Motion In Two and Three Dimensions

4-2	Average Velocity and Instantaneous Velocity

After reading this module, you should be able to …


4.04	Identify that velocity is a vector quantity and thus has both magnitude and direction and also has components.

4.05	Draw two-dimensional and three-dimensional velocity vectors for a particle, indicating the components along the axes of the coordinate system.

4.06	In magnitude-angle and unit-vector notations, relate a particle's initial and final position vectors, the time interval between those positions, and the particle's average velocity vector.

4.07	Given a particle's position vector as a function of time, determine its (instantaneous) velocity vector.

•

If a particle undergoes a displacement

in time interval

, its average velocity

for that time interval is

•

is shrunk to 0,

reaches a limit called either the velocity or the instantaneous velocity

which can be rewritten in unit-vector notation as

where

, and

•

The instantaneous velocity

of a particle is always directed along the tangent to the particle's path at the particle's position.

Average Velocity and Instantaneous Velocity

If a particle moves from one point to another, we might need to know how fast it moves. Just as in Chapter 2, we can define two quantities that deal with “how fast”: average velocity and instantaneous velocity. However, here we must consider these quantities as vectors and use vector notation.

If a particle moves through a displacement

in a time interval

, then its average velocity

(4-8)

This tells us that the direction of

(the vector on the left side of Eq. 4-8) must be the same as that of the displacement

(the vector on the right side). Using Eq. 4-4, we can write Eq. 4-8 in vector components as

(4-9)

For example, if a particle moves through displacement

in 2.0 s, then its average velocity during that move is

That is, the average velocity (a vector quantity) has a component of

along the x axis and a component of

along the z axis.

When we speak of the velocity of a particle, we usually mean the particle's instantaneous velocity

at some instant. This

is the value that

approaches in the limit as we shrink the time interval

to 0 about that instant. Using the language of calculus, we may write

as the derivative

(4-10)

Figure 4-3 shows the path of a particle that is restricted to the

plane. As the particle travels to the right along the curve, its position vector sweeps to the right. During time interval

, the position vector changes from

and the particle's displacement is

Figure 4-3
The displacement of a particle during a time interval , from position 1 with position vector at time t₁ to position 2 with position vector at time t₂. The tangent to the particle's path at position 1 is shown.

To find the instantaneous velocity of the particle at, say, instant t₁ (when the particle is at position 1), we shrink interval

to 0 about t₁. Three things happen as we do so. (1) Position vector

in Fig. 4-3 moves toward

so that

shrinks toward zero.

The direction of

(and thus of

) approaches the direction of the line tangent to the particle's path at position 1. (3) The average velocity

approaches the instantaneous velocity

at t₁.

In the limit as

, we have

and, most important here,

takes on the direction of the tangent line. Thus,

has that direction as well:

The direction of the instantaneous velocity

of a particle is always tangent to the particle's path at the particle's position.

The result is the same in three dimensions:

is always tangent to the particle's path.

To write Eq. 4-10 in unit-vector form, we substitute for

from Eq. 4-1:

This equation can be simplified somewhat by writing it as

(4-11)

where the scalar components of

are

(4-12)

For example,

is the scalar component of

along the x axis. Thus, we can find the scalar components of

by differentiating the scalar components of

Figure 4-4 shows a velocity vector

and its scalar x and y components. Note that

is tangent to the particle's path at the particle's position. Caution: When a position vector is drawn, as in Figs. 4-1 through 4-3, it is an arrow that extends from one point (a “here”) to another point (a “there”). However, when a velocity vector is drawn, as in Fig. 4-4, it does not extend from one point to another. Rather, it shows the instantaneous direction of travel of a particle at the tail, and its length (representing the velocity magnitude) can be drawn to any scale.

Figure 4-4
The velocity of a particle, along with the scalar components of .

The figure shows a circular path taken by a particle. If the instantaneous velocity of the particle is

, through which quadrant is the particle moving at that instant if it is traveling (a) clockwise and (b) counterclockwise around the circle? For both cases, draw

on the figure.