CALC You have two small, identical boxes that generate 440 Hz notes. While holding one, you drop the other from a 20-m-high balcony. How many beats will you hear before the falling box hits the ground? You can ignore air resistance.

Video transcript

Hey, everyone. So this problem is dealing with sound waves. Let's see what it's asking. Suppose you have a pair of identical tuning forks producing sound waves at a frequency of 523 Hertz holding one of the tuning forks steady. You let the other fall from a 15 m high ledge, ignoring air resistance, determine the total beats you will hear before the plummeting tuning fork impacts the surface. So our multiple choice answers here are a 22 beats. B 32 beats, C 36 beats or D beats. So at first glance, this is kind of a straightforward question how many beats, but it is a tricky problem that is going to require us to pull in some kinematics equations, some calculus. So it is tricky, but we will be able to answer this total beat by the end of this video. So stick with me. So as the fork falls, the speed changes, right? So the perceived frequency changes. So we're looking for the total beats and frequency as a function of speed, therefore, as a function of time. So this is all starting to sound like we're in an in a real situation, right? So we can write this as N or the total number of beats is the integral, our initial time to our final time of the change of frequency with respect to time. So that's delta F modified by G T. Now, we're trying to find um the final time. We don't know how long it will take for that plu plummeting tuning forward to impact the surface. So the first thing you can do is recall from our kinematics equations, distance is equal to our initial velocity multiplied by time plus one half, multiplied by acceleration multiplied by times square. And so we know our distance from the problem that's 15 m, our initial velocity is zero because you're holding the force steady right before you drop it. So that term goes to zero. And so now we have 15 m equals one half. Our acceleration is just gravity, it's 9.8 m per second squared. And I am writing that as a positive value um because we're also writing 15 m of positive. So we're kind of taking the drop the as the negative right direction to be positive. You like things you could, you could have um also just put negative in front of both of those and then it would cancel that. And so that's multiplied by T squared. And so we've just solved for T so T equals 1.75 seconds. Oh Sorry, that's not the final, it's not the final answer. Um It's, that would be nice. So, no, not the final answer. But it does tell us that T F is equal to 1.75 seconds. So we'll be using that when we integrate. So back to our integral, we know that delta F or the change in frequency is just the initial frequency minus the frequency of the um tuning fork as it's falling. And that's the um absolute value of that, of that difference. And so we can rewrite this and we can look at this frequency and we can think about if that frequency is falling, what equation might that look like? And so that's what we can recall the Doppler. So the Doppler effect equation when it's when the source is moving away is given by F equals F I multiplied by V divided by B plus BS where B is the speed of the object moving away. And BS is the speed of sound and F I is your initial frequency. And so plugging this in to the delta F equation will give us the equation that we are ultimately going to integrate. So delta F is equal to the absolute value of F I minus F I multiplied by the wanted TV, divided by V plus the subs. And so we can simplify that a little bit, we can pull out the um I that those initial frequency. So we have F I multiplied by the opposite value oh, the subs all over V plus BS. So that's when we pull out the initial frequency and get a common denominator. And so visa, we can calculate, I think originally, I think previously I I misspoken. Um And so that's not the um seed of the, of it's not the speed of sound in air. So I apologize if I misspoke earlier, but B of S is going to be the speed of the sounding, the tuning for it, the sound as it's falling. And so that is equal to our initial velocity plus acceleration multiplied by time again in our kinematics equations. And so VA s the initial velocity we've already established a zero. It's simply going to be G multiplied by T. And so we can plug that back into our delta F equation. So delta F equals F I and multiplied by the absolute value of G T divided by V plus G T. And now we have delta F in terms of T, we are ready to um to perform our integration. And so, and our number of beats, we'll, we'll pull that um constant F I out and then F I multiplied by the integral from to 1.75 seconds of G T plus or G T divided by B plus G T D T. And when we integrate that, we get our initial frequency multiplied by the absolute value of T minus V multiplied by L N of V plus G T to and that's from zero 21.757. So while this looks kind of ugly, we actually have everything we need here to solve four and our number of beats. So we have N is equal to that initial frequency was given to us in the problem as 523 Hertz multiplied by time, which is 1.75 seconds. We're going to do our final time minus R time zero as we saw this emerald. So 1.75 seconds minus V, that's the speed of sound in air. We can recall uh that constant at room temperature 343 m per second, multiplied by the natural log of again, speed 343 m per second plus G T. So that would be 9.8 m per second squared, multiplied by 1.75 seconds, all of that divided by G or 9.8 m per second squared. And so that's the first part of the integral. And then we are going to subtract when T equals zero. So we will have zero seconds minus 343 m per second for beam multiplied by the L N of m per second. And that's it because G multiplied by T is zero and all of that divided by 9.8 m per second squared. And so we plug all of this into our calculator. We did and is equal to 523 Hertz multiplied by 0. seconds. And that gives us 22.1 beats for approximately 22 beats. That is the answer to this question. The filing. So it was tricky. I appreciate you sticking with me through it. 22 beats is answer choice. A so a is the correct answer for this problem. Thanks everyone. We'll see you in the next video.