1 00:00:03,430 --> 00:00:08,039 We've now described what we call the average velocity for a time 2 00:00:08,039 --> 00:00:11,190 interval between when the runner started at time t 3 00:00:11,190 --> 00:00:13,790 to a later time at t plus delta t. 4 00:00:13,790 --> 00:00:16,760 And we described that as the component 5 00:00:16,760 --> 00:00:20,410 of the displacement vector divided by the time interval 6 00:00:20,410 --> 00:00:23,540 times a unit vector, i hat. 7 00:00:23,540 --> 00:00:25,640 Now what we'd like to ask is a separate question. 8 00:00:25,640 --> 00:00:27,370 So our question now is, what do we 9 00:00:27,370 --> 00:00:39,210 mean by the velocity at some specific time, T1? 10 00:00:39,210 --> 00:00:41,560 Now in order to understand that, let's just 11 00:00:41,560 --> 00:00:45,270 make a plot of the position function. 12 00:00:45,270 --> 00:00:48,140 So remember we called the component of the position 13 00:00:48,140 --> 00:00:49,570 function x of t. 14 00:00:49,570 --> 00:00:52,660 So we're going to plot the component of the position 15 00:00:52,660 --> 00:00:55,920 function with respect to time. 16 00:00:55,920 --> 00:00:58,410 Now let's just say that the runner started 17 00:00:58,410 --> 00:01:00,050 at the origin of time equal 0. 18 00:01:00,050 --> 00:01:04,560 So I can make some type of arbitrary plot of that position 19 00:01:04,560 --> 00:01:05,410 function. 20 00:01:05,410 --> 00:01:11,260 And let's indicate in particular, the time T1. 21 00:01:11,260 --> 00:01:16,690 So what this represents is x of T1. 22 00:01:16,690 --> 00:01:20,450 And so first I'd like to consider the interval T1 23 00:01:20,450 --> 00:01:25,080 and T1 plus some later time, delta T. 24 00:01:25,080 --> 00:01:30,670 So let's make this T1 plus delta t. 25 00:01:30,670 --> 00:01:36,220 This is the time delta t and up here 26 00:01:36,220 --> 00:01:41,170 we have our position function at T1 plus delta T. 27 00:01:41,170 --> 00:01:45,090 Then for this time interval, the average velocity, 28 00:01:45,090 --> 00:01:47,690 so for this particular time interval, 29 00:01:47,690 --> 00:01:54,500 the average represents delta x over delta t. 30 00:01:54,500 --> 00:01:59,100 So it's just rise over run. 31 00:01:59,100 --> 00:02:03,380 It's just the slope of this straight line. 32 00:02:03,380 --> 00:02:06,970 So for this particular interval, the average 33 00:02:06,970 --> 00:02:16,090 is the slope of the line shown here on the figure. 34 00:02:16,090 --> 00:02:18,350 Now this is just an average velocity 35 00:02:18,350 --> 00:02:21,320 and now what we would like to do is shrink down 36 00:02:21,320 --> 00:02:27,470 our interval delta T. So now let's make another case where 37 00:02:27,470 --> 00:02:32,840 we shrink delta t and let's again calculate 38 00:02:32,840 --> 00:02:33,990 the average velocity. 39 00:02:33,990 --> 00:02:39,050 So for instance, suppose we have a smaller delta t 40 00:02:39,050 --> 00:02:41,520 and we draw that line. 41 00:02:41,520 --> 00:02:48,360 Then our average velocity represents that slope. 42 00:02:48,360 --> 00:02:50,840 And again, we keep on taking a limit. 43 00:02:50,840 --> 00:02:55,420 So now we have another slope so we have one slope, two slopes, 44 00:02:55,420 --> 00:02:59,430 and now we shrink again to a new delta t 45 00:02:59,430 --> 00:03:03,760 and you can see that the slope is changing. 46 00:03:03,760 --> 00:03:08,690 And if we consider the limit as delta t 47 00:03:08,690 --> 00:03:14,210 goes to 0 of this sequence of slopes, 48 00:03:14,210 --> 00:03:17,640 then what are we getting, you can see graphically, 49 00:03:17,640 --> 00:03:24,190 that eventually we will get to a line which 50 00:03:24,190 --> 00:03:37,480 is the slope of the tangent line at time T1. 51 00:03:37,480 --> 00:03:40,200 And so in this particular case, what 52 00:03:40,200 --> 00:03:46,560 we mean by the instantaneous velocity, v at time T1, 53 00:03:46,560 --> 00:03:51,620 is the limit as delta x goes to 0 i hat here 54 00:03:51,620 --> 00:03:55,340 where we're taking, this is the limit, 55 00:03:55,340 --> 00:04:01,750 delta t goes to 0 x at T1 plus delta t minus x1 56 00:04:01,750 --> 00:04:08,480 of t divided by delta t and the whole thing is a vector, i hat. 57 00:04:08,480 --> 00:04:13,510 So what a limit is, is a sequence of numbers. 58 00:04:13,510 --> 00:04:17,660 So we take a fixed delta t, we calculate the slope. 59 00:04:17,660 --> 00:04:20,930 We take a smaller delta t, calculate the slope. 60 00:04:20,930 --> 00:04:23,250 And each time we do that, the slopes represent 61 00:04:23,250 --> 00:04:26,520 a sequence of numbers and the limit of that sequence 62 00:04:26,520 --> 00:04:29,120 you can see graphically, is the slope 63 00:04:29,120 --> 00:04:31,850 of the tangent line at time t 1. 64 00:04:31,850 --> 00:04:44,300 And so what we say is, v of T1 is the instantaneous velocity 65 00:04:44,300 --> 00:04:48,140 at time t equals t1. 66 00:04:48,140 --> 00:04:52,650 And that's how we describe instantaneous velocity 67 00:04:52,650 --> 00:04:54,590 at some specific time. 68 00:04:54,590 --> 00:04:57,460 If we were now being a little bit more general, 69 00:04:57,460 --> 00:05:01,180 we could just say that v at any time t 70 00:05:01,180 --> 00:05:04,630 is the limit delta t goes to 0 delta 71 00:05:04,630 --> 00:05:08,460 x over delta t ball direction i hat, 72 00:05:08,460 --> 00:05:11,210 and the only thing here is we're no longer considering 73 00:05:11,210 --> 00:05:14,730 T1 but an arbitrary time t. 74 00:05:14,730 --> 00:05:18,460 This quantity, the limit, is awkward to write every time. 75 00:05:18,460 --> 00:05:19,680 It has a name. 76 00:05:19,680 --> 00:05:24,400 And that's precisely what we call the derivative 77 00:05:24,400 --> 00:05:26,780 of the position function. 78 00:05:26,780 --> 00:05:30,250 So our instantaneous velocity is the time derivative 79 00:05:30,250 --> 00:05:35,330 of the position function at any instant in time.