Unit 4. Acceleration Vector of a Particle.

Learn the following words and word combinations by heart:

acceleration	ускорение
angular ~	угловое ускорение
average ~	среднее ускорение
instantaneous ~	мгновенное ускорение
angle of contiguity	смежный угол
axes of the natural trihedron	оси естественного трехгранника
be directed towards	быть направленным к…
binormal	бинормальный
~ curve	бинормальная кривая
~ space	бинормальное пространство
curvature	кривизна
~ of surface	кривизна поверхности
increase in velocity	увеличение скорости
limit sign	знак лимита, предела
limit direction	ограничивать направление
normal	нормаль
inward ~	нормаль, направленная в сторону выпуклости
two-dimensional ~	двухмерная нормаль
principal ~	основная нормаль
numerator and denominator of the fraction	числитель и знаменатель дроби
occupy a position	занимать положение
osculating plane	соприкасающаяся плоскость
respectively	соответственно
secant	секущая
second power of	в квадрате
set of coordinate axes	координатные оси
so-called	так называемый
component	составляющая, компонента, слагающая
normal ~	нормальная составляющая
rotational ~	вращательная составляющая
tangential ~	касательная составляющая

Part 1. Determination of the Acceleration of a Particle when its Motion is described by the Vector Method.

Acceleration characterizes the time rate of change of velocity in magnitude and direction.

Let a moving particle occupy a position M and have a velocity at a given time t, and let it at any time t₁ occupy a position M₁ and have a velocity (fig.4). The increase in velocity in the time interval t=t1-t is .

Note that vector is always directed towards the inside of the path. The ratio of the velocity increment vector to the corresponding time interval t defines the vector of average acceleration of the particle in the given time interval:

Fig.5

Obviously, the vector of average acceleration has the same direction as the vector , i.e. towards the inside of the path.

The instantaneous acceleration of a particle at a given time t is defined as the vector quantity towards which the average acceleration tends when the time interval t tends to zero:

Thus, the vector of instantaneous acceleration of a particle is equal to the first derivative of the velocity vector or the second derivative of the radius vector of the particle with respect to time. The vector lies in the so-called osculating plane and is directed towards the inside of the curve (path).

Comprehension check.

Ex 1. Answer the following questions.

1. What is vector always directed towards?

2. What direction does the vector of average acceleration have?

3. What is the vector of instantaneous acceleration of a particle equal to?

4. Where does the vector lie?

Ex 2. Find in the text (Part 1) the antonyms to the following words:

Whole body, retardation, still, eternal, decrease, indefinite time, outside, different, last, straight.