How To Solve Doppler Effect Physics Problems

Doppler Effect Overview

  • The Doppler effect refers to the change in frequency detected by an observer due to the motion of the source or observer.

  • If the source moves toward the observer (or vice versa), the frequency increases.

  • If the source moves away from the observer (or vice versa), the frequency decreases.

Key Concepts

  • Observed Frequency (f_o) vs. Source Frequency (f_s):

    • f_o > f_s when moving towards each other.

    • f_o < f_s when moving away from each other.

  • Example:

    • Source frequency = 800 Hz.

    • If source moves toward observer, heard frequency could be 850 Hz (increased).

    • If source moves away, heard frequency would decrease to values like 900 or 800 Hz.

Visualization

  • Wave Patterns:

    • When a source approaches, crests are closer (wavelength decreases, frequency increases).

    • When a source retreats, crests are farther apart (wavelength increases, frequency decreases).

Formula for Calculating Observed Frequency

  • Formula:[ f_o = f_s \times \frac{v + v_o}{v - v_s} ]

    • ( v ): speed of sound (around 343 m/s at 20°C).

    • ( v_o ): speed of observer (positive if moving towards, negative if away).

    • ( v_s ): speed of source (negative if moving towards, positive if away).

Speed of Sound Depending on Temperature

  • Speed can be calculated with:[ v = 331 + 0.6t ]

  • Where ( t ) is temperature in Celsius.

  • Example calculations:

    • At 0 °C, ( v \approx 331 ) m/s;

    • At 20 °C, ( v \approx 343 ) m/s;

    • At 25 °C, ( v \approx 346 ) m/s.

Sign Conventions for Using the Formula

  1. Source Moving Towards Observer:

    • ( v_s ): negative (decreases denominator, increases f_o).

  2. Source Moving Away from Observer:

    • ( v_s ): positive (increases denominator, decreases f_o).

  3. Observer Moving Towards Source:

    • ( v_o ): positive (increases numerator, increases f_o).

  4. Observer Moving Away from Source:

    • ( v_o ): negative (decreases numerator, decreases f_o).

Example Problems

Problem 1: Ambulance Truck

  • Frequency produced: 800 Hz.

  • If moving towards observer at 30 m/s: [ f_o = 800 \times \frac{343 + 0}{343 - 30} \approx 877 ext{ Hz} ]

  • If moving away from observer at 30 m/s: [ f_o = 800 \times \frac{343 + 0}{343 + 30} \approx 736 ext{ Hz} ]

Problem 2: Stationary Ambulance Truck

  • Frequency produced: 1200 Hz; observer moving towards at 25 m/s: [ f_o = 1200 \times \frac{343 + 25}{343 + 0} = 1287 ext{ Hz} ]

  • Observer moving away at 25 m/s: [ f_o = 1200 \times \frac{343 - 25}{343 + 0} \approx 1113 ext{ Hz} ]

Problem 3: Police Car and Driver

  • Police car emitting 900 Hz; moving at 20 m/s towards driver going 25 m/s: [ f_o = 900 \times \frac{343 + 25}{343 - 20} \approx 1258 ext{ Hz} ]

Conclusion

  • Understand how to manipulate the observed frequency based on the movement of the source and observer.

  • Practice with various scenarios to become comfortable with the formula and sign conventions.