1 00:00:00,000 --> 00:00:01,862 - [Voiceover] Alright, we can now do 2 00:00:01,862 --> 00:00:03,626 the math to solve for v. 3 00:00:03,626 --> 00:00:06,320 So let me just simplify the right hand side 4 00:00:06,320 --> 00:00:08,636 of this equation, v minus negative v. 5 00:00:08,687 --> 00:00:11,074 Well, that's just going to be two v. 6 00:00:11,230 --> 00:00:14,446 One minus negative of v squared over c squared. 7 00:00:14,446 --> 00:00:18,539 Well that's just one plus positive v squared over c squared. 8 00:00:18,939 --> 00:00:20,559 And let's see, what can we do next? 9 00:00:20,947 --> 00:00:24,227 Well we can multiply, we want to solve for v. 10 00:00:24,510 --> 00:00:26,642 We can multiply both sides of this equation by 11 00:00:26,740 --> 00:00:30,182 one plus v squared over c squared. 12 00:00:30,710 --> 00:00:31,819 So let's do that. 13 00:00:31,860 --> 00:00:34,931 One plus v squared over c squared. 14 00:00:35,539 --> 00:00:37,508 On the right hand side, those cancel. 15 00:00:37,793 --> 00:00:40,781 And on the left hand side, we can distribute the 0.8 16 00:00:41,215 --> 00:00:43,904 and we will get 0.8, 17 00:00:44,119 --> 00:00:46,616 or we can distribute the 0.8 c, I should say, 18 00:00:46,616 --> 00:00:51,616 0.8 c plus 0.8 c v squared over c squared. 19 00:00:53,034 --> 00:00:54,996 Well, the c in the numerator is going to cancel 20 00:00:54,996 --> 00:00:56,384 with one of these c's in the denominator. 21 00:00:56,471 --> 00:01:01,471 So it's going to be plus 0.8 v squared over c. 22 00:01:04,203 --> 00:01:08,214 And then, it is going to be equal to two v. 23 00:01:08,255 --> 00:01:10,885 And remember, we want to solve for v. 24 00:01:11,018 --> 00:01:13,404 So we're setting up essentially a quadratic in v. 25 00:01:15,347 --> 00:01:19,349 So let's now find ourselves some real estate, 26 00:01:19,349 --> 00:01:21,397 so let's go right over there. 27 00:01:21,809 --> 00:01:23,777 Let's subtract two v from both sides. 28 00:01:23,828 --> 00:01:26,296 And actually, I'm going to write it in order of degree. 29 00:01:26,348 --> 00:01:31,348 So on the left hand side, I have 0.8 over c v squared, 30 00:01:34,324 --> 00:01:37,615 and then minus two v, I subtracted two v from both sides, 31 00:01:37,784 --> 00:01:41,515 Plus 0.8 c, 32 00:01:41,585 --> 00:01:46,585 plus 0.8 c is equal to zero. 33 00:01:48,058 --> 00:01:49,399 I guess if we want to simplify a little bit, 34 00:01:49,405 --> 00:01:52,104 we can multiply both sides by c. 35 00:01:54,077 --> 00:01:55,833 If we multiply both sides by c, this will become 36 00:01:55,898 --> 00:01:59,847 0.8 v squared 37 00:02:00,178 --> 00:02:05,178 minus two c v plus 0.8c squared equals zero. 38 00:02:08,223 --> 00:02:10,800 And we could keep trying to algebraically manipulate this, 39 00:02:10,800 --> 00:02:13,233 but we could just go straight to the quadratic formula here 40 00:02:13,238 --> 00:02:14,313 to solve for v. 41 00:02:14,957 --> 00:02:16,576 I'll do that in a different color just for kicks. 42 00:02:16,982 --> 00:02:19,956 v is going to be equal to negative b. 43 00:02:20,286 --> 00:02:23,113 So this right over here is b. 44 00:02:25,799 --> 00:02:27,235 This is our a. 45 00:02:27,983 --> 00:02:30,136 And this is our c. 46 00:02:30,258 --> 00:02:32,550 We're talking about the different variables 47 00:02:32,555 --> 00:02:34,455 using the quadratic formula. 48 00:02:35,098 --> 00:02:36,753 So v is going to be negative b. 49 00:02:36,817 --> 00:02:38,448 So the negative of negative two c. 50 00:02:38,534 --> 00:02:41,884 So it's going to be two c, plus or minus the square root 51 00:02:42,074 --> 00:02:46,551 of b squared, so negative two c squared is positive 52 00:02:46,684 --> 00:02:51,486 four c squared, minus four, times a, 53 00:02:51,583 --> 00:02:56,583 so minus four times 0.8 times c, so times 0.8 c squared. 54 00:03:03,951 --> 00:03:08,951 All of that over two a, so 0.8 times two is 1.6. 55 00:03:10,796 --> 00:03:13,496 Now let's see this is going to be equal to two c 56 00:03:13,756 --> 00:03:16,084 plus or minus the square root -- 57 00:03:16,322 --> 00:03:18,905 now I can factor out a four c squared. 58 00:03:19,154 --> 00:03:24,154 Four c squared times one minus, 0.8 times 0.8 is 0.64. 59 00:03:29,080 --> 00:03:32,129 All I did was factor out a four c squared from both terms. 60 00:03:32,400 --> 00:03:35,786 Of course, all of that over 1.6 61 00:03:36,162 --> 00:03:37,642 Scroll down a little bit. 62 00:03:37,962 --> 00:03:39,383 And this is going to be equal to -- 63 00:03:39,400 --> 00:03:40,730 this is all algebra at this point -- 64 00:03:40,782 --> 00:03:42,844 two c plus or minus -- 65 00:03:42,952 --> 00:03:45,885 if I take the four c squared out of the radical, 66 00:03:45,993 --> 00:03:48,646 its going to be two c. 67 00:03:48,872 --> 00:03:52,083 So two c plus or minus two c times the square root 68 00:03:52,112 --> 00:03:53,883 of one minus 0.64. 69 00:03:54,040 --> 00:03:56,450 Well, that's 0.36. 70 00:03:56,953 --> 00:03:58,278 Things are getting nicely simple now. 71 00:03:58,278 --> 00:04:00,686 All over 1.6. 72 00:04:01,133 --> 00:04:05,005 The square root of 0.36, the principle root of that 0.6. 73 00:04:05,312 --> 00:04:07,890 So that's 0.6. 74 00:04:08,904 --> 00:04:10,001 And move down a little bit. 75 00:04:10,333 --> 00:04:12,741 So this is going to be equal to two c -- 76 00:04:13,049 --> 00:04:14,517 and I can see the end -- 77 00:04:14,604 --> 00:04:19,604 plus or minus 1.2 c, all of that over 1.6. 78 00:04:23,335 --> 00:04:25,581 So we got two possible values for v, 79 00:04:25,667 --> 00:04:29,730 but we can tell if we add this up here, 80 00:04:29,730 --> 00:04:34,014 you're going to end up with 3.2 c divided by 1.6, 81 00:04:34,014 --> 00:04:36,528 which is going to be a speed, a velocity, 82 00:04:36,534 --> 00:04:38,407 greater than the speed of light. 83 00:04:38,716 --> 00:04:41,149 So we can rule out the positive version of it. 84 00:04:41,293 --> 00:04:42,599 So we know the answer's going 85 00:04:42,627 --> 00:04:44,084 have to be negative version of it. 86 00:04:44,114 --> 00:04:47,998 So two c minus 1.2 c over 1.6, 87 00:04:48,177 --> 00:04:53,177 which is equal to 0.8 c over 1.6. 88 00:04:54,754 --> 00:04:59,754 Well 0.8 is half of 1.6, so this is 0.5 c. 89 00:05:00,626 --> 00:05:02,803 That was really, really, really cool. 90 00:05:02,947 --> 00:05:04,254 Because what do we know now? 91 00:05:04,468 --> 00:05:09,096 We know now that A in A's frame of reference 92 00:05:09,275 --> 00:05:11,301 feels like it is stationary. 93 00:05:11,493 --> 00:05:14,121 We have A's friend moving with a relative velocity 94 00:05:14,150 --> 00:05:15,724 of eight-tenths of the speed of light. 95 00:05:15,973 --> 00:05:17,802 There could be a third party, C, 96 00:05:17,808 --> 00:05:20,403 that defines a frame of reference moving away from A, 97 00:05:20,408 --> 00:05:23,038 with the velocity of half the speed of light. 98 00:05:24,054 --> 00:05:26,393 Half the speed of light. 99 00:05:26,630 --> 00:05:30,397 And from A's frame of reference, it looks like C's velocity 100 00:05:30,426 --> 00:05:32,522 is closer to B's than it is to A's. 101 00:05:32,608 --> 00:05:34,849 But we're dealing in this relativistic, 102 00:05:34,849 --> 00:05:39,469 this Einstein-ian world because if we then go into 103 00:05:39,469 --> 00:05:42,321 C's frame of reference, it actually looks like A and B 104 00:05:42,347 --> 00:05:46,731 are both leaving C with a speed of 0.5 c. 105 00:05:49,692 --> 00:05:52,351 So here you can view the velocity as positive 0.5 c, 106 00:05:52,576 --> 00:05:57,576 and here you could say the velocity is negative 0.5 c. 107 00:06:01,097 --> 00:06:01,951 This was really cool. 108 00:06:01,979 --> 00:06:03,523 We were able to find a frame of reference 109 00:06:03,523 --> 00:06:05,085 that you could think of it as right in between. 110 00:06:05,440 --> 00:06:07,146 Now what's going to be really cool about this, 111 00:06:07,146 --> 00:06:08,684 is that we can actually -- 112 00:06:08,794 --> 00:06:11,162 you know what some people don't like about Minkowski 113 00:06:11,162 --> 00:06:14,350 space-time diagrams is that it looks asymmetric. 114 00:06:14,541 --> 00:06:17,455 Even though B is moving in A's frame of reference 115 00:06:17,455 --> 00:06:19,777 with a velocity of positive eight-tenths 116 00:06:19,777 --> 00:06:20,798 of the speed of light, 117 00:06:20,798 --> 00:06:22,237 if you were in B's frame of reference, 118 00:06:22,237 --> 00:06:25,994 A is moving to the left with a speed of 0.8 times c. 119 00:06:26,532 --> 00:06:29,568 But now we can define a neutral frame of reference, 120 00:06:29,620 --> 00:06:31,594 where they're both moving in different directions 121 00:06:31,594 --> 00:06:33,109 with the same speed. 122 00:06:33,219 --> 00:06:36,435 Which actually makes interpreting the space-time 123 00:06:36,435 --> 00:06:37,764 diagram a little bit easier. 124 00:06:37,781 --> 00:06:39,337 We will do that in future videos. 125 00:06:39,337 --> 00:00:00,000 But this was a fun problem in it of itself.