A radar device emits microwaves with a frequency of 4.73E+9 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 751 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 = 3.00×108 m/s.) ... please show work.
To find the speed of the van, we need to use the concept of the Doppler effect. The Doppler effect describes the change in frequency of a wave (in this case, microwaves) when the source and observer are in motion relative to each other.
Given:
- Frequency of the source wave (f₀): 4.73 × 10^9 Hz
- Beat frequency (f_beat): 751 beats per second
The beat frequency (f_beat) can be calculated using the following formula:
f_beat = |f₀ - f_r|
Where:
- f_r is the frequency of the reflected wave
Now, the frequency of the reflected wave (f_r) can be expressed in terms of the speed of the van (v_van) and the speed of light (c):
f_r = (c ± v_van) / λ
Where:
- c is the speed of light: 3.00 × 10^8 m/s
- λ is the wavelength of the microwaves, which can be calculated using the formula: λ = c / f₀
Substituting the values and rearranging the equations, we get:
f_r = (c ± v_van) / λ
f_beat = |f₀ - ((c ± v_van) / λ)|
Since we know the beat frequency (f_beat) and the frequency of the source wave (f₀), we can solve for the speed of the van (v_van):
f_beat = |f₀ - ((c ± v_van) / λ)|
⇒ 751 = |4.73 × 10^9 - ((3.00 × 10^8 ± v_van) / ((3.00 × 10^8) / (4.73 × 10^9)))|
Now we can solve for v_van:
1. Let's assume the plus sign for the Doppler shift due to a moving source (van moving away):
751 = |4.73 × 10^9 - ((3.00 × 10^8 + v_van) / ((3.00 × 10^8) / (4.73 × 10^9)))|
⇒ 751 = |1 - ((1 + v_van / 3.00 × 10^8) / (1 / 4.73 × 10^9)))|
2. Evaluate the expression inside the absolute value brackets:
1 - ((1 + v_van / 3.00 × 10^8) / (1 / 4.73 × 10^9)) = ±751
3. Simplify and solve for v_van:
1 - ((1 + v_van / 3.00 × 10^8) / (1 / 4.73 × 10^9)) = ±751
⇒ 1 - (1 + v_van / 3.00 × 10^8) / (1 / 4.73 × 10^9) = ±751
⇒ 1 - (v_van / 3.00 × 10^8) / (1 / 4.73 × 10^9) = ±751
⇒ (v_van / 3.00 × 10^8) / (1 / 4.73 × 10^9) = -750 or 752
Now we can solve for v_van:
Case 1: (v_van / 3.00 × 10^8) / (1 / 4.73 × 10^9) = -750
⇒ (v_van / 3.00 × 10^8) * (4.73 × 10^9) = -750
⇒ v_van = -750 * 3.00 × 10^8 / (4.73 × 10^9)
Case 2: (v_van / 3.00 × 10^8) / (1 / 4.73 × 10^9) = 752
⇒ (v_van / 3.00 × 10^8) * (4.73 × 10^9) = 752
⇒ v_van = 752 * 3.00 × 10^8 / (4.73 × 10^9)
Calculating these values will give us the speed of the van.
To solve this problem, we can use the formula for beat frequency:
Beat Frequency = |f1 - f2|
Where f1 is the frequency of the source wave, and f2 is the frequency of the reflected wave.
In this case, we know the beat frequency (751 beats per second) and the frequency of the source wave (4.73E+9 Hz). We need to find the frequency of the reflected wave, which will allow us to calculate the speed of the van.
Let's express the beat frequency and the frequency of the source wave in terms of the frequency of the reflected wave.
751 = |f1 - f2|
f1 = frequency of the source wave = 4.73E+9 Hz
f2 = frequency of the reflected wave
To find f2, we can rearrange the equation:
f2 = f1 - Beat Frequency
f2 = 4.73E+9 Hz - 751 Hz
f2 = 4.73E+9 Hz - 751 Hz
f2 = 4.729249E+9 Hz
Now we have the frequency of the reflected wave.
Next, we can use the Doppler effect formula to relate the frequency of the reflected wave to the speed of the van:
f2 = f1 * (c + v) / (c - v)
Where c is the speed of light (3.00×108 m/s) and v is the speed of the van.
Let's rearrange the formula to solve for v:
f2 * (c - v) = f1 * (c + v)
f2 * c - f2 * v = f1 * c + f1 * v
(f2 * c - f1 * c) = (f1 * v + f2 * v)
(f2 - f1) * c = (f1 + f2) * v
v = [(f2 - f1) / (f1 + f2)] * c
v = [(4.729249E+9 Hz - 4.73E+9 Hz) / (4.73E+9 Hz + 4.729249E+9 Hz)] * (3.00×10^8 m/s)
v = [(-7.51 Hz) / (9.459249E+9 Hz)] * (3.00×10^8 m/s)
v = -2.6637E-17 m/s
The speed of the van is approximately -2.6637E-17 m/s.