Question

Greg is cruising down the street in his new Ferrari SRS (Special Relativistic Series) 2000, when he absent-mindedly drives straight through a red light. Unfortunately for him, officer Buckingham is sitting at the intersection and, upon witnessing the infraction, pulls him over. When asked why he decided to run the red light, Greg — a knowledgeable astrophysi- cist and renowned smooth-talker — responds, “Officer, I do apologize for acting in such blatant disregard of the law. However, you must understand that, while you saw me pass through a red light, the light actually appeared green to me due to our relative motion. I was thus completely unaware that I was breaking any law.”

1. The equation relating the wavelength of light seen by Greg, λG, and the wavelength seen by officer Buckingham, λB, is given by

where v is the velocity of Greg’s Ferrari as measured by officer Buckingham and c = 3.0×108 m/s is the speed of light. Given that the light observed by officer Buckingham was red and thus had a wavelength of λB = 700 nm (the unit nm = nanometers, i.e., billionths of a meter, which is often used to measure wavelengths), while the light observed by Greg was green and had a wavelength of λG = 500 nm, calculate the speed v at which Greg was traveling (leave your answer in units of c and show your work)

2. ) Equation (1) is actually only valid for speeds that are much less than the speed of light. Einstein’s special theory of relativity shows that the true relation — correct even when Greg is moving very near the speed of light — is given by Given that λB = 700 nm and using equation (2), calculate the wavelength that Greg would have observed if he had been moving at 95% the speed of light. How does this compare to the answer you obtain by using equation (1)? Into what regime of the electromagnetic spectrum does this wavelength (the one calculated using equation 2) fall?

Answer 1

Lambda_G = lambda_B (1 - v/c) 500 nm = 700 nm (1 - v/c) (1 -v/c) = 0.7/y = 1 V = 0.286 C lambda_G = 70 ohm squareroot 1 - 0.95/1 + 0.95 (v = 0.95 C) (Given) lambda_G = 112.08 nm (6y eq/n (2)) lambda_G = lambda_B (1 - v/c) = 700 times (1 - 0.95) nm lambda_G = 35 nm (by eq_n (1)) lambda_G (2) > lambda_G (1) lambda_G = 112.08 nm (Near v/travoilet)