the same direction as the change in speed Δ in extremely short periods of time during which the speed changes. The acceleration vector can be specified using projections onto the corresponding coordinate axes in a given reference system (projections a X, a Y, a Z).

With accelerated linear motion, the speed of the body increases in absolute value, i.e. v 2 > v 1 , and the acceleration vector has the same direction as the velocity vector 2 .

If the speed of a body decreases in absolute value (v 2< v 1), значит, у вектора ускорения направление противоположно направлению вектора скорости 2 . Другими словами, в таком случае наблюдаем slowing down(acceleration is negative, and< 0). На рисунке ниже изображено направление векторов ускорения при прямолинейном движении тела для случая ускорения и замедления.

If movement occurs along a curved path, then the magnitude and direction of the speed changes. This means that the acceleration vector is depicted as two components.

Tangential (tangential) acceleration They call that component of the acceleration vector that is directed tangentially to the trajectory at a given point of the motion trajectory. Tangential acceleration describes the degree of change in speed modulo during curvilinear motion.

U tangential acceleration vectorτ (see figure above) the direction is the same as that of linear speed or opposite to it. Those. the tangential acceleration vector is in the same axis with the tangent circle, which is the trajectory of the body.