Answer:
the observer can conclude that the object is moving with a radial velocity of [tex]vs = 2.76 * 10^8 m/s[/tex]
Explanation:
In relation to Doppler effect for light, the formula can be represented as:
[tex]\lambda = \frac {\lambda_0\sqrt{(1 - \beta)}} { \sqrt{(1 + \beta)}}[/tex]
where,
[tex]\lambda[/tex] = wavelength of the light emitted by an object = 607.5 nm
[tex]\lambda_0[/tex] = wavelength of ultraviolet Lyman-alpha line of hydrogen by astronomical object = 121.5 nm
[tex]\beta[/tex] =[tex]\frac { vs }{c}[/tex]
It is clear that the 'positive sign usually denotes "approaching" and the 'negative sign usually denotes "receding".
However, Since the object and the source are receding. Then, we have :
[tex]\lambda = \frac {\lambda_0 (1 + \beta)^{1/2}}{ (1 - \beta)^{1/2}}[/tex]
[tex]\frac {\lambda }{ \lambda_0} = \frac{(1 + \beta)^1/2 }{1 - \beta^1/2}[/tex]
[tex]\frac {607.5 \ nm }{ 121.5 \ nm} = \frac{(1 + \beta)^1/2 }{1 - \beta^1/2}[/tex]
[tex]5 = \frac{(1 + \beta)^1/2 }{1 - \beta^1/2}[/tex]
Squaring on both sides & we have:
[tex]25 = \frac{1 + \beta }{1 - \beta}[/tex]
[tex]25*{(1 - \beta)} = {1 + \beta }[/tex]
[tex]25- 25 \beta = {1 + \beta }[/tex]
[tex]- 25 \beta - \beta = 1 -25[/tex]
[tex]- 26 \beta = -24[/tex]
[tex]\beta = \frac{ -24}{- 26 }[/tex]
[tex]\beta = \frac{ 12}{13 }[/tex]
[tex]\frac{vs}{c} = \frac{ 12}{13 }[/tex]
[tex]\frac{vs}{c} = 0.9231[/tex]
[tex]{vs} = 0.9231 *{c}[/tex]
[tex]{vs} = 0.9231 * 3*10^8 \ m/s[/tex]
[tex]{vs} = 276930000[/tex]
[tex]vs = 2.76 * 10^8 m/s[/tex]
Therefore; the observer can conclude that the object is moving with a radial velocity of [tex]vs = 2.76 * 10^8 m/s[/tex]
