Part 2. Determination of the Velocity of a Particle when its Motion is described by the Coordinate Method

The velocity vector of particle is = d /dt. Hence, taking into account that rx = x; ry = y; rz = z; we have:

Vx = dx/dt; Vy = dy/dt; Vz = dz/dt;

or Vx = x; Vy = y; Vz = z, where the dot over the letter is a symbol of differentiation with respect to time.

Knowing the projections of the velocity, we can find the magnitude and direction (i.e. the angels a, b, c which vector makes with the coordinate axes) from the equations:

V=

cos a = Vx /V; cos b = Vy /V; cos c = Vz /V.

Part 3. Determination of the Velocity of the Particle when its Motion is described by the Natural Method

The path of a particle and the law of motion along it is in the form S = f(t).

If in time interval t=t1-t a particle moves from position M to position M1, the displacement along the arc of the path being S=S1-S, the numerical value of the average velocity will be Vav = (S1-S)/(t1-t)= S/ t

Fig.4

Passing to the limit, we obtain the numerical value of the instantaneous velocity for a given time t:

Thus, the numerical value of the instantaneous velocity of a particle is equal to the first derivative of the displacement (of the arc coordinate) S of the particle with respect to time.

The velocity vector is tangent to the path.

As the sign of V is the same as that of S, it will be readily appreciated that if V>0, the velocity vector is in the positive direction of S, if V<0, V is in the negative direction of S. Thus, the numerical value of the velocity defines simultaneously the modulus and the direction of the velocity vector. This vector quantity is called algebraic velocity.

Ex 3. Answer the following questions.

1. What is the numerical value of the instantaneous velocity of a particle equal to?

2. What is the velocity vector tangent to?

3. What is the direction of the velocity vector?

4. What is defined by the numerical value of the velocity?

Ex.4 .Say whether the following statements are True or False.

1. Vector and vector have different directions.

2. The average velocity tends to the instantaneous velocity of a particle when the time interval t tends to zero.

3. The limiting direction of the secant MM₁ is a curve line.

4. If V<0, the velocity vector V is in the positive direction of S.

5. The numerical value of the velocity defines simultaneously the modulus and the direction of the velocity vector.

Ex.5 .Complete the sentences with the following prepositions:

To, towards , over , along, by, with , during, in.

1. The position M is defined …. the radius vector .

2. The vector is directed …. a chord.

3. The average velocity of the particle M …. the given time interval is called vector quantity.

4. The average velocity tends….. the vector quantity .

5. The vector of instantaneous velocity of a particle is tangent…. the path of the particle … the direction of motion.

6. The dot … the letter is a symbol of differentiation …… respect to time.

Ex.6. Complete the following table:

Noun	Verb	Adjective
Quickness
	------	Rapid
	Direct
Determination
	Calculate
--------		Obtainable
Tendency		------
	Differentiate
		Average

Ex.7. Complete the sentences with the following words :

Let, then, obviously, thus, hence, as, finally.

1. …. , is a variable vector (a vector function) depending on the argument t.

2. …. the projections of this vector on the coordinate axes are equal to the coordinate of the particle, we can obtain this equation.

3. …. a particle M be moving relatively to any frame of reference.

4. First we should show the positive direction of this vector and …. the negative one.

5. …., taking into account this value , we can calculate the speed.

6. It is ………that the smaller the time interval, the more precisely will characterize the particle’s motion.

7. Having established all the dependence on the original data, we can…….indicate the procedure for the analysis.

Ex.8. Translate into English.

1. Скорость частицы – это векторная величина. Она характеризует скорость и направление изменения положения частицы.

2. Вектор смещения частицы направлен вдоль хорды.

3. Для рассмотрения этой проблемы необходимо ввести понятие мгновенной скорости частицы.

4. Числовое значение мгновенной скорости можно определить по этой формуле.

5. Положение частицы может определяться радиус вектором.

6. Вектор мгновенной скорости частицы равен первой производной радиус вектора частицы по отношению ко времени.