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Chapter 37: 30P (page 1147)

Stellar system Q1 moves away from us at a speed of 0.800c. Stellar system Q2, which lies in the same direction in space but is closer to us, moves away from us at speed 0.400c. What multiple of c gives the speed of Q2 as measured by an observer in the reference frame of Q1?

Short Answer

Expert verified

The multiple of c is 0.588.

Step by step solution

01

Describe the expression for the velocity of the particle

The relativistic velocity of the particle is given by,

u=u'+v1+u'vc 鈥︹�� (1)

Here, data-custom-editor="chemistry" u=0.400c is the velocity of stellar Q2 in reference frame, and data-custom-editor="chemistry" u=0.800c is the velocity of the reference frame of the observer on stellar Q1 relative to the reference frame.

02

Determine the multiple of c

Rearrange the equation (1) for u'.

u1+u'vc2=u'+vuu'vc-u'=v-uu'uvc2-1=v-uu'=v-uuvc2-1 鈥︹�� (2)

Substitute 0.800c for v, and 0.400c for u in equation (2).

u'=0.800c-0.400c0.800c0.400cc2-1=0.400c0.32-1=-0.588c

Here, the negative sign indicates the opposite direction.

Therefore, the multiple of c is 0.588.

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Most popular questions from this chapter

A spaceship of rest length 130 m races past a timing station at a speed of 0.740c. (a) What is the length of the spaceship as measured by the timing station? (b) What time interval will the station clock record between the passage of the front and back ends of the ship?

What is the momentum in MeV/c of an electron with a kinetic energy of 2.00MeV ?

A rod is to move at constant speedvalong thexaxis of reference frame S, with the rod鈥檚 length parallel to that axis. An observer in frame Sis to measure the lengthLof the rod. Which of the curves in Fig. 37-15 best gives length L(vertical axis of the graph) versus speed parameter尾?

The mean lifetime of stationary muons is measured to be 2.2000渭s. The mean lifetime of high-speed muons in a burst of cosmic rays observed from Earth is measured to be 16.000渭s. To five significant figures, what is the speed parameter 尾of these cosmic-ray muons relative to Earth?

The car-in-the-garage problem. Carman has just purchased the world鈥檚 longest stretch limo, which has a proper length of Lc=30.5鈥尘. In Fig. 37-32a, it is shown parked in front of a garage with a proper length of Lg=6.00鈥尘. The garage has a front door (shown open) and a back door (shown closed).The limo is obviously longer than the garage. Still, Garageman, who owns the garage and knows something about relativistic length contraction, makes a bet with Carman that the limo can fit in the garage with both doors closed. Carman, who dropped his physics course before reaching special relativity, says such a thing, even in principle, is impossible.

To analyze Garageman鈥檚 scheme, an xc axis is attached to the limo, with xc=0 at the rear bumper, and an xg axis is attached to the garage, with xg=0 at the (now open) front door. Then Carman is to drive the limo directly toward the front door at a velocity of 0.9980c(which is, of course, both technically and financially impossible). Carman is stationary in the xcreference frame; Garageman is stationary in the role="math" localid="1663064422721" Xgreference frame.

There are two events to consider. Event 1: When the rear bumper clears the front door, the front door is closed. Let the time of this event be zero to both Carman and Garageman: tg1=tc1=0. The event occurs at xg=xc=0. Figure 37-32b shows event 1 according to the xg reference frame. Event 2: When the front bumper reaches the back door, that door opens. Figure 37-32c shows event 2 according to the xg reference frame.

According to Garageman, (a) what is the length of the limo, and what are the spacetime coordinates (b) xg2 and (c) tg2 of event 2? (d) For how long is the limo temporarily 鈥渢rapped鈥� inside the garage with both doors shut? Now consider the situation from the xc reference frame, in which the garage comes racing past the limo at a velocity of 鈭�0.9980c. According to Carman, (e) what is the length of the passing garage, what are the spacetime coordinates (f) Xc2and (g) tc2 of event 2, (h) is the limo ever in the garage with both doors shut, and (i) which event occurs first? (j) Sketch events 1 and 2 as seen by Carman. (k) Are the events causally related; that is, does one of them cause the other? (l) Finally, who wins the bet?
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