1 00:00:00,005 --> 00:00:10,126 [music] Maybe I don't know my position and my velocity, but I know my acceleration. 2 00:00:10,126 --> 00:00:16,990 Maybe my acceleration at sometime t is just a constant 8 units per second 3 00:00:16,990 --> 00:00:21,838 squared, right? So, I'm going to accelerate at a constant 4 00:00:21,838 --> 00:00:25,432 rate. Is that enough information to determine my 5 00:00:25,432 --> 00:00:28,465 velocity? We have to think back to how velocity and 6 00:00:28,465 --> 00:00:33,036 acceleration are related. My definition and acceleration is the 7 00:00:33,036 --> 00:00:36,422 change to the derivative in velocity, right. 8 00:00:36,423 --> 00:00:39,477 Acceleration is rate of change in velocity. 9 00:00:39,478 --> 00:00:44,680 Or in other words, right, velocity is an anti-derivative of acceleration. 10 00:00:44,680 --> 00:00:49,336 So, I know that a of t is 8. Let's now solve for v. 11 00:00:49,336 --> 00:00:56,329 Okay, so a of t, my acceleration, is just constant function 8, and v of t is the 12 00:00:56,329 --> 00:01:02,359 anti-derivative of my acceleration, right. If I anti-differentiate acceleration, I'm 13 00:01:02,359 --> 00:01:06,084 going to get velocity. It means I'm anti-differentiating 8. 14 00:01:06,085 --> 00:01:12,611 Well what anti-differentiates to 8 or 8t, plus some constant C? 15 00:01:12,611 --> 00:01:15,824 It's that constant again. Right? 16 00:01:15,824 --> 00:01:20,252 Knowing my acceleration doesn't determine my velocity, it only determines my 17 00:01:20,252 --> 00:01:24,260 velocity up to some constant. I could be going really fast or really 18 00:01:24,260 --> 00:01:27,167 slow, but still accelerating at the same rate. 19 00:01:27,168 --> 00:01:30,834 Well, in any case, I've at least got a formula for my velocity with that 20 00:01:30,834 --> 00:01:33,661 constant. Now, can I use that formula to determine 21 00:01:33,661 --> 00:01:35,880 my position? Same kind of game. 22 00:01:35,880 --> 00:01:39,795 My velocity is the rate in change in my position, right? 23 00:01:39,795 --> 00:01:44,376 My velocity is the derivative of position. And that means position is an 24 00:01:44,376 --> 00:01:49,259 anti-derivative of velocity. So, let's anti-differentiate. 25 00:01:49,259 --> 00:01:54,766 So, p of t is an anti-derivative of my velocity, and I figured out my velocity a 26 00:01:54,766 --> 00:01:58,644 minute ago. My velocity is 8t plus c. 27 00:01:58,645 --> 00:02:02,084 So, I want to anti-differentiate 8t plus c. 28 00:02:02,085 --> 00:02:05,136 Well, it's an anti-derivative of a sum, so it's the sum of the anti-derivatives. 29 00:02:05,136 --> 00:02:09,082 And what's an anti-derivative for 8 times something? 30 00:02:09,082 --> 00:02:15,154 Well, that's a constant multiple, so it's 8 times the anti-derivative of t, plus an 31 00:02:15,154 --> 00:02:19,849 anti-derivative for C. Now, 8 times, what's an anti-derivative 32 00:02:19,849 --> 00:02:22,955 for t? Well one of them is t squared over 2. 33 00:02:22,956 --> 00:02:28,054 And what's an anti-derivative for C? Well, C times t is an anti-derivative for 34 00:02:28,054 --> 00:02:30,570 C. Then, I should add some constant here. 35 00:02:30,570 --> 00:02:33,583 I'm going to call that constant big D. Right. 36 00:02:33,584 --> 00:02:36,774 So, here's my position. I guess I could write this a little bit 37 00:02:36,774 --> 00:02:40,870 more nicely, coz the 8 and the dividing by 2 simplify that I could write it as 4t 38 00:02:40,870 --> 00:02:45,162 squared, plus Ct, plus D. Well, that's kind of weird, right? 39 00:02:45,162 --> 00:02:51,105 Why are there two constants in my answer? Just knowing that my acceleration is 8 40 00:02:51,105 --> 00:02:56,135 units per second squared, doesn't tell me my initial velocity, right? 41 00:02:56,136 --> 00:03:01,185 This quantity C here, is really v of 0. Right? 42 00:03:01,185 --> 00:03:06,914 It's my initial velocity. And I could be accelerating at a rate of 8 43 00:03:06,914 --> 00:03:11,106 units per second squared. But starting with any of a range of 44 00:03:11,106 --> 00:03:15,926 possible initial velocities, right? I could be going really fast at first or 45 00:03:15,926 --> 00:03:19,820 really slow at first. But still, always accelerating at 8 units 46 00:03:19,820 --> 00:03:24,298 per second squared. So, knowing this doesn't nail down C. 47 00:03:24,298 --> 00:03:29,183 Similarly, just knowing my velocity doesn't nail down my initial position. 48 00:03:29,183 --> 00:03:32,741 Right? This D here is really p of 0. 49 00:03:32,741 --> 00:03:37,037 Right? If I plug in t equals 0, I just get D. 50 00:03:37,038 --> 00:03:41,911 And that means that D is really providing my initial position. 51 00:03:41,912 --> 00:03:48,827 So, if I know my initial position and my initial velocity and my acceleration, then 52 00:03:48,827 --> 00:03:52,550 I can nail down an exact formula for my position. 53 00:03:52,550 --> 00:03:57,526 I can get rid of these mystery constants. The important lesson to take away here is 54 00:03:57,526 --> 00:04:00,182 that anti-differentiating can actually be useful. 55 00:04:00,182 --> 00:04:05,295 Anti-differentiating let's us take, say, velocity information and produce the 56 00:04:05,295 --> 00:04:10,277 position data. Or even better, in this case, it let us 57 00:04:10,277 --> 00:04:19,548 take acceleration information, figure out the velocity, and then, take that velocity 58 00:04:19,548 --> 00:04:23,773 information to figure out my position.