{"id":329,"date":"2014-08-08T16:38:16","date_gmt":"2014-08-08T20:38:16","guid":{"rendered":"http:\/\/advancedmathyoungstudents.com\/blog\/?p=329"},"modified":"2021-07-25T17:15:46","modified_gmt":"2021-07-25T21:15:46","slug":"a-new-look-at-exponential-decay","status":"publish","type":"post","link":"https:\/\/advancedmathyoungstudents.com\/blog\/?p=329","title":{"rendered":"A NEW LOOK AT EXPONENTIAL DECAY"},"content":{"rendered":"

And a first look at differential equations<\/strong><\/p>\n

At some point in the past, I believe you learned a little about radioactive decay and half-life.<\/p>\n

Here are some things that you may remember:<\/p>\n

1. The size of a radioactive sample is can be expressed as a function of time:<\/p>\n

\u00a0\u00a0\u00a0\u00a0 N(t)=No<\/sub>e-kt<\/sup><\/b><\/p>\n

2. The graph of this function looks like this:<\/p>\n

\"post21pic1\"<\/a><\/p>\n

Note that at time = 0, the value is No<\/sub> and then the value decays asymptotically toward zero.<\/p>\n

3. Graphs like this show up in many other contexts.\u00a0 For example, this looks a lot like the graph of current vs. time for a charging capacitor.\u00a0 It also looks like the graph of acceleration vs. time for an object in free-fall with air resistance.\u00a0 And there are a number of graphs associated with motors and generators that also have this shape.<\/p>\n

4. When a quantity decays exponentially, in equal time intervals it will decrease by the same factor.\u00a0 Most notably, the time it takes to decrease by 50% is called the \u201chalf-life\u201d.\u00a0\u00a0There is a useful formula relating the half-life to the decay constant:<\/p>\n

\"post21pic2\"<\/a><\/p>\n

Do NOT just memorize this \u2013 derive it for yourself.\u00a0 If you need help getting started, you can begin by letting N(t)= \u00bd No<\/sub>.<\/p>\n

*** To see that this claim about half-life is true, you can play with Part 1 of the desmos activity here:\u00a0 https:\/\/www.desmos.com\/calculator\/ooewvdzrmq<\/a><\/strong><\/p>\n

However, all of this assumes that we already know that the function is N(t)=No<\/sub>e-kt<\/sup><\/b> . But how do we know that?\u00a0 When you see exponential decay, you should suspect that somewhere behind the scenes lurks something called a differential equation.\u00a0 And that brings us (finally) to the topic of this post.<\/p>\n

WHAT IS A DIFFERENTIAL EQUATION?<\/strong><\/p>\n

\u201cThe rate at which a given radioactive sample decays is proportional to the size of the sample.\u201d<\/em><\/p>\n

I hope that sentence makes sense to you, intuitively.\u00a0 It is saying that when there are more nuclei available to decay, the decay rate is faster.\u00a0 As the sample shrinks in size, there are fewer available nuclei and so the rate slows down.<\/p>\n

*** This claim can be investigated further in Part 2 of that same Desmos activity: https:\/\/www.desmos.com\/calculator\/ooewvdzrmq<\/a>\u00a0 Really, go do this now!\u00a0 I’ll wait here…<\/strong><\/p>\n

OK, so the rate is proportional to the current value.<\/p>\n

But that sentence can be re-written using mathematical symbols:<\/p>\n

\"post21pic3\"<\/a><\/p>\n

The resulting equation relates a function, N(t), to one or more of its derivatives.\u00a0 In this case, the highest order derivative is the first derivative so this is called a \u201cfirst order differential equation\u201d.\u00a0 We are going to encounter a number of these in AP Physics and also a handful of second order differential equations as well.\u00a0 You will learn how to solve some of these in math class this year.\u00a0 If you continue in math and science, you may spend a number of semesters learning more about this topic.\u00a0 For now, I am going to show you a method that will be sufficient for our specific needs.<\/p>\n

\u00a0WHAT IS A SOLUTION?<\/strong><\/p>\n

When you solve an algebraic equation, you find a number that you can use in place of the variable, thus obtaining a true statement.\u00a0 For example, x= 3 is a solution to the equation 2x + 4 = 10 because when you replace x with 3 in that equation, you get a true result.\u00a0 And even if you didn\u2019t learn the step-by-step method of solving that equation, you could still verify that x=3 works.\u00a0 It wouldn\u2019t matter if the solution came to you in a dream!\u00a0 Once you verified it, you\u2019d know you were right.<\/p>\n

Now we have a different kind of equation:<\/p>\n

\"post21pic4\"<\/a><\/p>\n

We are not looking for a number.\u00a0 We are looking for a function<\/span>, one that will make a true statement when we use it to replace N(t) in that differential equation.\u00a0 Here is how we are going to do this:<\/p>\n

(Don\u2019t worry if this doesn\u2019t \u201cclick\u201d at first.\u00a0 We will walk through this several times.)<\/p>\n

1. Based on our intuition, draw a graph of what the function N(t) will look like.<\/p>\n

2. Use our extensive knowledge of pre-calculus to guess a general form of a function that has that a graph shaped like the one we just drew.<\/p>\n

3. Take the derivative of our guess (and second derivative if needed).<\/p>\n

4. Substitute back into the equation we are trying to solve to see if we get a true statement.<\/p>\n

If we have guessed correctly (as we often will), we will actually pick up some bonus information.\u00a0 Follow along with me and see what I mean.\u00a0 We\u2019ll start from the beginning.<\/p>\n

We are seeking a function N(t) that will satisfy this differential equation:<\/p>\n

\"post21pic5\"<\/a><\/p>\n

1. Our intuition about radioactive decay suggests that when we find the solution, its graph, as we already noted, will look like this:<\/p>\n

\"post21pic1\"<\/a><\/p>\n

2. There are a number of functions that have that shape.\u00a0 For example, it could be that:<\/p>\n

\"post21pic6\"<\/a><\/p>\n

It\u2019s not a terrible guess. It has the right value at t=0 and it approaches zero asymptotically.\u00a0 I don\u2019t remember seeing that function used to model decay before and it would be all wrong for t approaching -1, but let\u2019s try it anyway.<\/p>\n

3. Taking the derivative, we get\u2026<\/p>\n

\"post21pic7\"<\/a><\/p>\n

4. Now, substituting the function and its derivative back into the original differential equation, we get:<\/p>\n

\"post21pic8\"<\/a><\/p>\n

Hmm\u2026can this be true?\u00a0 For a given value of k, it is true at some particular<\/i> time. But we want a solution that is ALWAYS true.\u00a0 There is nothing we can do to fix this one.\u00a0 It turns out that our initial guess, though not terrible, was wrong.\u00a0 OK, new guess.\u00a0 Let\u2019s try an exponential function.\u00a0 Our memories from pre-calc \u00a0give us high hopes for this one:<\/p>\n

\"post21pic9\"<\/a><\/p>\n

In this case, we would have the derivative:<\/p>\n

\"post21pic10\"<\/a><\/p>\n

(because the derivative of \u201ce to the something\u201d is \u201ce to the something\u201d and then there is that chain rule multiplier.)<\/p>\n

But what happens when we substitute these expressions back into the original differential equation?<\/p>\n

\"post21pic11\"<\/a><\/p>\n

Can this be true? Why yes, but only if c = k.\u00a0 In other words, we have just learned that we had correctly guessed the general form of the solution but it can\u2019t be just any old exponentially decaying function.\u00a0 The decay constant in the function has to match the constant of proportionality in the original differential equation. \u00a0So with that adjustment, we have our solution:<\/p>\n

\"post21pic12\"<\/a><\/p>\n

This kind of thing happens a lot when we use this technique.\u00a0 At the end of the process, when you ask, \u201ccan this equation be true?\u201d you get as the answer: \u201cyes, but only if\u2026\u201d followed by some new information that tells you the required value of some constant.<\/p>\n

Closing remarks<\/strong><\/p>\n

Please do not worry if you have found this post to be challenging.\u00a0 In the next practice set, I will give you a couple of additional examples and walk you through this process step-by-step.\u00a0 And we will also do a couple together in class.\u00a0 But review this a few times and you may realize that it is just another way to make use of derivatives and we still have not used any rules beyond the ones I showed you in the earlier posts.\u00a0 But that does not mean that this was easy!<\/p>\n","protected":false},"excerpt":{"rendered":"

And a first look at differential equations At some point in the past, I believe you learned a little about radioactive decay and half-life. Here are some things that you may remember: 1. The size of a radioactive sample is … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false},"categories":[1],"tags":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4uvY7-5j","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/329"}],"collection":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=329"}],"version-history":[{"count":6,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/329\/revisions"}],"predecessor-version":[{"id":15979,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/329\/revisions\/15979"}],"wp:attachment":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=329"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=329"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=329"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}