Doppler-shifted frequency problem

[hitman0097]

Homework Statement

A radar "speed gun" emits microwaves of frequency f_o=36.0 GHz. When the gun is pointed at an object moving toward it at speed v, the object senses the microwaves at the Doppler-shifted frequency f. The moving object reflects these microwaves at this same frequency f. The stationary radar apparatus detects these reflected waves at a Doppler-shifted f'. The gun combines its emitted wave at f_o and its detected wave at f'. These waves interfere, creating a beat pattern whose beat frequency is f_beat=f'-f_o.

a.) Show that: v≈cf_beat/2f_o,

if f_beat <<f_o. If f_beat=6670 Hz, what is v(km/h)?

b.) If the object's speed is different by v, show that the difference in in beat frequency Δf_beat is given by : Δf_beat=2f_oΔv / c

c.) If the accuracy of the speed gun is to be 1 km/h, to what accuracy must the beat frequency be measured?

Homework Equations

f=c/λ = f_o[(c+v/c-v)^1/2]
λ=λ_o[(c-v/c+v)^1/2]
1/f_o=λ_o/c

The Attempt at a Solution

I am kind of lost on this problem, I think I heard my teacher say I'd need a double doppler shift. Any help and hints are appreciated!

Thanks

 

[hitman0097]

The first Doppler shift that occurs should be in the form
[ f=36.0 G hz [(c+v)/(c-v)]^.5]

The 2nd Doppler shift would be in the form of the first Doppler shift
[ f'=(f [(c+v/c-v)]^.5)] = 36 Ghz [(c+v)/(c-v)]

fbeat can be related in terms of f not [fbeat = 36.0 Ghz [(c+v)/(c-v)] - 36.0 Ghz]

 

[hitman0097]

Still confused to as v= c (fbeat)/ 2 fo If: fbeat<<fo. Also, I think the last fbeat is not correctly expressed or related by me. Also getting getting 27.34 m/s for part (a). Where the answer in the back of my book is 1.00 x 10^2 km/hr

 

[Simon Bridge]

There is no difference between the moving-source and moving-observer doppler-shift expressions for light. For waves that need a medium, the wave-speed gets measured wrt the medium which is why there is a difference there.

The usual idea is that a moving car reflects the waves as it receives them - so the reflected wavelength is shifted wrt the transmitted one.
The vehicle then acts as a source for the reflected wave.

https://www.physicsforums.com/showthread.php?t=223186
... subs in water - sonar doppler-shift illustrating the double-shift.

https://www.physicsforums.com/archive/index.php/t-335219.html
... for radar gun!

Not following your reasoning - what is the relationship between the beat frequency and the doppler shifted frequency?
We need to see how you are getting these results in order to help you properly.