Relative Motion in One Dimension

4.8	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 depends on the reference frame of 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 A in Fig. 4-21) is parked by the side of a highway, watching car P (the “particle”) speed past. Barbara (at the origin of frame B) 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. 4-21 we see that


	(4-40)

The equation is read: “The coordinate

of P as measured by A is equal to the coordinate

of P as measured by B plus the coordinate

of B as measured by A.” Note how this reading is supported by the sequence of the subscripts.

FIGURE 4-21

Alex (frame A) and Barbara (frame B) watch car P, as both B and P move at different velocities along the common x axis of the two frames. At the instant shown,

is the coordinate of B in the A frame. Also, P is at coordinate

in the B frame and coordinate

in the A frame.

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

Thus, the velocity components are related by

(4-41)

This equation is read: “The velocity

of P as measured by A is equal to the velocity

of P as measured by B plus the velocity

of B as measured by A.” The term

is the velocity of frame B relative to frame A.

Here we consider only frames that move at constant velocity relative to each other. In our example, this means that Barbara (frame B) drives always at constant velocity

relative to Alex (frame A). Car P (the moving particle), however, can change speed and direction (that is, it can accelerate).

To relate an acceleration of P as measured by Barbara and by Alex, we take the time derivative of Eq. 4-41:

Because

is constant, the last term is zero and we have

(4-42)

In other words,


	Observers on different frames of reference that move at constant velocity relative to each other will measure the same acceleration for a moving particle.

Sample Problem 4-11

In Fig. 4-21, suppose that Barbara’s velocity relative to Alex is a constant

and car P is moving in the negative direction of the x axis.

(a) If Alex measures a constant

for car P, what velocity

will Barbara measure?



KEY IDEA

We can attach a frame of reference A to Alex and a frame of reference B to Barbara. Because the frames move at constant velocity relative to each other along one axis, we can use Eq. 4-41

to relate

and

Calculation: We find

Thus,


	(Answer)

Comment: If car P were connected to Barbara’s car by a cord wound on a spool, the cord would be unwinding at a speed of 130 km/h as the two cars separated.

(b) If car P brakes to a stop relative to Alex (and thus relative to the ground) in time

at constant acceleration, what is its acceleration

relative to Alex?



KEY IDEA

To calculate the acceleration of car P relative to Alex, we must use the car’s velocities relative to Alex. Because the acceleration is constant, we can use Eq. 2-11

to relate the acceleration to the initial and final velocities of P.

Calculation: The initial velocity of P relative to Alex is

and the final velocity is 0. Thus,


	(Answer)

of car P relative to Barbara during the braking?



KEY IDEAS

To calculate the acceleration of car P relative to Barbara, we must use the car’s velocities relative to Barbara.

Calculation: We know the initial velocity of P relative to Barbara from part (a)

. The final velocity of P relative to Barbara is −52 km/h (this is the velocity of the stopped car relative to the moving Barbara). Thus,


	(Answer)

Comment: We should have foreseen this result: Because Alex and Barbara have a constant relative velocity, they must measure the same acceleration for the car.