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Chapter 26: Problem 68

Suppose that as you travel away from Earth in a spaceship, you observe another ship pass you heading in the same direction and measure its speed to be $0.50 c .$ As you look back at Earth, you measure Earth's speed relative to you to be \(0.90 c .\) What is the speed of the ship that passed you according to Earth observers?

Short Answer

Expert verified
Answer: The speed of the ship that passed you according to Earth observers is approximately \(0.966c\).

Step by step solution

01

List the known values

We know the following speeds: - Speed of the spaceship relative to Earth: \(u = 0.90c\) - Speed of the other ship relative to you: \(v = 0.50c\) - Speed of light: \(c = 1\) (in natural units)
02

Substitute the values into the relativistic velocity addition formula

Now, we can substitute the values into the formula: $$ w = \frac{0.90c + 0.50c}{1 + \frac{(0.90c)(0.50c)}{c^2}} $$
03

Simplify the equation

To simplify the equation, we can cancel out \(c\) since it's equal to \(1\) in natural units: $$ w = \frac{0.90 + 0.50}{1 + (0.90)(0.50)} $$
04

Calculate the result

Now, we can perform the calculations: $$ w = \frac{1.40}{1 + 0.45} = \frac{1.40}{1.45} $$
05

Convert the result to the speed of light

To convert the result back to the speed of light, we can multiply by \(c\) (in this case, \(c=1\), so no need to multiply): $$ w = \frac{1.40}{1.45}c $$
06

Final answer

The speed of the ship that passed you according to Earth observers is: $$ w \approx 0.966c $$

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Most popular questions from this chapter

A neutron (mass 1.00866 u) disintegrates into a proton (mass $1.00728 \mathrm{u}\( ), an electron (mass \)0.00055 \mathrm{u}$ ), and an antineutrino (mass 0 ). What is the sum of the kinetic energies of the particles produced, if the neutron is at rest? $\left(1 \mathrm{u}=931.5 \mathrm{MeV} / \mathrm{c}^{2} .\right)$
A lambda hyperon \(\Lambda^{0}\) (mass \(=1115.7 \mathrm{MeV} / \mathrm{c}^{2}\) ) at rest in the lab frame decays into a neutron \(n\) (mass \(=\) $939.6 \mathrm{MeV} / c^{2}\( ) and a pion (mass \)=135.0 \mathrm{MeV} / c^{2}$ ): $$\Lambda^{0} \rightarrow \mathrm{n}+\pi^{0}$$ What are the kinetic energies (in the lab frame) of the neutron and pion after the decay? [Hint: Use Eq. \((26-11)\) to find momentum.]
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.
Particle \(A\) is moving with a constant velocity \(v_{\mathrm{AE}}=+0.90 c\) relative to an Earth observer. Particle \(B\) moves in the opposite direction with a constant velocity \(v_{\mathrm{BE}}=-0.90 c\) relative to the same Earth observer. What is the velocity of particle \(B\) as seen by particle \(A ?\)
Event A happens at the spacetime coordinates $(x, y, z, t)=(2 \mathrm{m}, 3 \mathrm{m}, 0,0.1 \mathrm{s})$ and event B happens at the spacetime coordinates \((x, y, z, t)=\left(0.4 \times 10^{8} \mathrm{m}\right.\) $3 \mathrm{m}, 0,0.2 \mathrm{s}) .$ (a) Is it possible that event A caused event B? (b) If event B occurred at $\left(0.2 \times 10^{8} \mathrm{m}, 3 \mathrm{m}, 0,0.2 \mathrm{s}\right)$ instead, would it then be possible that event A caused event B? [Hint: How fast would a signal need to travel to get from event \(\mathrm{A}\) to the location of \(\mathrm{B}\) before event \(\mathrm{B}\) occurred?]
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