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Doppler Effect with math
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Class notes to accompany introduction to Doppler effect with equations derived.
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Materials Required: Computer and internet access Calculator Pen/pencil Digital camera or scanner Download and print the Hubble Diagram Sheet (as an additional option, you can create your graph with the Excel program or create your own graph by hand) Total Time Required: Approximately 2-3 Hours Part 1. The Doppler Effect Note: For your lab report, only include your clearly labeled answers to the below questions in all parts. Copy/paste in your photos or diagrams when needed. Among the great achievements of Einstein was his understanding of the speed of light. The speed of light, in a vacuum, is a constant at ~ 300,000 kilometers/second (the actual velocity is 299,792.458 km/s). The speed of light is essential to the viability of both Einstein’s theories of Special and General Relativity (since the speed of light is a constant it has been given its own mathematical symbol, c). If the speed of light is not constant than neither of Einstein’s theories are credible and would not be accurate in describing physics at the larger-scales of the Universe and objects moving at high velocities close to the speed of light. Therefore, since the speed of light is a constant any motion by an object emitting light has no effect on the lights velocity nor does an object seeing light from a source moving towards it measure any change in the speed of the light coming towards it. For example, a car is driving at night with its headlights on at a speed of 75 miles per hour. What is the speed of the light coming from the headlights? Common sense would give its speed as the speed of light plus 75 miles per hour (c + 75) but the measured speed is still the speed of light ( c ). Something had to change in this situation however and in in this part of the lab you will be investigating the change that is occurring here which is known as the Doppler Effect. Use this link to the Doppler Shift Demonstrator Animation. Click on the ‘Help’ button for instructions on how to run the animation. (Below is a screenshot of the Doppler Shift Demonstrator). Click and move the emitting source towards the middle, left side of the screen and click and move the observer to the opposite side. You can control the frequency of the emitted wave with the rate slider bar and can move either the source or object by left-clicking, holding, and dragging the object towards the direction you want it to move. Answer the following questions based on the simulations being viewed. With the emitting source and the observer on the opposite side of the screen press the ‘start emission’ button. Record your observations of the wave and its wavelength as seen by both the emitting source and the observer (be as detailed as possible). Now click, hold, and drag the observer so it is moving to the left, towards the emitting source. Record your observations of the wave and its wavelength as seen by both the emitting source and the observer (try to make the motion as uniform as poss.
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Background: SONAR is a bit more complicated, so we\'ll assume the easy case based on the information given. This question involves the Doppler effect with a reflection. In most problems, either the source or the observer is moving. In this case, the source is moving, sends a signal, which is perfectly reflected by a moving object, and observed with a Doppler shift. Typically, a beat (interference) is produced and the relative velocity of an object can be ascertained. Equations needed: [lambda_(apparent)=(c+-v_s)/f_s] Where f is the source frequency, v_s is the speed of the source and c is the speed of sound in the medium. [f_(apparent)=(c+-v_o)/lambda_(apparent)] where v_o is the speed of the observer. We choose + for observer moving towards, and - for moving away. Putting these together gets us: [f_(apparent)=(c+-v_o)/(c+-v_s)f_s] Approach: We will apply the equation above twice. Once for the original signal, and again for the bounced signal. In the first part, the source is the submarine and the observer is the torpedo. In the second part, the roles reverse, using the observed frequency (perfectly bouncing off of the torpedo) as the source frequency. The final result will be the answer. We will also see if there is a faster/general way of doing this. Plan and Calculate: 1st part: Submarine is source. Torpedo is observer. [f_(apparent)=(c+v_o)/(c-v_s)f_s] Here, we chose + for observer and - for source because they are moving towards each other. [f_(apparent1)=(1551 m/s+30 m/s)/(1551 m/s-15 m/s)15000 Hz=15439...Hz] I would leave this number in my calculator to continue. This is the frequency of the sound as it hits the torpedo. Now for the reflection where f_apparent becomes f_s and the v_o is now the submarine. [f_(apparent2)=(c+v_o)/(c-v_s)f_(apparent1)] again, we chose the appropriate signs. [f_(apparent2)=(1551 m/s+15 m/s)/(1551 m/s-30 m/s)15439...Hz=15896...Hz] [~~15900Hz] Evaluate: A sonar operator would observe a beat frequency of 900 Hz. If we go back and look at the math, we see that we can create a general equation for this, so we can do it in one step. [f_(text(received))=((c+v_text(target))(c+v_text(sender)))/((c-v_text(target))(c-v_text(sender)))] Answer: 15900 Hz Solution Background: SONAR is a bit more complicated, so we\'ll assume the easy case based on the information given. This question involves the Doppler effect with a reflection. In most problems, either the source or the observer is moving. In this case, the source is moving, sends a signal, which is perfectly reflected by a moving object, and observed with a Doppler shift. Typically, a beat (interference) is produced and the relative velocity of an object can be ascertained. Equations needed: [lambda_(apparent)=(c+-v_s)/f_s] Where f is the source frequency, v_s is the speed of the source and c is the speed of sound in the medium. [f_(apparent)=(c+-v_o)/lambda_(apparent)] where v_o is the speed of the observer. We choose + for observer moving towards, and - for moving a.
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Doppler Effect with math
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![An Introduction to the Doppler Effect ,[object Object],[object Object]](https://image.slidesharecdn.com/doppler2-1234803650445678-1/85/Doppler-Effect-with-math-1-320.jpg)
![What is the Doppler Effect? ,[object Object],[object Object],[object Object],[object Object],[object Object]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)