Video: Finding the Doppler Shift Produced by Two Objects Moving toward Each Other

Two cars have horns, and each horn’s frequency is 800 Hz. The cars move toward one another, both sounding their horns. Car A is moving at 65.00 mph, and car B is at 75.00 mph. Use a value of 343 m/s for the speed of sound. What is the beat frequency heard by car A’s driver? What is the beat frequency heard by car B’s driver?

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Video Transcript

Two cars have horns, and each horn’s frequency is 800 Hz. The cars move toward one another, both sounding their horns. Car A is moving at 65.00 miles per hour, and car B is at 75.00 miles per hour. Use a value of 343 meters per second for the speed of sound. What is the beat frequency heard by car A’s driver? What is the beat frequency heard by car B’s driver?

Let’s start by highlighting some of the important information we’re given. We’re told that the frequency of each car’s horn is 800 Hz; we’ll call that value 𝑓 sub 𝑠. We’re told that car A is moving at a speed of 65.00 miles per hour, which we’ll call 𝑣 sub 𝐴. And car B is moving at 75.00 miles per hour, 𝑣 sub 𝐵.

We were told that in this scenario, the speed of sound is 343 meters per second, what we’ll call 𝑣. We want to solve for the beat frequency heard by the driver in car A, what we’ll call 𝐵 sub 𝐴. And we want to solve for the same thing, the beat frequency, heard by the driver in car B, what we’ll call 𝐵 sub 𝐵.

To get started with our solution, let’s draw a diagram of the situation. As car A and car B approach one another, each one sounds its horn, and each horn has a source frequency of 800 Hz. To solve for the beat frequency observed by the driver in car A and the driver in car B, we first need to solve for the perceived frequency shift for those two drivers.

Because the cars are in motion, the perceived frequency of their horns by the opposite driver will shift as described by the Doppler equation, which says that the observed frequency 𝑓 sub 𝑜 is equal to the source frequency 𝑓 sub 𝑠 multiplied by the speed of sound 𝑣 plus the observer’s speed 𝑣 sub 𝑜 divided by the speed of sound minus the source of speed 𝑣 sub 𝑠.

When we apply this relationship to our scenario, ultimately looking to solve for 𝐵 sub 𝐴, we find that the frequency observed by the driver in car A, what we’ll call 𝑓 sub 𝐴, is equal to the speed of sound 𝑣 plus the speed of car A divided by the speed of sound 𝑣 minus the speed of car B all multiplied by the source frequency of 800 Hz.

Before we plug in to this equation, we’ll want to convert 𝑣 sub 𝐴 into units of meters per second so that it agrees with the units of the other values in the equation. If we take the fraction 65.00 miles per hour, multiply that by the fraction one hour per 3600 seconds, and multiply that by the conversion from miles to meters, 1609.34 meters per mile, we see that when we look at that units, hours cancels out as does miles, and we’re left with units of meters per second.

When we multiply these three fractions together, we find that, in units of meters per second, 𝑣 sub 𝐴 is 29.06. While we’re at it, converting numbers in miles per hour to units in meters per second, we can also convert 𝑣 sub 𝐵, the speed of car B. In that case, we multiply 75.00 miles per hour by our two conversion ratios. Multiplying through, we find that 𝑣 sub 𝐵 is 33.53 meters per second.

We’re now ready to plug in for the values in this equation solving for 𝑓 sub 𝐴, the frequency observed by the driver in car A. The speed of sound 𝑣 is 343 meters per second; 𝑣 sub 𝐴 is 29.06 meters per second; 𝑣 sub 𝐵 is 33.53 meters per second; and 𝑓 sub 𝑠 is 800 Hz. Entering these values on our calculator, we find 𝑓 sub 𝐴 to be 961.8 Hz.

Now we can recall the definition for beat frequency, which says that the frequency of beats between two frequencies, 𝑓 sub one and 𝑓 sub two, is equal to the magnitude of their difference. Applying this to our situation, this means that 𝐵 sub 𝐴 is equal to the magnitude of 𝑓 sub 𝐴 minus 𝑓 sub 𝑠. Plugging in for those values, we find that, to three significant figures, 𝐵 sub 𝐴 is 162 Hz.

That’s the beat frequency heard by the driver in car A. Next we want to solve for the beat frequency observed by the driver in car B. We’ll follow a similar process, starting by solving for the observed frequency of car A’s horn to the driver in car B, 𝑓 sub 𝐵.

Using the Doppler equation, that’s equal to the speed of sound 𝑣 plus the speed of car B divided by the speed of sound minus the speed of car A all multiplied by the source frequency 𝑓 sub 𝑠. Plugging in for those values, when we enter these numbers on our calculator, we find that to the driver in car B, the horn from car A will sound like it 959.5 Hz.

This means, using our equation for beat frequency, that the beat frequency observed by the driver in car B equals the magnitude of 959.5 Hz minus 800 Hz. To three significant figures, this equals 160 Hz. That’s the beat frequency observed by the driver in car B, the difference in frequencies between the horn of car B and the perceived frequency of the horn of car A.

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