How to Calculate the Speed of a Moving Object Using Radar and Beat Frequency?

[k77i]

Homework Statement

A radar device emits microwaves with a frequency of 3.20e+09 Hz. When the waves are reflected from a van moving directly away from the emitter, the beat frequency between the source wave and the reflected wave is 838 beats per second. What is the speed of the van? (Note: microwaves, like all forms of electromagnetic radiation, propagate at the speed of light c = 3.00e+08 m/s.)

Homework Equations

frequency of beat = |f1-f2|

f' = [(v-vo)/v]fo
where v is the speed of emitted frequency (the speed of light on this case), vo is the speed of the observer and fo is the frequency of the observer
The above equation is for when the observer is moving away from the source and the source is stationary

The Attempt at a Solution

First i found the frequency of the observer (van) by using the first equation:

f(beat) = |f1-f2|
838 = |3.20e+09 - f2|
So f2 = 3 199 999 162

Then i used the second equation to find v0 (speed of the van):

f' = [(v-vo)/v]fo
3 199 999 162 = [(3.00e+08 - vo)3.00e+08]3.20e+09
this gave me vo = 78.56m/s, but according to the homework it's not right

Is it because I am not supposed to consider f2 as f'? If not I am not exactly sure how to solve for vo.

 

[AtticusFinch]

There are actually two doppler shifts here. Since these are radio waves I'm not sure if you can use the classical doppler equations (due to relativity).

The van is first a moving observer. The incident frequency is reflected at some frequency f'. You only know the incident frequency.

Now the van becomes a moving source of reflected microwaves (can you see why?) Apply the appropriate doppler equation using the f' from earlier as your incident frequency and 3 199 999 162 as your observed frequency.

 

[k77i]

The van is first a moving observer. The incident frequency is reflected at some frequency f'. You only know the incident frequency.

Now the van becomes a moving source of reflected microwaves (can you see why?) Apply the appropriate doppler equation using the f' from earlier as your incident frequency and 3 199 999 162 as your observed frequency.

ok so i get why the van is first the observer then becomes the source.
so if i do it like that, I should set up the first equation as:

f' = [(v-vo)/v]fo = [(1 - vo/3.00*10^8)]3.2*10^9

Is this right? because then i still didn't find the f' from the first equation which you're telling me to use as the incident frequency (fo) in the second equation when the van is the source

 

[AtticusFinch]

ok so i get why the van is first the observer then becomes the source.
so if i do it like that, I should set up the first equation as:

f' = [(v-vo)/v]fo = [(1 - vo/3.00*10^8)]3.2*10^9

Is this right? because then i still didn't find the f' from the first equation which you're telling me to use as the incident frequency (fo) in the second equation when the van is the source

If you're teacher wants you to ignore relativity you are correct so far.

Now, like you are saying, write another equation for the van as a moving source. The initial frequency is f' and the observed frequency is given as 3 199 999 162. If i remember correctly it should look like this:

3199999162=(v/(v+vs))f'

but you have already expressed f' in terms of vo which is equal to vs.

So through substitution you'll get a equation with only one unknown.

 

[k77i]

If you're teacher wants you to ignore relativity you are correct so far.

Now, like you are saying, write another equation for the van as a moving source. The initial frequency is f' and the observed frequency is given as 3 199 999 162. If i remember correctly it should look like this:

3199999162=(v/(v+vs))f'

but you have already expressed f' in terms of vo which is equal to vs.

So through substitution you'll get a equation with only one unknown.

Alright then basically the final equation should look something like this?
3199999162 = ((9.6*10^17)/(3*10^8+x))-(9.6*10^17x)/(9.6*10^16+3*10^8x), where x is the value of the speed of the van.

Because this gives me 'no solution'

 

[AtticusFinch]

Alright then basically the final equation should look something like this?
3199999162 = ((9.6*10^17)/(3*10^8+x))-(9.6*10^17x)/(9.6*10^16+3*10^8x), where x is the value of the speed of the van.

Because this gives me 'no solution'

Ok that's fine. Remember beats can mean the incident frequency is either lower than the emitted by 838 Hz or higher than the emitted by 838 Hz. You know the lower one didn't work so try the higher one.