# Chapter 20 - Traveling Waves - Exercises and Problems: 75

The galaxy is receding from the earth with a speed of $1.5\times 10^6~m/s$

#### Work Step by Step

Since the wavelength of the light from the galaxy is longer than the wavelength measured in a laboratory, the galaxy is receding from the earth. We can find the speed of the galaxy as: $\lambda' = \lambda_0~\frac{\sqrt{1+v/c}}{\sqrt{1-v/c}}$ $1.005~\lambda_0 = \lambda_0~\frac{\sqrt{1+v/c}}{\sqrt{1-v/c}}$ $1.005 = \frac{\sqrt{1+v/c}}{\sqrt{1-v/c}}$ $(1.005)^2 = \frac{1+v/c}{1-v/c}$ $(1.005)^2(c-v) = c+v$ $(1.005)^2~v + v = (1.005)^2~c - c$ $v = \frac{(1.005)^2 - 1}{(1.005)^2 + 1}~c$ $v = \frac{(1.005)^2 - 1}{(1.005)^2 + 1}~(3.0\times 10^8~m/s)$ $v = 1.5\times 10^6~m/s$ The speed of the galaxy is thus $1.5\times 10^6~m/s$.

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