37-30

This problem discusses how the measurements of the speed of an object as made by two different observers are related. Since nothing can go faster than light as seen by any observer, we know the the relationship isn't going to be the same as in Galilean relativity. In the diagram below (fig.8) we have time axis from 3 different observers. The $ct$ axis is the 'stationary' observer, $S$. The $S'$ observer ($ct'$ axis) moves with speed $\beta_1$ as measured by S, and the observer $S''$ ($ct''$ axis) moves with speed $\beta_2$ as measured in S. We will then let $\beta_2'$ be the speed of $S''$ as measured by $S'$.

Figure 8: Space-time diagram for 37-30.

Note that we have labeled the angles between the time axis in the figure. We have

$$\theta_2=\theta_1+\theta_2'.$$

Now, remember from the section on space-time diagrams that the angle $\theta$ is related to the velocity parameter $\beta$ by $\beta=\tan\theta$. So, lets take the tangent of the above equation relating the various angles and try to convert it all to $\beta$'s:

$$\begin{eqnarray*} \theta_2 &=&\theta_1+\theta_2'\\ \tan\theta_2 &=&\tan(\theta_... ...ta_2'} \\ \beta_2&=&\frac{\beta_1+\beta_2'}{1-\beta_1\beta_2'}, \end{eqnarray*}$$

where we used a trig identity from the back of the book and in the last line substitute each $\tan\theta$ for the corresponding $\beta$. This is the formula for converting speed measurements of a third object (here $S''$) between two frames ($S$ and $S'$).

The data given is

$$\begin{eqnarray*} \beta_1 &=& .62\\ \beta_2' &=& .47. \end{eqnarray*}$$

That is, we know the speed of the second frame as measured by the first, and the speed of the third as measured by the second. So we can use the above formula to find $\beta_2$, the speed of the third as measured by the first. We have

$$\beta_2=\frac{.47+.62}{1-(.47)(.62)}=.844.$$

The classical counter part to this formula is the numerator only

$$\beta_2=\beta_1+\beta_2'=.47+.62=1.09,$$

which gives a velocity greater than that of light. We then see that it is the denominator in our formula which ensures that no velocity can ever be measured to be greater than light.

In the next part the third object is moving in the opposite direction as seen by $S'$, that is, $\beta_2'=-.47$ now. But he calculations are the same:

$$\beta_2=\frac{-.47+.62}{1-(-.47)(.62)}=.21,$$

while the classical calculation is

$$\beta_2=-.47+.62=.15,$$

and agrees more closely with the relativistic calculation since the speed is not as close to the speed of light.