9781118230725.pdf

78 CHAPTER 4 MOTION IN TWO AND THREE DIMENSIONS

4-6RELATIVE MOTION IN ONE DIMENSION

frames that move relative to each other at constant velocity

and along a single axis.

Learning Objective

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

4.18Apply the relationship between a particle’s position, ve-
locity, and acceleration as measured from two reference

where is the velocity of Bwith respect to A. Both ob-

servers measure the same acceleration for the particle:

a:PA:aPB.

:vBA

:vPAv:PB:vBA,

Key Idea

●When two frames of reference AandBare moving relative
to each other at constant velocity, the velocity of a particle P
as measured by an observer in frame Ausually differs from
that measured from frame B. The two measured velocities are
related by

Relative Motion in One Dimension

Suppose you see a duck flying north at 30 km/h. To another duck flying alongside,

the first duck seems to be stationary. In other words, the velocity of a particle de-

pends on the reference frameof whoever is observing or measuring the velocity.

For our purposes, a reference frame is the physical object to which we attach our

coordinate system. In everyday life, that object is the ground. For example, the

speed listed on a speeding ticket is always measured relative to the ground. The

speed relative to the police officer would be different if the officer were moving

while making the speed measurement.

Suppose that Alex (at the origin of frame Ain Fig. 4-18) is parked by the side

of a highway, watching car P(the “particle”) speed past. Barbara (at the origin of

frameB) is driving along the highway at constant speed and is also watching car P.

Suppose that they both measure the position of the car at a given moment. From

Fig. 4-18 we see that

xPAxPBxBA. (4-40)

The equation is read: “The coordinate xPAofPas measured by A is equal tothe

coordinatexPBofPas measured by B plusthe coordinate xBAofBas measured

byA.” Note how this reading is supported by the sequence of the subscripts.

Taking the time derivative of Eq. 4-40, we obtain

Thus, the velocity components are related by

vPAvPBvBA. (4-41)

This equation is read: “The velocity vPAofPas measured by A is equal tothe

d

dt

(xPA)

d

dt

(xPB)

d

dt

Figure 4-18Alex (frame A) and Barbara (xBA).

(frameB) watch car P, as both BandP

move at different velocities along the com-

monxaxis of the two frames. At the

instant shown,xBAis the coordinate of B

in the Aframe. Also,Pis at coordinate xPB

in the Bframe and coordinate xPAxPB

xBAin the Aframe.

x

FrameA FrameB

vBA

P

x

y y

xBA xPA=xPB+xBA

xPB

FrameB moves past

frameA while both

observeP.

To find the period Tof the motion, first note that the final

velocity is the reverse of the initial velocity. This means the

aircraft leaves on the opposite side of the circle from the ini-

tial point and must have completed half a circle in the given

24.0 s. Thus a full circle would have taken T48.0 s.

Substituting these values into our equation for a, we find

a (Answer)

2

 (640.31 m/s)

48.0 s

83.81 m/s^2 8.6g.

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