{"id":188,"date":"2014-05-19T12:34:16","date_gmt":"2014-05-19T16:34:16","guid":{"rendered":"http:\/\/advancedmathyoungstudents.com\/blog\/?p=188"},"modified":"2016-02-13T13:16:46","modified_gmt":"2016-02-13T17:16:46","slug":"velocity-now","status":"publish","type":"post","link":"https:\/\/advancedmathyoungstudents.com\/blog\/?p=188","title":{"rendered":"Velocity NOW!"},"content":{"rendered":"

When we examine the graph of your accelerating hair growth, it\u2019s easy to show that the growth rate is increasing.<\/span>\u00a0 <\/span>We can examine <\/span><\/span>average<\/span><\/i> growth rates for successive time intervals.\"post14pic1\"<\/a><\/span><\/span><\/p>\n

 <\/p>\n

For example, we could choose to look at 10-week intervals, calculating rise over run for each interval.<\/span>\u00a0 <\/span>You should check for yourself to see that the average growth rate increased from just under 0.5 inches\/week over the first 10-week interval to nearly triple that rate over the fifth 10-week interval.<\/span><\/span><\/p>\n

Those are still just average rates.<\/span>\u00a0\u00a0<\/span> But what if you wanted to know how fast your hair was growing right NOW!?\u00a0Not over a finite\u00a0interval but at a specific moment.\u00a0<\/span><\/span>\u00a0 <\/span>There is a way to answer that question too.<\/span><\/span>\u00a0 <\/span>But you may have philosophical or mathematical objections.<\/span><\/span>\u00a0 <\/span>\u201cIf no time passes, how can ANYTHING have a rate of change?\u201d<\/span><\/span>\u00a0 <\/span><\/span><\/span><\/p>\n

The way we answer this kind of question is by making use of the idea of a limit.<\/span>\u00a0 <\/span>Essentially, to find the slope at a given point, we start by finding the slope between two different points: the point we were asked about and another point a little later. But then we move that second point closer to the first one.\u00a0 How close?\u00a0\u00a0Reeeeeeeally close but not all the way there. \u00a0And as\u00a0it gets closer, we look to see if there is a value that\u00a0the slope approaches.\u00a0 If so, we use that value as the instantaneous rate of change.<\/span><\/span><\/p>\n

\u00a0<\/span>There are many websites that explore this concept in greater depth.<\/span>\u00a0 <\/span>And they have some nice animations.\u00a0 Here are a few:<\/span><\/span><\/p>\n

http:\/\/clas.sa.ucsb.edu\/staff\/lee\/Secant,%20Tangent,%20and%20Derivatives.htm<\/span><\/span><\/a><\/p>\n

http:\/\/www.math.umn.edu\/~garrett\/qy\/Secant.html<\/span><\/span><\/a><\/p>\n

http:\/\/www.slu.edu\/classes\/maymk\/Applets\/SecantTangent.html<\/span><\/span><\/a><\/p>\n

Here is an animation taken from the first website listed above:\"post14gif1\"<\/a><\/span><\/p>\n

\u00a0UPDATE:\u00a0 Now that I have discovered Paul Seeburger’s applet, here is a nicer look at that same idea:<\/span><\/p>\n

\n\t\t\t
\n\t\t\t\t
\n\t\t\t\t<\/span>\n\t\t\t\tsecanttangent<\/span>\n<\/div>\n\t\t<\/div>\n

 <\/p>\n

I will leave most of the details for your math teachers to explain.<\/span>\u00a0 <\/span>But I will point out that this is exactly why they want you to learn about limits.<\/span><\/span>\u00a0 <\/span>Limits are the mathematical way to say “reeeeally close but not all the way there”.\u00a0 It\u2019s reasonable to say that limits were invented as a way of dealing with precisely this question: how do we find the slope of a line <\/span><\/span>at a given point?<\/span><\/span><\/span><\/i><\/p>\n

\u00a0<\/span>As your physics teacher, I want you to know two ways to deal with this question.<\/span>\u00a0 <\/span><\/span><\/span><\/p>\n

1. WITHOUT CALCULUS<\/span><\/p>\n

Here is a graphical method that will give us an approximate answer:<\/span><\/p>\n

(This is the method I use with my 11<\/span>th<\/span><\/sup> graders.)<\/span><\/span><\/p>\n

Use a ruler to draw a tangent line.<\/span>\u00a0 <\/span>The slope of that tangent line is also the instantaneous rate of change.<\/span><\/span><\/strong><\/p>\n

So let\u2019s say you wanted to how fast your hair was growing at the end of week #20.<\/span>\u00a0 <\/span>(Note carefully: we are not looking for the average<\/span> rate over the first 20 weeks.<\/span><\/span>\u00a0 <\/span>We are looking for the instantaneous<\/span> rate exactly at t = 20 weeks!<\/span><\/span>\u00a0 <\/span>\u00a0\u00a0<\/span>It\u2019s a different question.)<\/span><\/span><\/p>\n

\"post14pic2\"<\/a>After you draw the tangent line, you choose any two points on that line <\/i>and use them to calculate rise over run.\u00a0 So, using the values we get from the graph above, we calculate that the slope of the tangent line is (15 inches)\/(20 weeks) = .75 inches\/week.\u00a0So now we know:<\/p>\n

At week #20, the instantaneous velocity of hair growth was .75 inches\/week.<\/strong><\/p>\n

\u00a0<\/b>2. WITH CALCULUS<\/b><\/p>\n

The point of the entire discussion up until this point has really been to remind you about the idea of slope as a rate of change and to introduce (or re-introduce) what we mean by instantaneous rate of change.\u00a0 But I don\u2019t expect that we will be drawing many tangent lines by hand during AP Physics.\u00a0 When we need to know the slope of a graph, there are formulas for that kind of thing.\u00a0 We\u2019ll start looking at those formulas\u00a0in the next post.<\/p>\n

\u00a0<\/b><\/p>\n","protected":false},"excerpt":{"rendered":"

When we examine the graph of your accelerating hair growth, it\u2019s easy to show that the growth rate is increasing.\u00a0 We can examine average growth rates for successive time intervals.   For example, we could choose to look at 10-week … 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\/p4uvY7-32","jetpack_sharing_enabled":true,"_links":{"self":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/188"}],"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=188"}],"version-history":[{"count":8,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/188\/revisions"}],"predecessor-version":[{"id":623,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=\/wp\/v2\/posts\/188\/revisions\/623"}],"wp:attachment":[{"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=188"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=188"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/advancedmathyoungstudents.com\/blog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=188"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}