Step 1: Understand the colour change.
When you drive fast toward a light, its wavelength looks shorter (it shifts toward blue). Red light has a longer wavelength than green, so moving toward the red signal can make it look green.
Step 2: Write the Doppler shift for light.
For a source approaching, the change in wavelength is: \[ \Delta\lambda = \lambda\,\frac{v}{c} \] where $v$ is the car speed and $c = 3\times10^{8}\ ms^{-1}$.
Step 3: Find the needed wavelength change.
The red wavelength is $6200$ and the green is $5400$ (in angstrom units): \[ \Delta\lambda = 6200 - 5400 = 800 \]
Step 4: Rearrange for the speed.
\[ \frac{\Delta\lambda}{\lambda} = \frac{v}{c} \;\Rightarrow\; v = c\,\frac{\Delta\lambda}{\lambda} \]
Step 5: Put in the numbers.
Using the original red wavelength as $\lambda = 6200$: \[ v = (3\times10^{8})\frac{800}{6200} = (3\times10^{8})\frac{4}{31} \]
Step 6: Work out the value.
\[ v = \frac{12\times10^{8}}{31} \approx 3.9\times10^{7}\ ms^{-1} \] \[ \boxed{v \approx 3.9\times10^{7}\ ms^{-1}} \]