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18 CHAPTER 2 MOTION ALONG A STRAIGHT LINE

Instantaneous Velocity and Speed

You have now seen two ways to describe how fast something moves: average

velocity and average speed, both of which are measured over a time interval t.

However, the phrase “how fast” more commonly refers to how fast a particle is

moving at a given instant—its instantaneous velocity(or simply velocity)v.

The velocity at any instant is obtained from the average velocity by shrinking

the time interval tcloser and closer to 0. As tdwindles, the average velocity

approaches a limiting value, which is the velocity at that instant:

(2-4)

Note that vis the rate at which position xis changing with time at a given instant;

that is,vis the derivative of xwith respect to t. Also note that vat any instant is

the slope of the position – time curve at the point representing that instant.

Velocity is another vector quantity and thus has an associated direction.

Speedis the magnitude of velocity; that is, speed is velocity that has been

stripped of any indication of direction, either in words or via an algebraic sign.

(Caution:Speed and average speed can be quite different.) A velocity of 5 m/s

and one of 5 m/s both have an associated speed of 5 m/s. The speedometer in a

car measures speed, not velocity (it cannot determine the direction).

v lim

t: 0

x

t



dx

dt

2-2INSTANTANEOUS VELOCITY AND SPEED

After reading this module, you should be able to...

2.07Given a particle’s position as a function of time,
calculate the instantaneous velocity for any particular time.

2.08Given a graph of a particle’s position versus time, deter-

mine the instantaneous velocity for any particular time.

2.09Identify speed as the magnitude of the instantaneous

velocity.

●The instantaneous velocity (or simply velocity) vof a moving
particle is

wherexx 2 x 1 andtt 2 t 1.

v lim

t: 0

x

t



dx

dt

,

●The instantaneous velocity (at a particular time) may be

found as the slope (at that particular time) of the graph of x

versust.

●Speed is the magnitude of instantaneous velocity.

Checkpoint 2

The following equations give the position x(t) of a particle in four situations (in each

equation,xis in meters,tis in seconds, and t0): (1) x 3 t2; (2) x 4 t^2 2;

(3)x2/t^2 ; and (4) x2. (a) In which situation is the velocity vof the particle con-

stant? (b) In which is vin the negative xdirection?

Calculations:The slope of x(t), and so also the velocity, is

zero in the intervals from 0 to 1 s and from 9 s on, so then

the cab is stationary. During the interval bc, the slope is con-

stant and nonzero, so then the cab moves with constant ve-

locity. We calculate the slope of x(t) then as

(2-5)

x

t

v

24 m4.0 m

8.0 s3.0 s

4.0 m/s.

Sample Problem 2.02 Velocity and slope of xversust, elevator cab

Figure 2-6ais an x(t) plot for an elevator cab that is initially
stationary, then moves upward (which we take to be the pos-
itive direction of x), and then stops. Plot v(t).

KEY IDEA

We can find the velocity at any time from the slope of the
x(t) curve at that time.

Learning Objectives

Key Ideas