Doppler Effect: A highway patrol officer uses a device that measures the speed of
vehicles by bouncing radar off them and measuring the Doppler shift. The outgoing radar
has a frequency of 100 GHz and the returning echo has a frequency 15.0 kHz higher. What
is the velocity of the vehicle? Note that there are two Doppler shifts in echoes. Be certain
not to round off until the end of the problem, because the effect is small.

To calculate the velocity of the vehicle based on the Doppler shift, we'll use the Doppler Effect formula. Since radar waves are reflected, there are two Doppler shifts: one when the radar wave reaches the vehicle and one when the reflected wave returns to the radar device.

The formula for the Doppler shift \( f' \) when the source is moving towards the observer is:

\[
f' = f \left( \frac{v + v_s}{v} \right)
\]

Where:
- \( f' \) is the observed frequency (frequency of the reflected wave),
- \( f \) is the emitted frequency,
- \( v \) is the speed of the wave (in this case, the speed of light, \( c = 3.00 \times 10^8 \, \text{m/s} \)),
- \( v_s \) is the velocity of the vehicle (which we're solving for).

Since there are two Doppler shifts, the total observed frequency shift due to the moving vehicle is:

\[
\Delta f = 2 f \frac{v_s}{v}
\]

### Step-by-step process:

1. **Given data**:
   - \( f = 100 \, \text{GHz} = 100 \times 10^9 \, \text{Hz} \) (emitted frequency),
   - \( \Delta f = 15.0 \, \text{kHz} = 15.0 \times 10^3 \, \text{Hz} \) (frequency shift),
   - \( v = c = 3.00 \times 10^8 \, \text{m/s} \) (speed of light).

2. **Apply the Doppler shift formula**:
   \[
   \Delta f = 2 f \frac{v_s}{v}
   \]
   Solve for \( v_s \):
   \[
   v_s = \frac{\Delta f \cdot v}{2 f}
   \]

3. **Substitute the known values**:
   \[
   v_s = \frac{(15.0 \times 10^3) \cdot (3.00 \times 10^8)}{2 \cdot (100 \times 10^9)}
   \]

Now, let's calculate the velocity \( v_s \).The velocity of the vehicle is \( 22.5 \, \text{m/s} \).

Would you like further details or have any questions?

Here are 5 related questions to expand on this concept:
1. How does the Doppler effect change if the vehicle is moving away from the radar?
2. How would the calculation change if the frequency shift were given in megahertz (MHz)?
3. What are some practical applications of the Doppler effect besides speed measurement?
4. How would air temperature affect the speed of sound in a Doppler shift calculation?
5. What is the difference between the Doppler effect in sound and electromagnetic waves?

**Tip**: In Doppler shift problems, always be mindful of whether the source or observer is moving toward or away from each other, as this affects the sign of the frequency change.

Doppler Effect: A highway patrol officer uses a device that measures the speed of vehicles by bouncing radar off them and measuring the Doppler shift. The outgoing radar has a frequency of 100 GHz and the returning echo has a frequency 15.0 kHz higher. What is the velocity of the vehicle? Note that there are two Doppler shifts in echoes. Be certain not to round off until the end of the problem, because the effect is small.

Solve a Doppler Effect problem involving radar waves. Learn how to calculate the velocity of a vehicle using the Doppler shift formula, with an outgoing radar frequency of 100 GHz and a 15.0 kHz frequency shift. This problem covers electromagnetic wave principles and velocity calculation, suitable for undergraduate physics students.

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