37-9

In this problem we are concerned with measurement of the length of a moving object. We measure an object by noting where the front and back of the object is at the same time and then find the length between these two places. When we say the 'rest length' we mean the length of an object as measure when the object is at rest, that is, how long the object judges itself to be (since anything is always at rest with respect to itself).

In the diagram (fig.6), the moving frame is that of the space ship. The $ct'$ axis will denote the rear of the space ship (the rear is at $x'=0$ at every time $t'$), and a line parallel to the $ct'$ axis through $x'=L'$ will denote the front of the ship (the front is at $x'=L'$ at every time $t'$). We then note that $L'$, the length of the ship as measured in the ship's frame, is the rest length of the ship.

Figure 6: Space-time diagram for 37-9.

Now, what we want to do is measure the length of the ship in the stationary frame. How do we do this? We pick some time, say $t=0$, and measure where the front and back of the spaceship are at those times and take the difference. We know that at $t=0$ the rear of the spaceship is at the origin, so $x_R=0$ (the rear). Where's the front end? Well, we know that $x'=L'$ always, so lets see if we can use the Lorentz transformation to find out where this is in the stationary frame.

$$\begin{eqnarray*} x' &=&\gamma(x-\beta ct)\\ x' &=&\gamma x\\ L' &=&\gamma L, \end{eqnarray*}$$

since we measure at $t=0$, and $L$ and $L'$ are the lengths in the lab and ship frames respectively. We have $\gamma=(1-.740^2)^{-1/2}=1.4868$, so

$$L=\frac{L'}{\gamma}=\frac{130\;\rm m}{1.4868}=87.44 \;\rm m.$$

We have just deduced the relativistic length contraction formula from the space-time diagram.

Next we want to know how long it takes for the ship to pass a given point as measured by the lab. Well, we have a measurement of how fast the ship is, and we have a measurement of how long it is (we just found it), so we can find the time by dividing:

$$ct=\frac{L}{\beta} = \frac{87.44\;\rm m}{.740}=118.162\;\rm m,$$

or $t=3.94\times 10^{-7}\;\rm s$.