Problem in Taylor and Wheeler's Spacetime Physics

[jlowe22]

The problem is 5-5 part d.  As worded is:

"One radar gun used by the Massachusetts Highway Patrol operates at a frequency of 10.525 X 10⁹ cycles/second. By how many cycles/second is the reflected beam shifted in frequency when reflected from a car approaching at 100 kilometers/ hour?"

The book gives the answer 3136, and I can't for the life of me get this answer.  I've done this multiple ways and get the same wrong answer every time.  Doing similar problems elsewhere I'm getting right answers, so I can only conclude that either I'm overlooking something really, really silly, or there's a typo somewhere in the book.  If someone can confirm that 3136 is indeed the right answer, I'll go on trying to figure out where I screwed up.

 

The link to the full problem is here:

https://phys.libretexts.org/Bookshelves/Relativity/Spacetime_Physics_(Taylor_and_Wheeler)/05%3A_Trekking_through_Spacetime/5.E%3A_Trekking_through_Spacetime_(Exercises)

[Genady]

I'm getting about 2000.

[studiot]

I made it 2328

[jlowe22]

1 hour ago, Genady said:

I'm getting about 2000.

I get ~ 1951 every time I try.

[studiot]

3 hours ago, jlowe22 said:

I get ~ 1951 every time I try.

I agree that this would be the best Excel can do, and Excel is the best way I have for handling multidigit numbers.

 

However since wee are talking about the difference between the last four digits to two eleven digit numbers, not to mention the loss involved in the quotients, there will be a lot of variation depending upon calculation method.

 

 

[Genady]

You get the book answer if you use 100 mph instead of 100 km/hour.

[studiot]

15 minutes ago, Genady said:

You get the book answer if you use 100 mph instead of 100 km/hour.

I wondered that and agree.

Here is the modifed spreadsheet

[jlowe22]

5 hours ago, Genady said:

You get the book answer if you use 100 mph instead of 100 km/hour.

Of course.  I knew something silly was going on, I could get the book answer using 44.6 m/s but I didn't make the connection to mph.

Thanks guys.

[iNow]

2 hours ago, jlowe22 said:

I knew something silly was going on, I could get the book answer using 44.6 m/s but I didn't make the connection to mph.

Don't be too hard on yourself. NASA made a remarkably similar mistake a little while back: https://www.wired.com/2010/11/1110mars-climate-observer-report/

[Sensei]

@studiot Shouldn't the relativistic Doppler effect formula be used instead of the classical one formula for sound waves? (i.e. sqrt() is needed)

 

Edited August 22, 2023 by Sensei

[swansont]

At such a low speed the difference between the results would be negligible

[Genady]

sqrt() is not needed because it is multiplied twice.

[studiot]

2 hours ago, Sensei said:

@studiot Shouldn't the relativistic Doppler effect formula be used instead of the classical one formula for sound waves? (i.e. sqrt() is needed)

 

Here is the Wiki article on the subject of police doppler radar guns.

Quote

The formula for radar Doppler shift is the same as that for reflection of light by a moving mirror.^[2] There is no need to invoke Albert Einstein's theory of special relativity, because all observations are made in the same frame of reference.^[

https://en.wikipedia.org/wiki/Doppler_radar

[Sensei]

On 8/21/2023 at 6:30 PM, studiot said:

I agree that this would be the best Excel can do, and Excel is the best way I have for handling multidigit numbers.

..or you can write a Python script..

#!/bin/python

import sys
import math

args = len( sys.argv )
if( args != 2 ):
	print( "The required argument is missing!" )
	sys.exit( 1 )

v = float( sys.argv[ 1 ] )

c = 299792458.0 # in vacuum
#c = 299705000 # in air
fin = 10.525e+9
gamma = ( c + v ) / ( c - v )
fout = fin * gamma
df = fout - fin

print( "Fs", fin )
print( "v", v )

print()
print( "Classic:" )
print( "gamma", gamma )
print( "Fo", fout )
print( "dF", df )
print( "dF (short equation)", 2 * v * fin / c )

gamma_sr = ( math.sqrt( gamma ) - 1 ) * 2 + 1
fout_sr = fin * gamma_sr
df_sr = fout_sr - fin

print()
print( "Special Relativity:" )
print( "gamma", gamma_sr )
print( "Fo", fout_sr )
print( "dF", df_sr )

 

[Genady]

Doing it in SR,

f1 = emitted frequency

f2 = received and reflected frequency by the car

f3 = measured frequency:

f2 = f1 * sqrt(...)

f3= f2 * sqrt(...) = f1 * sqrt(...) * sqrt(...) = f1 * ...

No sqrt(...) in the final calculation.

[jlowe22]

On 8/21/2023 at 7:29 PM, iNow said:

Don't be too hard on yourself. NASA made a remarkably similar mistake a little while back: https://www.wired.com/2010/11/1110mars-climate-observer-report/

In my defense the book uses km/hr, but yes I certainly should have caught the problem before spending so much time on it.