Question

You may be familiar with the bumper sticker below, the punch line of which is based on the Doppler effect. Assuming the bumper sticker is affixed to a parked car, and that the red color corresponds to a wavelength of 650 nm, how fast would you have to be traveling for the sticker to appear blue (lambda = 475 nm)?

Answer 1

Blue shift equation for wavelength is given as

$\\ \lambda_{obs}=\frac{\lambda_{0}}{\sqrt{\frac{1+\beta}{1-\beta}}} \\ Where \ \beta=\frac{v}{c} \\ where, v=speed \ of \ body \ ( + \ if \ body \ is \ approaching \, -if \ body \ is \ moving \ away) \\ for \ blue \ shift \ we \ consider \ approaching \ body \\ \sqrt{\frac{1+\beta}{1-\beta}}=\frac{\lambda}{\lambda_{obs}} \\ \frac{1+\beta}{1-\beta}=\frac{650}{475}\Rightarrow 1.873 \\ \beta=0.304 \\ \beta=\frac{v}{c} \\ or, v=0.912\times 10^{8} \ m/s \\ v=0.304c$