Two spacecraft A and B are moving in opposite directions, as shown in Figure 39.15. An observer on the Earth measures the speed of craft A to be 0.750c and the speed of craft B to be 0.850c. Find the velocity of craft B as observed by the crew on craft A.

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To conceptualize this problem, we carefully identify the observers and the event. The two observers are on the Earth and on spacecraft A. The event is the motion of spacecraft B. Because the problem asks to find an observed velocity, we categorize this problem as one requiring the Lorentz velocity transformation. To analyze the problem, we note that the Earth observer makes two measurements, one of each spacecraft. We identify this observer as being at rest in the S frame. Because the velocity of spacecraft B is what we wish to measure, we identify the speed [latex] u_{x} ext { as }-0.850 c [/latex]. The velocity of spacecraft A is also the velocity of the observer at rest in the S" frame, which is attached to the spacecraft, relative to the observer at rest in S. Thus, v ! 0.750c. Now we can obtain the velocity [latex] u_{x}^{\prime} [/latex] of craft B relative to craft A
by using Equation 39.16:

[latex] u_{x}^{\prime}=\frac{u_{x}-v}{1-\frac{u_{x} v}{c^{2}}} [/latex] (39.16).

[latex] u_{x}^{\prime}=\frac{u_{x}-v}{1-\frac{u_{x} v}{c^{2}}}=\frac{-0.850 c-0.750 c}{1-\frac{(-0.850 c)(0.750 c)}{c^{2}}} [/latex].

[latex] =-0.977 c [/latex].

To finalize this problem, note that the negative sign indicates that craft B is moving in the negative x direction as observed by the crew on craft A. Is this consistent with your expectation from Figure 39.15? Note that the speed is less than c. That is, an object whose speed is less than c in one frame of reference must have a speed less than c in any other frame. (If the Galilean velocity transformation equation were used in [latex] ext { this example, we would find that } u_{x}^{\prime}=u_{x}-v=-0.850 c- [/latex][latex] 0.750 c=-1.60 c [/latex], which is impossible. The Galilean transformation equation does not work in relativistic situations.)

Question 39.7: Two spacecraft A and B are moving in opposite directions, as...

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