An ambulance’s siren has a frequency of 1.00 kilohertz. The ambulance is approaching an accident scene at a speed of 70.00 miles per hour. A nurse is approaching the scene from the opposite direction to the ambulance at a speed of 7.00 meters per second. Determine the frequency of the siren as measured by the nurse. Use a value of 343.00 meters per second for the speed of sound in the region around the accident scene.
Let’s start our solution by highlighting some of the vital information given in the statement. We know the frequency of the ambulance’s siren as given by the source, 1.00 kilohertz. We also know the ambulance is moving at a speed of 70 miles per hour. In the other direction, a nurse is approaching the same accident scene at a speed of 7.00 meters per second. If we use 343.00 meters per second for the speed of sound, we want to figure out the frequency of the siren as measured by the nurse. We’ll call that frequency 𝑓 sub 𝑜, for observed frequency.
Let’s begin by drawing a diagram of our scenario. A nurse and an ambulance both approach one another moving towards the scene of an accident. While they move, the ambulance emits sound waves from its siren. And the waves are produced at a frequency we can call 𝑓 sub 𝑠, for 𝑓 source, of 1.00 kilohertz. The speed of the ambulance is given as 70.00 miles per hour. We call that 𝑣 sub 𝑎. The nurse approaches in the opposite direction, moving at a speed of 7.00 meters per second. That’s 𝑣 sub 𝑛. We’re also told that the speed of sound, that is the speed with which the ambulance’s siren sound waves move out, is exactly 343.00 meters per second. We’ll call that 𝑣 sub 𝑠, for the speed of sound.
What’s interesting about this situation is that the observed frequency, that is the frequency that the nurse hears, is not the same as the source frequency 𝑓 sub 𝑠. Because of the motion of the source and the motion of the nurse, we’ll find a different number. And that result is determined by the Doppler equation. This equation states that 𝑓 sub zero, the observed frequency, is equal to 𝑣 plus 𝑣 zero [𝑣 𝑜], where 𝑣 by itself is the speed at which sound travels. 𝑣 sub 𝑜 is the speed of the observer. And in the denominator, 𝑣 again is the speed of sound. And 𝑣 sub 𝑠 is the speed of the source. All multiplied by 𝑓 sub 𝑠, the frequency emitted by the source. So we see that the source frequency, 𝑓 sub 𝑠, and the observed frequency, 𝑓 sub 𝑜, are connected by the speed of sound and the speed of the observer and source.
Let’s apply this relationship to our scenario, being careful to keep track of our subscripts. In our situation, the frequency observed by the nurse is equal to the speed of sound plus the nurse’s speed divided by the speed of sound minus the ambulance’s speed, all multiplied by the source frequency 𝑓 sub 𝑠. Each one of these values is given to us in the statement.
But before we enter them into this equation, they need to be all in the same set of units. Our given value for 𝑣 sub 𝑎, the speed of the approaching ambulance, is in units of miles per hour. We’ll want to convert that to a value in meters per second. Let’s do that as our step towards using the Doppler equation.
We want to convert 70.00 miles per hour to some number in units of meters per second. So we’ll convert miles into meters and hours into seconds. Well, let’s start with converting miles. One mile is equal to 5280 feet. So we can multiply this fraction by 5280 feet divided by one mile. Because that fraction is equal to one, we haven’t changed the value of the number. We see that by multiplying these fractions, the units of miles cancel out. But we don’t have meters yet as our unit of distance. We have feet. So we can convert from feet to meters by multiplying by that conversion ratio. One meter is equal to 3.281 feet. This multiplication means that our units of feet cancel out. And we’re left with the distance units of meters, just as we wanted.
Now let’s turn to the time unit, hours. To convert that value into seconds, we recognize that in one hour, there are 3600 seconds. Multiplying by this fraction, which again is equal to one, we see that the units of hours cancel out. And overall, we now have a value in units of meters per second. When we carry through multiplying each of these four fractions together, we get a result for 𝑣 sub 𝑎, the speed of the ambulance, of 31.29 meters per second. Now we’re ready to insert the values we’ve been given into the Doppler equation for our problem. Let’s do that now.
We find that the observed frequency is equal to the speed of sound plus the speed of the nurse divided by the speed of sound minus the speed of the ambulance, all multiplied by the frequency of the siren. Calculating this result, we come up with an observed frequency of 1.12 kilohertz. Calculating this result, we find an observed frequency of 1.12 kilohertz. So the approaching nurse hears a higher-pitched sound than is created by the ambulance’s siren.