Solving Doppler Shift to Calculate Cyclist's Speed

[Beatdiz]

Homework Statement

A man riding his bicycle was caught driving through a red traffic light. The man was taken to court and trialled where he claimed he was cycling so quickly that the light had appeared to be green to him due to the Doppler effect. The prosecution accepted his excuse but decided to find him 1 Euro for every km hr by which the cyclist was traveling at.

Combine and re-arrange the following two equations to give an equation that relates to the speed of the cyclist to the shift in the wavelength of the light emitted by the traffic light.

Homework Equations

z=[Δλ] / [λ0] ... Where z is the red/blueshift, λ0 is the original wavelength, Δλ is the change in wavelength

v= z x c ... Where v is the speed of the galaxy, c is the speed of light, z is as above.

The Attempt at a Solution

s = Speed of motorist

s = [(c)(Δλ/λ0)2-c] / [(Δλ/λ0)2+1]

Does my equation look correct as I'm not convinced? Any help would be great.

 

[Beatdiz]

Is anyone able to help with this please? :)

 

[I like Serena]

The Attempt at a Solution[/h2]
s = Speed of motorist

s = [(c)(Δλ/λ0)2-c] / [(Δλ/λ0)2+1]

Does my equation look correct as I'm not convinced? Any help would be great.

Hi Beatdiz, welcome to PF!

I do not understand how you arrived at your equation for s.
In your case you do not have to work with "the speed of the galaxy v".
But your v is simply the speed of the bicyclist.
No need to introduce a new symbol s, of which I do not understand what you did with it.

So your equation should simply read:
v = (Δλ/λ0) x c

Cheers!

 

[Quinzio]

Is anyone able to help with this please? :)

I think you need to start with more realistic examples, like that one:
in a train station, one train arrives exactly each $$n$$ minutes, always in the same direction.
All the train travels at $$v$$ speed
Now you take a train in the opposite direction running at $$v'$$ , you cross a train in the opposite direction each $$n'$$ minutes.

State $$n'$$ as a function of the other data.

 

[rwisz]

I would first like to point out that the equations provided are for calculating the speed of a galaxy, not a cyclist. However, the general concept of using the Doppler effect to calculate speed is applicable in both cases.

To solve this problem, we can use the equation v = z x c, where v is the speed of the cyclist, z is the red/blueshift, and c is the speed of light. In this case, we are looking for the speed of the cyclist, so we can rearrange the equation to solve for v:

v = z x c

We can then substitute the value for z from the first equation, z = (Δλ/λ0), into the second equation:

v = (Δλ/λ0) x c

Now, we need to rearrange this equation to solve for the speed of the cyclist. We can do this by multiplying both sides by λ0 and then dividing by Δλ:

v = (Δλ/λ0) x c

v = c x (Δλ/λ0)

v = c x (Δλ/λ0) / Δλ

v = c x (1/λ0)

Finally, we can substitute the values for the original wavelength, λ0, and the change in wavelength, Δλ, to get our final equation:

v = c / (λ0 + Δλ)

This equation relates the speed of the cyclist to the shift in the wavelength of the light emitted by the traffic light. To find the speed of the cyclist, we would need to know the original wavelength of the light and the change in wavelength caused by the Doppler effect.