WEBVTT 00:00:00.000 --> 00:00:03.160 This is Giancoli Answers with Mr. Dychko. 00:00:03.760 --> 00:00:07.600 This car initially drives to the west— 225 kilometers— 00:00:07.600 --> 00:00:11.540 and then turns at a 45 degree angle because that's what south-west is 00:00:11.540 --> 00:00:13.540 and drives 98 kilometers. 00:00:13.820 --> 00:00:17.160 And the resultant displacement 00:00:17.160 --> 00:00:19.780 is gonna be from the origin where it started 00:00:19.780 --> 00:00:21.940 to where it finishes here 00:00:22.200 --> 00:00:28.000 so this in red line is d subscript R for resultant. 00:00:29.180 --> 00:00:33.220 And we are gonna figure out the components of the resultant and then 00:00:33.300 --> 00:00:36.560 based on those components, figure out all the details about the resultant. 00:00:36.560 --> 00:00:39.000 So we are gonna need to find this angle Θ in here 00:00:39.000 --> 00:00:42.460 its angle by which it's south of west 00:00:42.660 --> 00:00:45.400 and we'll need to find the length of this resultant 00:00:45.400 --> 00:00:50.740 and we'll use Pythagoras using the x-component of the resultant 00:00:50.740 --> 00:00:53.300 which will be d 1 plus 00:00:53.300 --> 00:00:55.140 the x-component of d 2 00:00:55.520 --> 00:00:58.800 and then the y-component of the resultant is 00:00:58.800 --> 00:01:02.080 the y-component of d 2. 00:01:02.900 --> 00:01:05.460 So let's get started. 00:01:05.780 --> 00:01:07.720 So x-component of resultant is 00:01:07.720 --> 00:01:09.900 those two added together as I was just saying 00:01:09.900 --> 00:01:14.640 so this resultant is along this line here 00:01:14.640 --> 00:01:17.220 with length d 1 plus d 2 x 00:01:17.220 --> 00:01:18.920 and then the resultant is 00:01:18.920 --> 00:01:22.080 since d 1 has no component in the y-direction, 00:01:22.080 --> 00:01:26.280 the y-component of the resultant is entirely due to 00:01:26.500 --> 00:01:29.300 the y-component of d 2. 00:01:30.440 --> 00:01:34.260 Now we know cos of 45 degrees is adjacent over hypotenuse 00:01:34.260 --> 00:01:37.100 and so that's d 2 x over d 2 00:01:37.320 --> 00:01:40.000 and multiply both sides by d 2 00:01:40.700 --> 00:01:43.480 and then switch the sides around and then you get 00:01:43.560 --> 00:01:48.220 d 2 x is d 2 times cos 45 or 98 kilometers 00:01:48.380 --> 00:01:51.260 times cos 45 00:01:51.380 --> 00:01:53.260 and make sure your calculator's in degree mode 00:01:53.260 --> 00:01:55.260 if you are using 45 degrees; 00:01:55.900 --> 00:02:00.000 if it's in radian mode, you will have to use π over 4 instead of 45 degrees. 00:02:01.720 --> 00:02:08.320 And then we get 69.296 kilometers 00:02:08.320 --> 00:02:14.860 and I should write that in here I guess. 00:02:16.780 --> 00:02:19.940 And for the y-component sin 45 00:02:19.940 --> 00:02:23.240 is equal to cos 45 so I didn't put that in the calculator 00:02:23.240 --> 00:02:27.620 and you get the same answer, 69.296 kilometers. 00:02:28.580 --> 00:02:31.840 And the resultant in the x-direction then is this 00:02:32.060 --> 00:02:35.580 component of d 2 in the x-direction 00:02:35.580 --> 00:02:38.820 plus d 1—225 kilometers— and that gives 00:02:38.820 --> 00:02:44.560 294.296 kilometers to the west is the x-component of the resultant. 00:02:44.880 --> 00:02:49.020 And its y-component is then the y-component of d 2. 00:02:49.820 --> 00:02:52.320 And define the angle of the resultant 00:02:52.500 --> 00:02:54.860 that's this angle in red Θ here 00:02:54.960 --> 00:03:00.320 the tangent of this Θ is the y-component divided by the x-component 00:03:01.100 --> 00:03:05.660 and so that's the y-component of the resultant— 00:03:05.660 --> 00:03:08.020 69.296—divided by 00:03:08.200 --> 00:03:11.800 the x-component of the resultant— 294.296 kilometers—and then 00:03:11.800 --> 00:03:14.940 take the inverse tangent of that result 00:03:14.940 --> 00:03:18.740 and you get 13.25 degrees 00:03:18.800 --> 00:03:22.700 and that's the angle below the horizontal there. 00:03:24.460 --> 00:03:28.060 And then the length of the resultant is given by using Pythagoras 00:03:28.060 --> 00:03:29.640 and so we take the square root of 00:03:29.780 --> 00:03:31.600 the x-component of the resultant squared 00:03:31.600 --> 00:03:33.720 plus the y-component of the resultant squared 00:03:33.720 --> 00:03:35.240 and we get 302. 00:03:35.480 --> 00:03:39.320 And so our final answer then is 302 kilometers 00:03:39.320 --> 00:03:41.580 for the resultant displacement 00:03:41.580 --> 00:03:43.600 13 degrees south of west 00:03:43.600 --> 00:03:45.600 and I'll explain south of west a little bit. 00:03:46.020 --> 00:03:51.980 So if this angle was zero, the thing would be going entirely west 00:03:51.980 --> 00:03:56.240 but it's deflected downwards slightly, it's deflected towards the west 00:03:56.240 --> 00:03:58.240 or sorry, let me say that again, 00:03:58.240 --> 00:04:01.480 it's deflected towards the south compared to west. 00:04:01.680 --> 00:04:04.700 So when you say south of west, you are saying 00:04:04.880 --> 00:04:07.100 to the south with respect to west 00:04:07.100 --> 00:04:10.580 or you know, to the south of west. 00:04:11.100 --> 00:04:14.780 Hopefully that kind of makes sense so there we go.