Question

A spaceship is moving away from Earth at speed 0.20c. A source on the rear of the ship emits light at wavelength 450 nm according to someone on the ship. What (a) wavelength and (b) color (blue, green, yellow, or red) are detected by someone on Earth watching the ship?

Short answer

(a) The radial speed NGC7319 relative to Earth is 0.023c.

(b) The color of light detected on Earth is Yellow.

Step-by-step solution

Identification of given data

The speed of spaceship is $\mathrm{v}=0.20\mathrm{c}$

The wavelength of light emitted from rear end of ship is ${\mathrm{\lambda }}_{\mathrm{o}}=450\mathrm{nm}$

The speed for the variation in the colour of light from red to green is found by using the formula for Doppler shift.

Determination of wavelength of light detected on Earth

(a)

The wavelength of light detected on Earth is given as:

$\mathrm{\lambda }={\mathrm{\lambda }}_{\mathrm{o}}\sqrt{\frac{1+\left({\displaystyle \frac{\mathrm{v}}{\mathrm{c}}}\right)}{1-\left({\displaystyle \frac{\mathrm{v}}{\mathrm{c}}}\right)}}$

Here, c is the speed of light.

Substitute all the values in the above equation.

$\begin{array}{rcl}\mathrm{\lambda }& =& \left(450\mathrm{nm}\right)\sqrt{\frac{1+\left({\displaystyle \frac{0.20\mathrm{c}}{\mathrm{c}}}\right)}{1-\left({\displaystyle \frac{0.20\mathrm{c}}{\mathrm{c}}}\right)}}\\ & =& 550\mathrm{nm}\end{array}$

Therefore, the wavelength of light detected on Earth is 550 nm.

Identification color of light detected on Earth

(b)

The wavelength of the light detected on Earth falls under the yellow color of the visible spectrum so the color of light detected on Earth is Yellow.