∵Baseball scouts often use a radar gun to measure the speed...

$\because$ Baseball scouts often use a radar gun to measure the speed of a pitch. One particular model of radar gun emits a microwave signal at a frequency of 10.525 GHz. What will be the increase in frequency if these waves are reflected from a $95.0-$ mi/h fastball headed straight toward the gun? (Note: $1 \mathrm{mi} / \mathrm{h}=0.447 \mathrm{m} / \mathrm{s}$ )

$\because$ Baseball scouts often use a radar gun to measure the speed of
a pitch. One particular model of radar gun emits a microwave
signal at a frequency of 10.525 GHz. What will be the increase in
frequency if these waves are reflected from a $95.0-$ mi/h fastball
headed straight toward the gun? (Note: $1 \mathrm{mi} / \mathrm{h}=0.447 \mathrm{m} / \mathrm{s}$ )

Key Concepts

Doppler Effect

The Doppler effect describes the change in frequency or wavelength of a wave in relation to an observer who is moving relative to the wave source. It is a fundamental concept used to determine how the observed frequency of a wave is affected by the motion of the source or reflector, which is central to understanding how radar guns measure object speeds.

Reflection of Waves from a Moving Object

When a wave reflects off a moving object, the process involves two successive Doppler shifts: one when the wave is incident on the moving object and another when it is reflected back. This double shift is a critical concept in radar applications, as it influences the overall frequency change that is detected and used to compute the object's velocity.

Electromagnetic Waves

Electromagnetic waves, including microwaves, travel at the speed of light and are characterized by their frequency and wavelength. Understanding these properties is essential in radar technology, as the frequency of the emitted signal and its subsequent shift upon reflection are used to measure the speed of objects.

Unit Conversion

Accurate unit conversion is a crucial concept in problem-solving, particularly when converting speeds from one unit system to another. It ensures that all physical quantities are expressed in consistent units, which is necessary for applying formulas correctly in calculations like determining frequency shifts in radar applications.

Transcript

00:01 All right, what's going on here can be a little bit tricky because we're actually dealing with a double doppler shift.

00:08 So what's happening is we have a radar gun and it is sending out a signal at a particular frequency.

00:16 And the signal is being seen by a baseball coming towards the radar gun.

00:22 And so because of the doppler shift, the baseball sees a different incoming frequency, which we're going to label f prime and then it reflects the signal and the signal comes back out towards the radar gun and the radar gun is going to see a signal back that is a doppler shift of the doppler shifted signal.

00:49 It's kind of crazy.

00:52 Okay, so we need to remember our equation for the doppler shift.

00:55 So it's f prime is equal to to the original frequency times one plus, because these two things are getting closer and they're coming towards each other, it's u over c, okay? but we actually don't care about this quantity.

01:20 We're really interested in the doppler shifted, doppler shifted frequency.

01:26 We wanna know the frequency that comes back to the gun.

01:29 So this is equal to f, the shifted frequency that the baseball saw times 1 plus u over c and the u in both of these equations is going to be the same because the baseball and the radar gun are still getting closer at the same speed both times so f prime prime is going to be equal to 1 plus u over c times f prime which is going to be 1 plus u over c times the original frequency that leaves the radar gun.

02:19 Okay? and this problem actually doesn't care about f prime, prime.

02:23 It wants to know the change in frequency, delta f prime prime, which is just equal to f prime prime minus what the original frequency was, f.

02:38 That's going to be equal to f, and pull out f because it's going to be in both expressions, times 1 plus u over c squared minus 1.

02:55 Okay? so this part is defined f prime, and this part is to subtract off the original f.

03:03 Okay, so our change in frequency, delta f is our original frequency, but the radar gun shot out, which is 10 .5 to 5 times 10 to the 9th, because this is in gigahertz...

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