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29P

Expert-verifiedFound in: Page 1147

Book edition
10th Edition

Author(s)
David Halliday

Pages
1328 pages

ISBN
9781118230718

**Galaxy A is reported to be receding from us with a speed of 0.35c. Galaxy B, located in precisely the opposite direction, is also found to be receding from us at this same speed. What multiple of c gives the recessional speed an observer on Galaxy A would find for (a) our galaxy and (b) Galaxy B?**

- The multiple of c is 0.35.
- The multiple of c is 0.62.

**The relativistic velocity of the particle is given by,**

** **

** ${\mathbf{u}}{\mathbf{=}}\frac{{\mathbf{u}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{v}}{\mathbf{1}\mathbf{+}\frac{{\mathbf{u}}^{\mathbf{\text{'}}}\mathbf{v}}{{\mathbf{c}}^{\mathbf{2}}}}$**

** **

**Here, u is the velocity of the particle as measured in S, u^{'} is the velocity of the particle as measured in S^{'}, v is the velocity of the S^{'} relative to S, and c is the velocity of the light.**

If a person P is moving with speed v eastward from the person Q, then person P should observe that the person Q moving v away from person P.

In the similar way, if the Galaxy A is moving from our Galaxy with a speed of 0.35c then the observer in Galaxy A must see our Galaxy with the same speed of 0.35c or 0.35 multiple of c away from him.

Therefore the speed of an observer on Galaxy A from our galaxy is 0.35 multiples of c.

Rearrange the equation (1) as follows.

$\begin{array}{rcl}\mathrm{u}\left(1+\frac{{\mathrm{u}}^{\text{'}}\mathrm{v}}{{\mathrm{c}}^{2}}\right)& =& {\mathrm{u}}^{\text{'}}+\mathrm{v}\\ \mathrm{u}-\mathrm{v}& =& {\mathrm{u}}^{\text{'}}\left(1-\frac{\mathrm{uv}}{{\mathrm{c}}^{2}}\right)\\ \frac{\mathrm{u}\text{'}}{\mathrm{c}}& =& \frac{\frac{\mathrm{u}}{\mathrm{c}}-\frac{\mathrm{v}}{\mathrm{c}}}{1-\frac{\mathrm{uv}}{{\mathrm{c}}^{2}}}.......\left(2\right)\end{array}$

Substitute -0.35c for u and 0.35c for v in equation (2).

$\begin{array}{rcl}\frac{{\mathrm{u}}^{\text{'}}}{\mathrm{c}}& =& \frac{\frac{-0.35\mathrm{c}}{\mathrm{c}}-\frac{0.35}{\mathrm{c}}}{1-\frac{\left(-0.35\mathrm{c}\right)\left(0.35\mathrm{c}\right)}{{\mathrm{c}}^{2}}}\\ \frac{{\mathrm{u}}^{\text{'}}}{\mathrm{c}}& =& \frac{-2\left(0.35\right)}{1+{\left(0.35\right)}^{2}}\\ \frac{{\mathrm{u}}^{\text{'}}}{\mathrm{c}}& =& -0.62\\ {\mathrm{u}}^{\text{'}}& =& -0.62\mathrm{c}\end{array}$

Here, the negative sign indicates the direction of the galaxy B with respect to the observer in galaxy A.

Therefore, the multiple of c is 0.62.

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