# Doppler Effect Equation

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## Objective

Students use the Doppler effect equation to calculate apparent shifts in frequency or the velocity of a moving wave source or observer.

#### Big Idea

The Doppler effect can be used to calculate the velocity of either a wave source or observer of a wave.

## Context

After the lesson Observations on the Doppler Effect, students know and understand why frequencies shift when a wave source or observer move. Now it is time to apply the Doppler formula to quantify the effect. First students learn how to apply the Doppler formula to calculate the amount of frequency shift due to the motion of the observer or the wave source. Then students solve the equation for velocity and calculate the velocity given the original frequency and the shifted frequency. It is amazing how many applications there are for the Doppler formula, including radar detectors, Doppler radar for weather and the expansion of the universe discovered by Edwin Hubble. These applications begin with the Doppler formula.

This lesson applies CCSS Math Practice 2: Reason abstractly and quantitatively and Math Practice 4: Model with mathematics as well as NGSS Science Practice 5: Using mathematics and computational thinking and Science Practice 7: Engaging in argument from evidence. This is all in the context of the performance standard HS-PS4-1: Use mathematical representations to support a claim regarding relationships among the frequency, wavelength, and speed of waves traveling in various media.

## Doppler Formula Lecture

15 minutes

I start off the lesson with a review of the previous lesson.  I project the Doppler Effect Formula power point which has opening slides that show how the frequency changes on a car horn as a car drives by an observer. I tell students that today we quantify how the frequency changes based on the velocities of the observer or source and the speed of the wave.

I then display the Doppler equation on the power point and explain the variables. The easy one to explain is V, the speed of the wave. Typically, textbooks assign the variable f0 for the original frequency from the wave source. For my students, I teach it as the "source frequency", fs, so that the nomenclature is consistent. We use Vs for the velocity of the wave source and Vo for the velocity of the observer. Students are easily confused by all of the subscripts especially when they are not consistent.

The next task is to explain that the  signs for the velocities of the observer and the source. I train the students to keep the signs the way they are in the formula and to put the a new sign in front of the Vs and Vo depending on their motion. If the wave source is moving towards, the sign is positive; if it is moving away, the sign is negative.

Again this leads to frequent mistakes by the students. But if they consistently apply VFW, which stands for:

• Variables: students must list the variables that are given and for which they must sovle.
• Formulas: students must write the formula without the numbers.
• Work: students show their work and mathematical manipulations.

they are far less likely to make mistakes.

## Doppler Formula Practice Problems

35 minutes

Once the power point is over, I hand out the Doppler Effect Calculations for students to work on for the next 25 minutes. While students work, I go from table to table to help students and provide support. I have the Doppler Calculation Solutions in hand so that when students have questions or want to check their answers, I am prepared to help them (download this file to see all solutions as preview only gives the first page) .

While circulating the room, when a student provides an exemplary solution I put a star next to their solution and ask if they would be willing to present their solution to the class with the document camera. Things go like this until all of the problems are starred with sample student work.

We spend the last 10 minutes of class with students presenting their solutions for the class to see. They place their work under my document camera and talk through the steps they did to get a solution. I ask why they put a positive or negative sign in front of the velocity if they forget to mention it. I inform the students that there is to be a quiz on this material and that I expect them to correct any mistakes they have on their own handout. I collect their work to assess for a grade as they exit the room.