Speed and velocity

Active vocabulary

speed velocity instantaneous velocity

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швидкість швидкість (вектор) миттєва швидкість

average speed average velocity uniform motion

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середня швидкість середня швидкість рівномірний рух

Speed is a physical quantity which shows what distance is covered by a steadily moving material point in a unit of time,

It is expressed in meters per second (m/s) in SI. Speed is a quantity which has no direction but only magnitude, it means that speed is a scalar.

Fig. 1.4

However, the complete information is provided for this physical quantity by stating its direction in addition to its magnitude. In this case we say ‘ velocity ’ instead of «speed».

The distinction between speed and velocity can be made clear by reference to a point moving uniformly round a circular track. At every point on the track the speed is the same (fig 1.4). At every point, however, the velocity is different because it is pointed in different directions.

We shall define the velocity of an object moving steadily as the displacement per time unit:

.

We considered motions in which the velocity was constant for the whole time. Obviously, many motions do not occur at constant velocity. If an object is moving non-uniformly, we may deal with its average speed and average velocity (fig. 1.5):

Fig. 1.5

,

.

The modulus of average velocity is

.

The velocity at a given instant is called the instantaneous velocity. Weuse the word 'instantaneous' to distinguish the velocity at one moment from the average velocity along a time interval.

How can we find the velocity of an object at any particular time? We say that the velocity at a particular instant is given by the limit of as 'approaches zero', that is, as gets smaller and smaller. Symbolically this statement is written as

In the language of calculus, the instantaneous velocity is the rate at which a particle's position is changing with time at a given instant. The velocity of a particle at any instant is the slope of its position-time curve at the point representing that instant.

Substituting for yields

which can be rewritten as

where

; .

The modulus of velocity is calculated according to the Pythagorean Theorem, as vectors are mutually perpendicular:

Direction of velocity vector is set by direction cosines according to the known rule:

, , .

Acceleration

Active vocabulary

acceleration accelerated motion average acceleration total acceleration

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прискорення прискорений рух середнє прискорення повне прискорення

instantaneous acceleration centripetal or normal (center seeking) acceleration

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миттєве прискорення нормальне прискорення

In case of a non-uniform motion the speed of the moving object changes. The acceleration of a moving object at an instant is the rate of change of its speed at that instant.

If the speed changes in equal amounts at equal times, no matter how small the time intervals may be, the acceleration is said to be uniform, and the motion to be uniformly accelerated motion.

Let’s consider a non-uniform motion. We shall name two times and , and use symbols and for the corresponding velocities at these two times. The acceleration is then given by the relationship

,

this is the average acceleration for a time interval .

The acceleration is measured in m/sec².

To find the instantaneous acceleration at any particular time , we determine the average acceleration vector for shorter and shorter time intervals which include the time . We define the instantaneous acceleration at a given time as the limit of the average acceleration as the interval becomes smaller and smaller. This limiting vector has a definite size and direction:

This equation shows that the acceleration is the first derivative of velocity with respect to time.

Using expression for yields

Or

in which the three scalar components of the acceleration vector are expressed as

, .

Fig. 1.6

It is obvious that the acceleration vector has the same direction as the change of the velocity; since this change need not be in the same direction as , the acceleration may point at any direction with respect to the motion Fig.1.6.

When is small, the angle is small too. then points towards O, the centre of the circle and the acceleration related to the change of velocity vector is, thus, directed towards the centre. This acceleration is called centripetal (or center - seeking) acceleration, (Fig.1.7). It is given by

,

here is a unit vector directed along the radius towards the center and is an angular speed of an object.

In case of an object moving at non-constant speed round a circle, the acceleration related to the change in speed exists. It is called tangent acceleration, , (Fig.1.7) and is determined as

Fig. 1.7

,

here is a unit vector in the direction of tangent to the curve at a point.

Therefore acceleration vector mentioned above will be the vector sum of two vector accelerations, and ,

and its magnitude

The direction of the total acceleration may be found using the formula

Where is the angle between vectors and .

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Дата добавления: 2015-11-16