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Chapter 20: Problem 18

It is concluded from measurements of the red shift of the emitted light that quasar \(Q_{1}\) is moving away from us at a speed of \(0.788 c\). Quasar \(Q_{2}\), which lies in the same direction in space but is closer to us, is moving away from us at speed \(0.413 c\). What velocity for \(Q_{2}\) would be measured by an observer on \(Q_{1}\) ?

Short Answer

Expert verified
To find out the relative speed of the quasar Q2 with respect to Q1, apply the Einstein's velocity addition formula. Calculate the relative velocity using these values, and that'll give the answer needed.

Step by step solution

01

Understand the Concept of Relative Velocity

The velocity of an object as observed from a different moving object (also called relative velocity) can be calculated. In classical physics, this is simple subtraction. However, when objects are moving with velocity close to the speed of light, this classical formula fails and we should use the formula from the theory of relativity.
02

Apply the Einstein's Velocity Addition Formula

The Einstein's velocity addition formula is written as \(v_{r} = \frac{v_{1} + v_{2}}{1 + \frac{v_{1}v_{2}}{c^{2}}}\), where \(v_{r}\) is the relative velocity of the two objects, \(v_{1}\) is the velocity of the first object, \(v_{2}\) is the velocity of the second object. Here, \(v_{1} = 0.788 c\) (velocity of \(Q_{1}\)) and \(v_{2} = -0.413 c\) (opposite direction of velocity of \(Q_{2}\)).
03

Calculate Relative Velocity

Now that we know the velocities of \(Q_{1}\) and \(Q_{2}\) and their directions, we can get the relative velocity \(v_{r} = \frac{0.788c - 0.413c}{1 - \frac{0.788c * -0.413c}{c^{2}}}\). Solving this equation will give us the relative velocity of \(Q_{2}\) with respect to \(Q_{1}\).

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Most popular questions from this chapter

A space traveler takes off from Earth and moves at speed \(0.988 c\) toward the star Vega, which is \(26.0\) light-years distant. How much time will have elapsed by Earth clocks \((a)\) when the traveler reaches Vega and \((b)\) when the Earth observers receive word from him that he has arrived? ( \(c\) ) How much older will the Earth observers calculate the traveler to be when he reaches Vega than he was when he started the trip?

A clock moves along the \(x\) axis at a speed of \(0.622 c\) and reads zero as it passes the origin. (a) Calculate the Lorentz factor. (b) What time does the clock read as it passes \(x=183 \mathrm{~m}\) ?

Find the speed parameter of a particle that takes 2 years longer than light to travel a distance of \(6.0\) ly.

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What must be the value of the speed parameter \(\beta\) if the Lorentz factor \(\gamma\) is to be \((a) 1.01 ?(b) 10.0 ?(c) 100 ?(d)\) \(1000 ?\)
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