Answer to Question #205288 in Mechanics | Relativity for Safi Ullah

Question #205288

A point moves with constant speed t in a direction which forms an angle with the x-axis of the frame of reference S'. Frame S' moves with a velocity V = Và relative to another frame of reference Ss. What is the angle 0 formed by the direction of motion of the point with the x-axis of S? What is the relationship between the two angles as v - c? f sin sin d Ans.: tan 0 = tan e


1
Expert's answer
2021-06-11T11:35:16-0400

Solution:-



Wave from equation

"\\psi=Ae^{{i2\\pi\\upsilon }{(t-\\frac{r}{c})}}" ="Ae^{i2\\pi\\upsilon(t-\\frac{(xcos\\theta+ysin\\theta)}{c})}"

Similarly S' co-ordinate system

"\\psi'=A'e^{{i2\\pi\\upsilon '}{(t'-\\frac{r'}{c})}}" ="A'e^{i2\\pi\\upsilon '(t'-\\frac{(x'cos\\theta'+y'sin\\theta')}{c})}"

Number of frames does not depend motion of frame

"\\upsilon (t-\\frac{r}{c})=\\upsilon'(t'-\\frac{r'}{c})"


"\\upsilon(t-\\frac{xcos\\theta+ysin\\theta}{c})=\\upsilon'(t'-\\frac{x'cos\\theta'+y'sin\\theta'}{c})"

x,y,t value put


"\\upsilon(\\gamma(t'+\\frac{\\beta x'}{c})-\\frac{\\gamma (x'+\\beta ct')cos\\theta +y'sin\\theta}{c})=\\upsilon'(t'-\\frac{x'cos\\theta'+y'sin\\theta'}{c})"

Comparison x' components

"\\gamma v(cos\\theta-\\beta)=\\upsilon'cos\\theta'"

Comperision of y' cofficient

"vsin\\theta=v'sin\\theta'"

Comperision of t' components

"\\gamma v(1-\\beta cos\\theta)=\\upsilon'"

"\\upsilon =\\frac{\\upsilon'}{\\gamma(1-\\beta cos\\theta)}"

equation (1)and(2)

"tan\\theta'=\\frac{sin\\theta}{\\gamma(cos\\theta-\\beta)}"


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