The Radar Doppler Effect is a phenomenon observed when electromagnetic waves are reflected off a moving object. It causes a change in the frequency of the reflected waves relative to the transmitted waves. This effect is widely used in radar systems to measure the velocity of moving objects, such as vehicles, aircraft, or weather phenomena.

The formula for the Radar Doppler Effect is:

\[ f' = f \cdot \frac{c \pm v_r}{c \mp v_s} \]

Where:

- \( f' \) is the received frequency (Hz)
- \( f \) is the transmitted frequency (Hz)
- \( c \) is the speed of electromagnetic waves (speed of light, m/s)
- \( v_r \) is the radial velocity of the receiver relative to the source (m/s)
- \( v_s \) is the radial velocity of the source relative to the receiver (m/s)
- Use + for \( v_r \) and - for \( v_s \) if the object is moving towards the radar, and vice versa

Let's calculate the received frequency for a radar system detecting a moving car:

- Given:
- Transmitted frequency (\( f \)) = 10 GHz = 10,000,000,000 Hz
- Speed of light (\( c \)) = 299,792,458 m/s
- Car speed (\( v_s \)) = 30 m/s (towards the radar)
- Radar is stationary (\( v_r = 0 \))

- Apply the Doppler Effect formula: \[ f' = f \cdot \frac{c}{c - v_s} \]
- Substitute the known values: \[ f' = 10,000,000,000 \cdot \frac{299,792,458}{299,792,458 - 30} \]
- Perform the calculation: \[ f' \approx 10,000,001,000 \text{ Hz} = 10.000001 \text{ GHz} \]

Let's visualize the Radar Doppler Effect:

This diagram illustrates:

- The stationary radar (green triangle)
- The moving car (red rectangle)
- The transmitted wave (yellow dashed line)
- The reflected wave with changed frequency (blue dashed line)
- The velocity of the car towards the radar (\( v_s \))

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