Part 1. Vector Method of Describing Motion

Let a particle M be moving relatively to any frame of reference Oxyz. The position of the particle at any instant can be specified by a vector drawn from the origin O to the particle M (Fig.1). Vector is called the radius vector (position vector) of the particle.

When the particle moves, the vector changes with time both in magnitude and direction. Thus is a variable vector (a vector function) depending on the argument t.

= (t) (1)

Eq.1 describes the curvilinear motion of a particle in a vector form and can be used to construct a vector for any particular moment of time and to determine the position of the moving particle at that moment.

The locus of the tip of vector defines the path of the moving particle.

We can introduce unit vectors , , directed along the x, y, z axes respectively.

Fig. 1

As the projections of vector on the coordinate axes are equal to the coordinate of the particle M, i.e. r_x = x, r_y = y, r_z = z, we obtain

=x +y + z

Ex.6. Answer the following questions:

1. How can we specify the position of a particle?

2. What is the radius vector of the particle?

3. What happens with the vector when the particle moves?

4. What is the path of the moving particle defined by?

Part 2. Coordinate Method of Describing Motion

The position of a particle with respect to a given frame of reference Oxyz can be specified by its Cartesian coordinates x, y, z. When motion takes place, the three coordinates will change within the frame. If we want to know the equations of motion of a particle, i.e. its location in space at any instant, we must know its coordinates for any moment of time, i.e., the relations

x=f1(t); y=f2(t); z=f3(t) (2)

should be known.Eqs. (2) are the equations of motion of a particle in terms of the Cartesian rectangular coordinates.

Eqs. (2) are, at the same time, the equations of the particle’s path in parametric form, where the time t is the parameter. By eliminating time t from the equations of motion we can obtain the equation of the path in the usual form, i.e., in the form of a relation between the particle’s coordinates.

Ex.7. Answer the following questions:

1. What is the coordinate method of describing motion?

2. What happens when the motion takes place?

3. How can we obtain the equation of the path from the equation of motion?

Part 3. Natural Method of Describing Motion

The continuous curve described by moving particle with respect to a given frame of reference is called the path of that particle.

If the path is a straight line, the motion is said to be rectilinear, if the path is a curve, the motion is curvilinear.

The natural method of describing motion is convenient when the particle’s path is known at once.

In order to describe the motion of a particle by the natural method, a problem must state: (1) the path of the particle; (2) the origin of the path, showing the positive and negative directions; (3) the equations of the particle’s motion along the path in the form s = f(t) (Fig.2).

Fig. 2

Ex.8. Answer the following questions.

1. Give the definition of the path of a particle.

2. What motion is rectilinear?

3. What motion is curvilinear?

4. When is the natural method convenient?

Ex.9 .Say whether the following statements are True or False.

1. Kinematics doesn’t take into account the mass of bodies.

2. Kinematics takes into account the forces acting on bodies.

3. When the particle moves, the vector changes with time only in direction.

4. When motion takes place, the three coordinates remain the same within the frame of reference.

5. The motion is said to be rectilinear if the path of a moving particle is a straight line.

Ex.10. Complete the following sentences with the information from the text.

1. Coordinate system is…

2. The continuous curve described by moving particle with respect to a given frame of reference is…

3. The motion of the body is considered with respect to …

4. The body is said to be in motion relative to the given frame of reference if…

5. The locus of the tip of vector defines…

Ex.11. Put the questions to the following answers.

1. It is the continuous curve described by a moving particle with respect to a given frame of reference

2. The path of the rectilinear motion is a straight line.

3. The natural method of describing motion is convenient when we know the particle’s path .

4. We must know the coordinates of a particle for any moment of time.

Ex.12. Match the verbs and nouns or noun phrases they can be used with . Compose 5 sentences with any of these expressions.

	Verbs:		Nouns / noun phrases
1	be directed	a	the positive direction
2	calculate	b	a characteristic of a motion
3	describe	c	along a chord
4	introduce	d	the path of a particle
5	obtain	e	the motion
6	occupy	f	the average velocity
7	show	g	a position
8	state	h	a concept

Ex.13. Translate into English.

1. Чтобы определить положение частицы в любой момент времени относительно определенной системы координат, необходимо опустить вектор из точки начала координат к частице.

2. Это переменный вектор, потому что его значение и направление изменяются со временем.

3. Естественный метод описания движения применяется тогда, когда путь частицы, точка начала движения частицы и направление движения известны.

Unit 3.