Problem 46 - Physics

Question

Suppose that the police car in that example is moving to the right at \(27 \mathrm{~m} / \mathrm{s},\) while the speeder is coming up from behind at a speed of \(39 \mathrm{~m} / \mathrm{s}\), both speeds being with respect to the ground. Assume that the electromagnetic wave emitted by the radar gun has a frequency of \(8.0 \times 10^{9} \mathrm{~Hz}\). Find the difference between the frequency of the wave that returns to the police car after reflecting from the speeder's car and the original frequency emitted by the police car.

Step-by-Step Solution

Answer

The frequency difference is approximately 1040 Hz.

Step 1: Identify Known Variables

The speed of the police car is given as \( v_p = 27 \, \mathrm{m/s} \) and the speed of the speeder is \( v_s = 39 \, \mathrm{m/s} \). The frequency of the emitted wave is \( f_0 = 8.0 \times 10^{9} \, \mathrm{Hz} \).

Step 2: Set Up the Doppler Effect Formula for Moving Source

When the source is moving towards a stationary observer, the observed frequency \( f' \) is given by the formula: \[ f' = f_0 \left( \frac{c}{c - v_p} \right) \]Here, \( c \) is the speed of light \( 3.00 \times 10^8 \, \mathrm{m/s} \). Substitute \( v_p = 27 \, \mathrm{m/s} \).

Step 3: Calculate Frequency Reaching the Speeder

Substituting the values into the formula:\[ f' = 8.0 \times 10^9 \left( \frac{3.00 \times 10^8}{3.00 \times 10^8 - 27} \right) \]Calculate \( f' \).

Step 4: Set Up the Doppler Effect Formula for Moving Observer

Now the speeder becomes the source and the radar in the police car the observer. The new frequency \( f'' \) is given by:\[ f'' = f' \left( \frac{c + v_s}{c} \right) \]Substitute \( v_s = 39 \, \mathrm{m/s} \).

Step 5: Calculate Returning Frequency to Police Car

Substitute the previously calculated \( f' \) and plug in the values:\[ f'' = f' \left( \frac{3.00 \times 10^8 + 39}{3.00 \times 10^8} \right) \]Calculate \( f'' \).

Step 6: Find the Frequency Difference

The difference \( \Delta f \) between the returning frequency and the original frequency is \( f'' - f_0 \).Perform this subtraction based on the values you calculated.

Key Concepts

Radar Speed Detection

Radar speed detection is a common method used by law enforcement to monitor the speed of moving vehicles. This system relies on radar technology to emit electromagnetic waves towards a target, like a car.
When these waves hit the moving vehicle, they reflect back to the radar device. By analyzing the changes in the frequency of these returned waves, the radar can determine the speed of the target vehicle.

• The principle utilized here is the Doppler Effect, which describes how the frequency of a wave changes if the source or observer is moving relative to one another.
• In practice, the police car emits a frequency towards the speeder's vehicle, and the frequency of the reflected wave varies due to the relative motion between the car and the speeder.

By carefully calculating these frequency shifts, the radar system detects and displays the car's speed, aiding in traffic law enforcement.

Frequency Shift

The concept of frequency shift is central to understanding how radar speed detection works. When the radar wave, traveling at the speed of light, is directed at a moving car, the frequency of the reflected wave shifts depending on the movement of the car.

• As the car moves towards the radar, the frequency of the reflected waves increases. This is called an upward shift or a blue shift.
• If the car moves away, the frequency decreases, known as a downward shift or red shift.

Using the Doppler Effect formula, the frequency shift is calculated by comparing the emitted and returned frequencies. In our specific example:

• The radar wave is emitted at a frequency of \(8.0 \times 10^9 \mathrm{Hz}\).
• Due to the Doppler Effect, the speeding car, which is moving faster than the police car, causes the frequency to shift.
• The returning waves have altered frequencies \(f'\) and \(f''\) that are crucial in calculating the speed difference between the speeder and the police car.

This calculation provides a direct insight into how different speeds affect wave frequencies.

Wave Reflection

Wave reflection is the process by which waves, including light and sound waves, bounce off a surface. In radar speed detection, the radar emits waves that travel until they encounter and reflect off a moving vehicle.

• The central mechanism involves radar waves traveling towards the target car and then reflecting back towards the radar system.
• The waves undergo a frequency change due to relative motion, which is described by the Doppler Effect.

For the radar to effectively measure the vehicle's speed, the returning wave must be received and analyzed.

• This involves calculating the initial wave frequency \(f_0\), and the reflected wave frequencies \(f'\) and \(f''\).
• Reflected waves undergo a frequency alteration, which when analyzed, provides data about the speed and direction of the moving car.

By understanding wave reflection, we comprehend how radar systems discern the movement characteristics of vehicles, ultimately translating these observations into quantifiable speed readings.