Question

A meteor made of pure kryptonite (yes, we know that there really isn't such a thing as kryptonite ... ) is moving toward Earth. If the meteor eventually hits Earth, the impact will cause severe damage, threatening life as we know it. If a laser hits the meteor with light of wavelength \(560 \mathrm{nm}\), the meteor will blow up. The only laser on Earth powerful enough to hit the meteor produces light with a 532 -nm wavelength. Scientists decide to launch the laser in a spacecraft and use special relativity to get the right wavelength. The meteor is moving very slowly, so there is no correction for relative velocities. At what speed does the spaceship need to move so that the laser light will have the right wavelength, and should it travel toward or away from the meteor?

Short answer

Answer: The spaceship should travel at a speed of approximately 14.6 x 10^6 m/s away from the meteor for the laser light to have the desired wavelength of 560 nm in the meteor's frame.

Step-by-step solution

Express the problem using the relativistic Doppler effect formula.

First, let's recall the relativistic Doppler effect formula, which states: $$\frac{\lambda_{observed}}{\lambda_{emitted}} = \sqrt{\frac{1 + \beta}{1 - \beta}}$$ where \(\lambda_{observed}\) is the observed wavelength, \(\lambda_{emitted}\) is the emitted wavelength, and \(\beta = \frac{v}{c}\) is the ratio of the spacecraft's velocity (\(v\)) to the speed of light (\(c\)). In our case, \(\lambda_{emitted} = 532 \, nm\), and \(\lambda_{observed} = 560 \, nm\). We need to find the value of \(v\), and determine if it's positive (spaceship moving towards the meteor) or negative (spaceship moving away from the meteor).

Rearrange the formula to solve for \(\beta\)

We want to solve the relativistic Doppler effect formula for \(\beta\): $$\beta = \frac{v}{c} = \frac{\left(\frac{\lambda_{observed}}{\lambda_{emitted}}\right)^2 - 1}{\left(\frac{\lambda_{observed}}{\lambda_{emitted}}\right)^2 + 1}$$

Plug in the known values and solve for \(\beta\)

Now let's plug in the known values for \(\lambda_{observed} = 560 \, nm\) and \(\lambda_{emitted} = 532 \, nm\): $$\beta = \frac{\left(\frac{560}{532}\right)^2 - 1}{\left(\frac{560}{532}\right)^2 + 1}$$ Calculating this, we find: $$\beta \approx 0.0487$$

Calculate the spaceship's velocity \(v\)

Now that we have found the value of \(\beta\), we can find the required velocity \(v\) for the spaceship by multiplying \(\beta\) by the speed of light \(c \approx 3 \times 10^8 \, m/s\): $$v = \beta c \approx 0.0487 \times 3 \times 10^8 \, m/s \approx 14.6 \times 10^6 \, m/s$$

Determine the direction of the spaceship's motion

Since \(\lambda_{observed} > \lambda_{emitted}\), the observed light is redshifted (has a larger wavelength), which implies that the spaceship should move away from the meteor. This increases the wavelength of the emitted light as observed from the meteor's frame. In conclusion, the spaceship needs to travel at a speed of about \(14.6 \times 10^6 \, m/s\) away from the meteor for the laser light to have the desired wavelength of 560 nm in the meteor's frame.