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Chapter 37: Q8P (page 1145)

An electron of β=0.9999987moves along the axis of an evacuated tube that has a length of 3.00 m as measured by a laboratory observer S at rest relative to the tube. An observer S’ who is at rest relative to the electron, however, would see this tube moving with speed v ( =βC).What length would observer S’ measure for the tube?

Short Answer

Expert verified

The length measured by observer S’ for the tube is 0.0153m.

Step by step solution

01

Identification of given data

The speed parameter is β=0.999987β=0.999987

The length of the evacuated tube is L0=3m

The length contraction is used to find the length measured by the observer S’ for the tube.

02

Determination of length measured by observer S’ for the tube

The length measured by observer S’ for the tube is given as:

L=L01-β2

Substitute all the values in the equation.

L=3m1-0.9999872L=0.0153m

Therefore, the length measured by observer S’ for the tube is 0.0153m.

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Most popular questions from this chapter


To eight significant figures, what is speed parameter if the Lorentz factor γ is (a) 1.010 000 0, (b) 10.000 000, (c) 100.000 00, and (d) 1000.000 0?

Figure 37-20 shows the triangle of Fig 37-14 for six particles; the slanted lines 2 and 4 have the same length. Rank the particles according to (a) mass, (b) momentum magnitude, and (c) Lorentz factor, greatest first. (d) Identify which two particles have the same total energy. (e) Rank the three lowest-mass particles according to kinetic energy, greatest first.

Spatial separation between two events. For the passing reference frames of Fig. 37-25, events A and B occur with the following spacetime coordinates: according to the unprimed frame,(xA,tA)and role="math" localid="1663045013644" (xB,tB)according to the primed frame,(x'A,t'A) androle="math" localid="1663045027721" (x'B,t'B). In the umprimed frameΔt=tB−tA=1.00 μsandrole="math" localid="1663045143133" Δx=xB−xA=240 m.(a) Find an expression forin Δx'terms of the speed parameterβand the given data. GraphΔx'versusβfor two ranges ofβ: (b)0 to0.01and (c)0.1to 1. (d) At what value ofβisΔx'=0?

In the reaction p+F19→α+O16, the masses are m(p) = 1.007825 u, m(a) = 4.002603 u, m(F) = 18.998405 u, m(O) = 15.994915 u. Calculate the Q of the reaction from these data.

The car-in-the-garage problem. Carman has just purchased the world’s longest stretch limo, which has a proper length of Lc=30.5 m. In Fig. 37-32a, it is shown parked in front of a garage with a proper length of Lg=6.00 m. 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’s 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 “trapped” 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 −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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