WEBVTT 00:00:00.000 --> 00:00:03.240 This is College Physics Answers with Shaun Dychko. 00:00:03.840 --> 00:00:05.180 A Supernova occurred 00:00:05.180 --> 00:00:07.740 in the Magellanic Cloud in 1987 00:00:07.900 --> 00:00:11.100 and that cloud is 120,000 light years away 00:00:11.480 --> 00:00:14.200 and some neutrinos were detected on Earth 00:00:14.240 --> 00:00:17.040 as well as light photons, you know, 00:00:17.040 --> 00:00:19.500 light from the Supernova was also detected 00:00:19.640 --> 00:00:22.860 but there's a time difference between when they were detected. 00:00:22.920 --> 00:00:25.040 The neutrinos were slightly slower 00:00:25.260 --> 00:00:29.000 and that's on account of their mass; 00:00:29.260 --> 00:00:31.360 to the extent that they have mass, 00:00:31.500 --> 00:00:34.500 they are not capable of traveling at the speed of light. 00:00:35.560 --> 00:00:39.620 So we are going to estimate based on 00:00:39.660 --> 00:00:44.860 kinetic energy of 700 kiloelectron volts for the neutrinos 00:00:44.940 --> 00:00:49.000 and an estimated mass of 7 electron volts per c squared, 00:00:49.160 --> 00:00:53.140 what is the time difference between receiving the photons 00:00:53.180 --> 00:00:55.560 versus receiving the neutrinos? 00:00:56.960 --> 00:00:58.440 The question in part (a) asks 00:00:58.440 --> 00:01:00.960 what is this Lorentz factor γ? 00:01:01.300 --> 00:01:04.340 So we know that kinetic energy is γ minus 1 00:01:04.400 --> 00:01:06.720 times rest mass times c squared 00:01:06.760 --> 00:01:09.820 and then we can expand the bracket by multiplying through 00:01:09.820 --> 00:01:11.060 by mc squared here and we have 00:01:11.060 --> 00:01:13.320 γmc squared minus mc squared, 00:01:13.580 --> 00:01:16.520 we'll add mc squared to both sides 00:01:17.480 --> 00:01:21.740 and we have γ times mc squared equals kinetic energy plus mc squared 00:01:22.020 --> 00:01:24.540 then divide both sides by mc squared 00:01:25.800 --> 00:01:30.160 and then we get γ is kinetic energy plus mc squared all over mc squared 00:01:30.500 --> 00:01:33.340 and that's 700 times 10 to the 3 electron volts 00:01:33.380 --> 00:01:36.760 plus 7 electron volts per c squared times c squared 00:01:36.840 --> 00:01:40.380 divided by 7 electron volts per c squared times c squared 00:01:40.440 --> 00:01:45.240 and this works out to 100001— that's the value of gamma. 00:01:47.140 --> 00:01:50.060 In part (b), we have to find this time difference. 00:01:50.120 --> 00:01:52.820 Well, let's first figure out what is the distance 00:01:53.480 --> 00:01:57.400 and this is a distance of 120000 light years, 00:01:57.440 --> 00:01:59.980 which means it's the distance that 00:02:00.440 --> 00:02:04.520 light would travel in 1 year times 120000. 00:02:04.780 --> 00:02:09.940 So this l represents the speed of light and this y represents years 00:02:10.040 --> 00:02:15.320 so sometimes, I like to abbreviate this as c.y— 00:02:15.380 --> 00:02:19.000 the speed of light multiplied by a time period of 1 year. 00:02:19.700 --> 00:02:23.300 So we are going to turn this into units of c times s 00:02:23.300 --> 00:02:27.220 and this is 00:02:27.400 --> 00:02:30.280 also a bit strange because, you know, 00:02:30.340 --> 00:02:32.940 normally, I would convert this into a distance of meters 00:02:33.120 --> 00:02:34.380 but I am not going to do that here 00:02:34.380 --> 00:02:36.280 because I want this speed of light... 00:02:36.340 --> 00:02:40.040 I want this distance as a factor of 00:02:40.240 --> 00:02:42.520 or a factor times the speed of light 00:02:42.900 --> 00:02:46.640 because the speed of light was going to cancel conveniently later on. 00:02:46.900 --> 00:02:48.100 So we are just going to convert 00:02:48.100 --> 00:02:50.800 the year unit though into seconds 00:02:50.920 --> 00:02:54.200 so we multiply by 365.25 days per year and then 00:02:54.240 --> 00:02:57.820 24 hours per day and then 3600 seconds per hour 00:02:58.000 --> 00:03:02.920 and we are left with 3.786912 times 10 to the 12 00:03:02.960 --> 00:03:05.660 times the speed of light-seconds. 00:03:06.460 --> 00:03:07.600 Alright! 00:03:08.340 --> 00:03:10.080 So if that seems strange 00:03:10.100 --> 00:03:13.260 well, it's just as strange as light years and this is now just light seconds. 00:03:13.420 --> 00:03:14.740 Okay! 00:03:15.440 --> 00:03:18.740 The time for the neutrino to travel 00:03:18.740 --> 00:03:21.600 is going to be this distance divided by the speed of the neutrino 00:03:21.740 --> 00:03:23.120 and we need to 00:03:23.260 --> 00:03:28.360 solve for v in this expression for the Lorentz factor gamma. 00:03:29.380 --> 00:03:32.880 So we have... 00:03:33.340 --> 00:03:35.440 we can square both sides 00:03:36.560 --> 00:03:40.280 and so we have γ squared is 1 over 1 minus v squared over c squared 00:03:40.320 --> 00:03:42.400 then multiply both sides by 1 minus 00:03:42.400 --> 00:03:46.060 v squared over c squared divided by γ squared 00:03:50.500 --> 00:03:53.040 and then we get 1 minus v squared over c squared equals 00:03:53.040 --> 00:03:54.380 1 over γ squared 00:03:54.740 --> 00:03:58.140 and then we'll add v squared over c squared to both sides 00:03:59.060 --> 00:04:04.680 and subtract 1 over γ squared from both sides 00:04:05.700 --> 00:04:07.120 then switch the sides around 00:04:07.120 --> 00:04:08.820 and then we get v squared over c squared equals 00:04:08.820 --> 00:04:10.520 1 minus 1 over γ squared, 00:04:11.040 --> 00:04:14.420 multiply both sides by c squared and you get this line, 00:04:14.460 --> 00:04:16.200 take the square root of both sides 00:04:16.380 --> 00:04:18.720 and you have v then is the speed of light times square root 00:04:18.720 --> 00:04:20.960 1 minus 1 over γ squared. 00:04:21.560 --> 00:04:24.140 And I am not going to substitute in a number for c, 00:04:24.200 --> 00:04:25.920 I am just going to leave it as the factor c 00:04:25.980 --> 00:04:29.220 and for the same reason that I left this factor c 00:04:29.260 --> 00:04:30.540 in this distance as well, 00:04:30.540 --> 00:04:32.540 it's going to cancel away later on. 00:04:33.020 --> 00:04:38.460 So we have the square root of 1 minus 1 over 100001 squared 00:04:38.460 --> 00:04:41.480 is this number and this number comes from 00:04:41.540 --> 00:04:43.500 spreadsheet program Excel 00:04:43.660 --> 00:04:46.860 because the calculator isn't capable of handling 00:04:46.860 --> 00:04:48.200 so many digits but 00:04:48.240 --> 00:04:50.340 but Excel spreadsheet can 00:04:50.640 --> 00:04:52.520 so this is square root 1 minus 00:04:52.680 --> 00:04:55.280 1 over 100001 squared 00:04:55.360 --> 00:04:58.560 and that makes 0.9 with ten 9s 00:04:58.600 --> 00:05:01.080 a 5 and then 0001 00:05:02.220 --> 00:05:05.360 so it's that factor times c is the speed. 00:05:06.120 --> 00:05:07.340 Okay! 00:05:07.800 --> 00:05:09.300 Now the time difference then 00:05:09.340 --> 00:05:11.980 is going to be the time it takes for the neutrino to arrive 00:05:12.020 --> 00:05:13.320 minus the time it takes 00:05:13.320 --> 00:05:16.340 for a photon to arrive traveling at the speed of light c. 00:05:17.100 --> 00:05:20.240 So that's the distance divided by the speed of the neutrino minus 00:05:20.240 --> 00:05:21.940 the distance divided by the speed of light 00:05:21.980 --> 00:05:25.260 and we can factor the d out and we have d times 1 over 00:05:25.260 --> 00:05:27.920 speed of the neutrino minus 1 over speed of light. 00:05:28.580 --> 00:05:30.900 Now I don't want to write in the number for 00:05:30.900 --> 00:05:32.460 the speed of light because we are dealing with 00:05:32.500 --> 00:05:34.660 such small... 00:05:35.240 --> 00:05:37.580 we are dealing with numbers that are so precise 00:05:38.440 --> 00:05:41.580 I would be introducing a problem by 00:05:41.580 --> 00:05:45.180 entering in a speed for the speed of light unless I had 00:05:45.180 --> 00:05:47.200 you know, this many digits in the speed of light 00:05:47.200 --> 00:05:50.160 which I don't feel like writing down because that's just 00:05:50.280 --> 00:05:52.260 inconvenient to write down so many digits 00:05:52.300 --> 00:05:55.400 and I did it here but I don't want to do it for the speed of light as well 00:05:55.400 --> 00:05:57.020 and there's a way to get around that issue 00:05:57.020 --> 00:05:58.580 by multiplying this by 00:05:58.640 --> 00:06:00.340 the number 1 00:06:00.480 --> 00:06:02.460 but number 1 is going to look funny— 00:06:02.460 --> 00:06:04.000 it's going to look like c over c 00:06:04.160 --> 00:06:07.160 and then leave the c in the denominator 00:06:07.220 --> 00:06:08.880 for this fraction and then multiply 00:06:08.880 --> 00:06:11.420 the top c into the brackets. 00:06:12.400 --> 00:06:13.960 So we have d over c 00:06:14.080 --> 00:06:17.260 times c over the speed of the neutrino 00:06:17.480 --> 00:06:19.320 c over c, which is 1. 00:06:20.320 --> 00:06:22.800 And this is convenient to have this expression because 00:06:22.820 --> 00:06:26.240 we have c divided by some factor multiplied by c 00:06:26.300 --> 00:06:28.100 and so the c's cancel 00:06:28.500 --> 00:06:32.260 and then this distance is some factor multiplied by c 00:06:32.260 --> 00:06:34.480 which cancels with this c here 00:06:34.560 --> 00:06:37.100 and we are left with seconds in our units here 00:06:37.320 --> 00:06:40.180 times 1 over this big number 00:06:40.960 --> 00:06:45.240 well, it's a small number but, you know, many digits... anyhow... 00:06:45.380 --> 00:06:46.800 minus 1. 00:06:47.440 --> 00:06:49.120 Now this again is something that 00:06:49.120 --> 00:06:51.300 a calculator brain can't really handle 00:06:51.560 --> 00:06:53.620 so we turn to the spreadsheet 00:06:54.280 --> 00:06:55.740 and it's going to be 00:06:56.860 --> 00:07:01.740 this number here, this time 00:07:01.840 --> 00:07:03.920 it was a distance but then 00:07:03.920 --> 00:07:06.160 the c canceled with the c in the denominator here 00:07:06.160 --> 00:07:10.740 so it's 3.786912 times 10 to the 12 seconds— 00:07:11.000 --> 00:07:12.620 that's this part here— 00:07:12.680 --> 00:07:17.480 times 1 over this denominator here 00:07:17.480 --> 00:07:18.180 (whoops) 00:07:18.600 --> 00:07:23.200 which is the cell A1 minus 1. 00:07:23.540 --> 00:07:26.020 So I have plugged this into the Excel 00:07:26.060 --> 00:07:32.420 and it gives the answer 189 so that's 189 seconds. 00:07:32.620 --> 00:07:35.000 So there's 189 seconds between when 00:07:35.000 --> 00:07:37.780 the photons are detected from the Supernova 00:07:37.820 --> 00:07:42.480 and the neutrinos are then detected 189 seconds later.