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Chapter 26: Problem 29

Rocket ship Able travels at \(0.400 c\) relative to an Earth observer. According to the same observer, rocket ship Able overtakes a slower moving rocket ship Baker that moves in the same direction. The captain of Baker sees Able pass her ship at \(0.114 c .\) Determine the speed of Baker relative to the Earth observer.

Short Answer

Expert verified
Answer: The velocity of rocket ship Baker relative to the Earth observer is approximately \(0.273c\).

Step by step solution

01

Write down the problem information

We have the following information: - Velocity of rocket ship Able relative to Earth Observer, \(v_A = 0.400c\) - Velocity of rocket ship Able relative to rocket ship Baker, \(v_{AB} = 0.114c\) We need to find the velocity of rocket ship Baker relative to Earth Observer, \(v_B\).
02

Write down the relativistic velocity addition formula

The relativistic velocity addition formula is given by \(v_{AB} = \cfrac{v_A - v_B}{1 - \cfrac{v_A v_B}{c^2}}\)
03

Rearrange the formula to isolate \(v_B\)

To isolate \(v_B\), we can first multiply both sides by the denominator: \(v_{AB}(1 - \cfrac{v_A v_B}{c^2}) = v_A - v_B\) Next, expand the left side and simplify: \((v_{AB} - \cfrac{v_{AB} v_A v_B}{c^2}) = v_A - v_B\) Now, move the terms with \(v_B\) to the left side and all other terms to the right side: \(v_B + \cfrac{v_{AB} v_A v_B}{c^2} = v_A - v_{AB}\) factor \(v_B\) from the left side: \(v_B(1 + \cfrac{v_{AB} v_A}{c^2}) = v_A - v_{AB}\) Finally, divide by \((1 + \cfrac{v_{AB} v_A}{c^2})\) to isolate \(v_B\): \(v_B = \cfrac{v_A - v_{AB}}{1 + \cfrac{v_{AB} v_A}{c^2}}\)
04

Substitute the given values and solve for \(v_B\)

Substitute \(v_A = 0.400c\) and \(v_{AB} = 0.114c\) into the formula for \(v_B\) and solve: \(v_B = \cfrac{0.400c - 0.114c}{1 + \cfrac{0.114c \cdot 0.400c}{c^2}}\) \(v_B = \cfrac{0.286c}{1 + \cfrac{0.0456c^2}{c^2}}\) \(v_B = \cfrac{0.286c}{1 + 0.0456}\) \(v_B = \cfrac{0.286c}{1.0456}\) \(v_B \approx 0.273c\)
05

Write down the final answer

The velocity of rocket ship Baker relative to the Earth observer is approximately \(0.273c\).

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Most popular questions from this chapter

As observed from Earth, rocket Alpha moves with speed \(0.90 c\) and rocket Bravo travels with a speed of \(0.60 c\) They are moving along the same line toward a head-on collision. What is the speed of rocket Alpha as measured from rocket Bravo? (tutorial: adding velocities)
Derivation of the Doppler formula for light. A source and receiver of EM waves move relative to one another at velocity \(v .\) Let \(v\) be positive if the receiver and source are moving apart from one another. The source emits an EM wave at frequency \(f_{\mathrm{s}}\) (in the source frame). The time between wavefronts as measured by the source is \(T_{\mathrm{s}}=1 / f_{\mathrm{s}}\) (a) In the receiver's frame, how much time elapses between the emission of wavefronts by the source? Call this \(T_{\mathrm{T}}^{\prime} .\) (b) \(T_{\mathrm{T}}^{\prime}\) is not the time that the receiver measures between the arrival of successive wavefronts because the wavefronts travel different distances. Say that, according to the receiver, one wavefront is emitted at \(t=0\) and the next at \(t=T_{\mathrm{o}}^{\prime} .\) When the first wavefront is emitted, the distance between source and receiver is \(d_{\mathrm{r}}\) When the second wavefront is emitted, the distance between source and receiver is \(d_{\mathrm{r}}+v T_{\mathrm{r}}^{\prime} .\) Each wavefront travels at speed \(c .\) Calculate the time \(T_{\mathrm{r}}\) between the arrival of these two wavefronts as measured by the receiver. (c) The frequency detected by the receiver is \(f_{\mathrm{r}}=1 / T_{\mathrm{r}} .\) Show that \(f_{\mathrm{r}}\) is given by $$f_{\mathrm{r}}=f_{\mathrm{s}} \sqrt{\frac{1-v / \mathrm{c}}{1+v / \mathrm{c}}}$$
Kurt is measuring the speed of light in an evacuated chamber aboard a spaceship traveling with a constant velocity of \(0.60 c\) with respect to Earth. The light is moving in the direction of motion of the spaceship. Siu- Ling is on Earth watching the experiment. With what speed does the light in the vacuum chamber travel, according to Siu-Ling's observations?
A rectangular plate of glass, measured at rest, has sides \(30.0 \mathrm{cm}\) and \(60.0 \mathrm{cm} .\) (a) As measured in a reference frame moving parallel to the 60.0 -cm edge at speed \(0.25 c\) with respect to the glass, what are the lengths of the sides? (b) How fast would a reference frame have to move in the same direction so that the plate of glass viewed in that frame is square?
A charged particle is observed to have a total energy of \(0.638 \mathrm{MeV}\) when it is moving at \(0.600 c .\) If this particle enters a linear accelerator and its speed is increased to \(0.980 c,\) what is the new value of the particle's total energy?
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