Near the center of our galaxy, hydrogen gas is moving directly away from us in its orbit about a black hole. We receive 1900 nm electromagnetic radiation and know that it was 1875 nm when emitted by the hydrogen gas. What is the speed of the gas?

Speed of gas = 3.973 x 10^(6) m/s

Explanation:

From the relativistic doppler effect, the observed frequency is given as;

λobs = λs√[(1 + (v/c))/(1-(v/c))]

Where;

λobs is the wavelength received by the observer

λs is the wavelength of the wave emitted from the source

v is the relative velocity between the observer and the source

c is the speed of light which is 3 x 10^(8) m/s

λobs = 1900 nm

λs = 1875 nm

Let's make the relative velocity(v) the subject of the equation.

λobs = λs√[(1 + (v/c))/(1-(v/c))]

(λobs/λs)²= [(1 + (v/c))/(1-(v/c))]

Let's call (λobs/λs) = p

Thus;

p² = [(1 + (v/c))/(1-(v/c))]

p²[(c-v)/c] = (c+v)/c

Multiply both sides by c to obtain ;

p²(c-v) = c + v

p²c - p²v = c + v

p²c - c = v(p² + 1)

v = c(p²-1)/(p²+1)

Now since p = (λobs/λs)

p = 1900/1875 = 1.0133

So, v = 3x10^(8) [1.0133²-1]/[1.0133²+1]

= 3.973 x 10^(6) m/s

its c i just checked it