EBK PHYSICS FOR SCIENTISTS AND ENGINEER
EBK PHYSICS FOR SCIENTISTS AND ENGINEER
9th Edition
ISBN: 8220100663987
Author: Jewett
Publisher: Cengage Learning US
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Chapter 39, Problem 39.69AP

A Doppler weather radar station broadcasts a pulse of radio waves at frequency 2.85 GHz. From a relatively small batch of raindrops at bearing 38.6° east of north, the station receives a reflected pulse after 180 µs with a frequency shifted upward by 254 Hz. From a similar batch of raindrops at bearing 39.6° east of north, the station receives a reflected pulse after the same time delay, with a frequency shifted downward by 254 Hz. These pulses have the highest and lowest frequencies the station receives, (a) Calculate the radial velocity components of both batches of raindrops. (b) Assume that these raindrops are swirling in a uniformly rotating vortex. Find the angular speed of their rotation.

(a)

Expert Solution
Check Mark
To determine

The radial velocity components of both batches of raindrops.

Answer to Problem 39.69AP

The radial velocity component of first batch of raindrops is 13.4m/s toward the station and the radial velocity component of second batch of raindrops is 13.4m/s away from the station.

Explanation of Solution

Given info: The frequency of radio waves is 2.85GHz , bearing of raindrops is 38.6° east of north, the time after which station receives pulse is 180μs , the frequency shifts upward by a value equal to 254Hz , the bearing of raindrops from a similar batch is 39.6° east of north, the time after which the station receives pulse is 180μs and the frequency shifts downward by a value equal to 254Hz .

Write the equation of frequency of radio waves received.

f'=fc+vcv

Here,

f' is the frequency of radio waves received which is reflected by the raindrops.

f is the frequency of radio waves.

c is the speed of light.

v is the speed of the raindrops.

Write the equation of frequency of radio waves received for another upward shift.

f1'=f'c+vcv

Here,

f1' is the frequency of radio waves received for another upward shift.

Substitute fc+vcv for f' in above equation.

f1'=(fc+vcv)c+vcvf1'=f(c+vcv)v=c(f1'ff1'+f) (1)

The frequency of radio waves received for upward shift is,

f1'=f+f1

Here,

f1 is the frequency shifted upward.

Substitute 2.85GHz for f and 254Hz for f1 in above equation to find f1' .

f1'=2.85GHz(109Hz1GHz)+254Hz=2,850,000,254Hz

Thus, the value of f1' is 2,850,000,254Hz .

Substitute 3.0×108m/s for c , 2,850,000,254Hz for f1' and 2.85GHz for f in equation (1) to find v .

v=(3.0×108m/s)[(2,850,000,254Hz){2.85GHz(109Hz1GHz)}(2,850,000,254Hz)+{2.85GHz(109Hz1GHz)}]=13.368m/s13.4m/s

Thus, the radial velocity of the radio waves with the upward shift is 13.4m/s towards the station.

The frequency of radio waves received for downward shift is,

f1'=f+f1

Here,

f1 is the frequency shifted downward.

Substitute 2.85GHz for f and 254Hz for f1 in above equation to find f1' .

f1'=2.85GHz(109Hz1GHz)254Hz=2,849,999,746Hz

Thus, the value of f1' is 2,849,999,746Hz .

Substitute 3.0×108m/s for c , 2,849,999,746Hz for f1' and 2.85GHz for f in equation (1) to find v .

v=(3.0×108m/s)[(2,849,999,746Hz){2.85GHz(109Hz1GHz)}(2,849,999,746Hz)+{2.85GHz(109Hz1GHz)}]=13.368m/s13.4m/s

The radial velocity of the radio waves with the downward shift is 13.4m/s away from the station.

Conclusion:

Therefore, the radial velocity component of first batch of raindrops is 13.4m/s toward the station and the radial velocity component of second batch of raindrops is 13.4m/s away from the station.

(b)

Expert Solution
Check Mark
To determine

The angular speed of the rotation.

Answer to Problem 39.69AP

The angular speed of the rotation is 0.0567rad/s .

Explanation of Solution

Given info: The frequency of radio waves is 2.85GHz , bearing of raindrops is 38.6° east of north, the time after which station receives pulse is 180μs , the frequency shifts upward by a value equal to 254Hz , the bearing of raindrops from a similar batch is 39.6° east of north, the time after which the station receives pulse is 180μs and the frequency shifts downward by a value equal to 254Hz .

Write the equation of angular speed of the vortex.

ω=vr (2)

Here,

ω is the angular speed of the vortex.

r is the radius of vortex.

Write the equation for radius of vortex.

r=12xθ

Here,

x is the one way distance covered by the rain.

θ is the angle made by raindrops.

The radio waves travels towards the rain and comes back.

Write the equation for one way distance covered by the rain.

x=12c(Δt)

Here,

Δt is the time after which the station receives pulse.

Substitute 12c(Δt) for x in above equation to find r .

r=12(12c(Δt))θr=14cΔtθ

Substitute 14c(Δt)θ for r in equation (2) to find ω .

ω=v(14c(Δt)θ)ω=4vcΔtθ

Substitute 13.4m/s for v , 3.0×108m/s for c , 180μs for Δt and 1° for θ in above equation to find ω .

ω=4(13.4m/s)(3.0×108m/s){180μs(1s106μs)}{1°(πrad180°)}=0.0567rad/s

Conclusion:

Therefore, the angular speed of the rotation is 0.0567rad/s .

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Chapter 39 Solutions

EBK PHYSICS FOR SCIENTISTS AND ENGINEER

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