To determine the difference in frequency between the emitted beam from the weather station and the frequency of the reflected beam from the approaching thunderstorm, we can use the Doppler effect. This phenomenon occurs when there is relative motion between a source of waves and an observer, leading to a change in frequency perceived by the observer.
Understanding the Doppler Effect
The Doppler effect explains how the frequency of a wave changes based on the relative motion of the source and the observer. In this scenario, the weather station emits a beam at a frequency of 200 MHz, and the thunderstorm is moving towards the station at a speed of 45 miles per hour.
Converting Units
First, we need to convert the speed of the thunderstorm from miles per hour to meters per second, as the standard unit for speed in physics is meters per second.
- 1 mile = 1609.34 meters
- 1 hour = 3600 seconds
Thus, the conversion is as follows:
Speed in meters per second = 45 miles/hour × (1609.34 meters/mile) / (3600 seconds/hour) ≈ 20.12 m/s.
Applying the Doppler Effect Formula
For a source emitting waves towards a moving observer, the frequency observed can be calculated using the formula:
f' = f (v + v₀) / (v - vₛ)
Where:
- f' = observed frequency
- f = emitted frequency (200 MHz)
- v = speed of the waves in the medium (for electromagnetic waves in air, this is approximately the speed of light, 3 x 10^8 m/s)
- v₀ = speed of the observer (thunderstorm, moving towards the source, so it's positive)
- vₛ = speed of the source (the weather station, which is stationary, so it's zero)
Calculating the Observed Frequency
Plugging in the values:
f' = 200 MHz × (3 x 10^8 m/s + 20.12 m/s) / (3 x 10^8 m/s - 0)
f' = 200 MHz × (3 x 10^8 + 20.12) / (3 x 10^8)
Since the speed of the thunderstorm is very small compared to the speed of light, we can approximate:
f' ≈ 200 MHz × (3 x 10^8) / (3 x 10^8) = 200 MHz.
Calculating the Frequency of the Reflected Beam
Now, when the thunderstorm reflects the beam, it acts as a new source of waves. The frequency of the reflected wave will also be affected by the Doppler effect, but now the thunderstorm is the source moving towards the weather station. We will use the same formula, but this time:
f'' = f' (v + v₀) / (v - vₛ)
Here, f' is the frequency we just calculated, and now:
- v₀ = speed of the thunderstorm (20.12 m/s, moving towards the observer)
- vₛ = 0 (the weather station is stationary)
Final Calculation
Substituting the values:
f'' = 200 MHz × (3 x 10^8 m/s + 20.12 m/s) / (3 x 10^8 m/s)
f'' ≈ 200 MHz × (3 x 10^8) / (3 x 10^8) = 200 MHz.
Finding the Frequency Difference
The frequency difference between the emitted frequency and the reflected frequency can be calculated as follows:
Difference = f'' - f = 200 MHz - 200 MHz = 0 MHz.
In this case, the frequency of the reflected beam remains the same as the emitted frequency due to the relatively small speed of the thunderstorm compared to the speed of light. Therefore, the difference in frequency is zero.
