1 00:00:00,380 --> 00:00:04,260 Let's do another example where we're projecting something and it lands at a different level 2 00:00:04,300 --> 00:00:06,740 Let's also figure out some other interesting things. We'll figure out 3 00:00:06,930 --> 00:00:09,670 what the actual velocity vector is when it's landing 4 00:00:09,670 --> 00:00:12,300 So both its magnitude and direction 5 00:00:12,340 --> 00:00:16,910 Let's say we're launching something from ground level 6 00:00:17,240 --> 00:00:20,830 and we're going to launch it at a pretty step angle over here 7 00:00:21,120 --> 00:00:39,920 Let's say we launch it with an angle of 80 degrees 8 00:00:39,940 --> 00:00:44,860 and it is going to be going at 30 meters per second 9 00:00:44,880 --> 00:00:48,050 So that's the length of the vector; that's the magnitude of that vector 10 00:00:48,080 --> 00:00:50,640 And let's say we want to make it land on this landing 11 00:00:50,650 --> 00:01:03,110 And this landing over here has a height of 10 meters 12 00:01:03,150 --> 00:01:13,250 So what I want to do first of all is figure out how far along the landing do I actually land? 13 00:01:13,270 --> 00:01:16,330 And maybe I'll add some other information right here 14 00:01:16,340 --> 00:01:20,170 From the launching point to the beginning of the landing-- 15 00:01:20,170 --> 00:01:24,260 let's say this right over here is 2 meters 16 00:01:24,280 --> 00:01:27,910 So we just want to know how far along the landing do we land 17 00:01:27,920 --> 00:01:35,420 So like we did before, we want to break this vector into its horizontal and vertical components 18 00:01:35,440 --> 00:01:37,210 I'm gonna go a little bit faster in this video 19 00:01:37,210 --> 00:01:39,240 hopefully we're getting hold of this type of thing 20 00:01:39,880 --> 00:01:46,020 So the vertical component of our velocity 21 00:01:46,170 --> 00:01:50,900 is going to be equal to the magnitude of our total velocity 30 m/s 22 00:02:01,540 --> 00:02:06,610 And the horizontal component of our velocity is going to be 23 00:02:13,480 --> 00:02:19,250 Again cosine is adjacent over hypotenuse 24 00:02:19,350 --> 00:02:23,780 I'm gonna skipping steps. In the last two videos I go into this in a much more detail 25 00:02:23,910 --> 00:02:30,050 So how much time do we spend in the air? 26 00:02:30,570 --> 00:02:37,940 So once again, in the last few videos, we saw that we can look at our displacement 27 00:02:39,090 --> 00:02:43,330 If we want to figure out time in the air, we know that displacement is equal to 28 00:02:43,550 --> 00:02:47,140 the initial velocity times time-- 29 00:02:47,500 --> 00:02:55,500 let me write change in time, that's technically more correct 30 00:02:55,950 --> 00:03:03,070 plus acceleration times change in time squared over two 31 00:03:03,450 --> 00:03:06,440 Now in our situation, we know what our initial velocity is 32 00:03:06,440 --> 00:03:08,690 We're talking about the vertical direction right over here 33 00:03:08,710 --> 00:03:10,820 So our initial velocity is going to be this 34 00:03:10,820 --> 00:03:12,630 We're trying to figure out how much time in the air 35 00:03:12,630 --> 00:03:14,710 and then the vertical component determines that 36 00:03:14,730 --> 00:03:17,390 because at some point when it hits back to the ground 37 00:03:17,390 --> 00:03:20,720 it's not going to be traveling anymore, so that's what determines this time in the air 38 00:03:20,910 --> 00:03:25,360 So we know the acceleration. Remember the convention when we're dealing with the 39 00:03:25,460 --> 00:03:29,350 vertical dimension is, up is positive, down is negative 40 00:03:29,500 --> 00:03:33,530 So this is -9.8 m/s squared 41 00:03:33,890 --> 00:03:37,430 And then what is the total displacement that we're gonna have? 42 00:03:37,460 --> 00:03:39,430 Well, we're starting at ground level 43 00:03:39,460 --> 00:03:41,500 And we're just talking about the vertical, remember that 44 00:03:41,500 --> 00:03:43,960 So our total displacement is going to be 10 m 45 00:03:43,990 --> 00:03:46,330 So this value right here is going to be 10 m 46 00:04:09,080 --> 00:04:13,560 Times our change in time 47 00:04:13,570 --> 00:04:31,120 So this is -9.8 divided by 2, so it's -4.9 m/s squared, times delta t squared 48 00:04:31,150 --> 00:04:33,920 And then we can subtract 10 from both sides and write this into a 49 00:04:33,920 --> 00:04:36,660 traditional quadratic equation form 50 00:04:36,680 --> 00:04:55,030 So we get -4.9 times delta t squared + 29.54 times delta t -10 is equal to zero 51 00:04:55,390 --> 00:04:58,980 And then we can use the quadratic formula to find the roots of this 52 00:04:59,010 --> 00:05:04,210 So the delta t's that satisfy this quadratic equation are going to be negative B 53 00:05:04,630 --> 00:05:07,780 So -29.54 54 00:05:10,490 --> 00:05:14,030 of 29.54 squared, B squared 55 00:05:14,640 --> 00:05:25,960 -4 times A which is -4.9, the negative times and negative is positive, so it's +4 times +4.9 56 00:05:26,210 --> 00:05:30,000 times--I shouldn't have jumped so fast to get rid of the negatives 57 00:05:30,000 --> 00:05:38,000 So it's gonna be -4 times A which is -4.9, times C which is -10 58 00:05:38,000 --> 00:05:45,550 So just A times C, -4.9 times -10 59 00:05:45,570 --> 00:05:48,900 These two guys, their signs are gonna cancel out 60 00:05:48,940 --> 00:05:58,190 All of that over 2A, over -4.9 times 2, so -9.8 61 00:05:58,550 --> 00:06:01,600 And like we saw in the last video, we want a positive value for this 62 00:06:01,600 --> 00:06:04,790 and negative time is nonsensical. That's kind of going into the past 63 00:06:05,010 --> 00:06:07,020 So we want a positive value 64 00:06:07,050 --> 00:06:11,340 And since we have a negative in the denominator, we want have a negative value up here 65 00:06:11,480 --> 00:06:14,900 And if we already have a negative value here and if we subtract from that negative value 66 00:06:14,900 --> 00:06:16,700 we'll definitely have a negative value here 67 00:06:16,740 --> 00:06:19,470 Then you divide by a negative value. You'll get a positive value 68 00:06:19,470 --> 00:06:22,250 So we can really focus on the subtracting the radical 69 00:06:22,560 --> 00:06:24,890 You can try it out. If you try the positive version 70 00:06:24,890 --> 00:06:27,450 you'll get a negative value for this entire thing 71 00:06:28,040 --> 00:06:30,600 You can try that out after this video 72 00:06:30,600 --> 00:06:32,690 just to verify that that will get a nonsensical answer 73 00:06:32,690 --> 00:06:35,180 So let's use the negative right over here. So we have 74 00:06:35,420 --> 00:06:45,200 -29.54 - square root of 75 00:06:45,220 --> 00:07:00,620 29.54 squared -4 times (-4.9)ĄÁ(-10) which is 49, times 49 76 00:07:00,780 --> 00:07:09,360 Actually I should add some parentheses. Okay 77 00:07:10,250 --> 00:07:14,910 Times 49. So this right over here would give me the numerator if I evaluate this 78 00:07:15,500 --> 00:07:22,110 We've got a negative value, and I divide that by -9.8 79 00:07:22,890 --> 00:07:31,350 Gives me 5.67 second 80 00:07:31,500 --> 00:07:35,790 And you can keep the units in there and make sure the dimensional analysis works 81 00:07:35,870 --> 00:07:37,550 You'll find that it does 82 00:07:37,700 --> 00:07:47,420 So our total time in the air is 5.67 seconds 83 00:07:47,610 --> 00:07:51,790 Now what I want to do, the whole point of this, is to figure out 84 00:07:51,790 --> 00:07:55,200 how far along the this landing we land 85 00:07:55,490 --> 00:07:58,890 Well, the horizontal component of our velocity is right over here 86 00:07:59,220 --> 00:08:05,870 We know that our displacement in the horizontal direction 87 00:08:05,890 --> 00:08:09,460 will be our velocity in the horizontal direction-- 88 00:08:09,460 --> 00:08:11,170 it's a constant velocity 89 00:08:11,170 --> 00:08:14,050 So it's the same thing as our average velocity in the horizontal direction 90 00:08:14,270 --> 00:08:20,100 Times the change in time 91 00:08:33,570 --> 00:08:38,250 I won't write the unit. This is m/s times s and it'll give us the answer in m 92 00:08:48,610 --> 00:08:52,200 Gives us 29.53 m 93 00:08:52,650 --> 00:09:04,580 So our total horizontal traveling displacement is 29.53 m 94 00:09:04,600 --> 00:09:11,270 It's a vector. That is our horizontal displacement, which is 29.53 m 95 00:09:11,690 --> 00:09:15,040 Now we've done a lot of deconstructing vectors 96 00:09:15,040 --> 00:09:17,840 What I'm interesting in this video is to construct a vector 97 00:09:17,940 --> 00:09:21,870 So we know our horizontal displacement; we also know our vertical displacement 98 00:09:21,890 --> 00:09:23,910 It's positive 10 m 99 00:09:23,930 --> 00:09:25,700 So what's our total displacement? 100 00:09:25,720 --> 00:09:27,160 Let me write this down 101 00:09:27,180 --> 00:09:34,860 So we have a horizontal displacement of 29.53 m 102 00:09:34,980 --> 00:09:42,330 and we have a vertical displacement of +10 m 103 00:09:42,350 --> 00:09:48,420 So what is our total displacement? 104 00:09:48,430 --> 00:09:51,070 We can use the Pythagorean theorem here 105 00:09:51,090 --> 00:09:53,740 The square of the magnitude of our total displacement is 106 00:09:53,740 --> 00:09:56,130 going to be equal to the sum of these two squares 107 00:09:56,300 --> 00:09:59,160 This is just the Pythagorean theorem 108 00:09:59,180 --> 00:10:06,440 So let me write it over here. So this is the magnitude of our displacement right over here 109 00:10:06,730 --> 00:10:26,060 The magnitude of our total displacement squared is going to be equal to 10^2 +29.53^2 110 00:10:26,080 --> 00:10:30,100 To solve for this, we just take the square root of both sides 111 00:10:30,100 --> 00:10:34,560 If we just take the square root of both sides 112 00:10:34,730 --> 00:10:37,990 we will get the magnitude of our total displacement-- 113 00:10:38,000 --> 00:10:40,390 let's get the calculator out to do that 114 00:10:40,740 --> 00:10:43,730 So the magnitude of our total displacement is 115 00:10:44,320 --> 00:10:49,370 the square root of 10^2 is just 100 + 29-- 116 00:10:49,410 --> 00:10:51,830 I can use all this information 117 00:10:51,830 --> 00:10:58,590 I'll use the Ans, which literally means the previous answer which is 29.53 118 00:10:58,830 --> 00:11:00,410 squared 119 00:11:00,870 --> 00:11:09,360 Gives us a total displacement of 31.18 m 120 00:11:09,400 --> 00:11:14,660 Of course it's a vector. This is only the magnitude. We also need the direction 121 00:11:14,990 --> 00:11:19,640 So one way to specify direction is to give you the angle with the horizontal 122 00:11:19,660 --> 00:11:22,120 And let's call that angle theta 123 00:11:22,190 --> 00:11:27,410 And once again we can use our trig functions over here 124 00:11:27,410 --> 00:11:30,190 We can use pretty much any of the trig functions 125 00:11:30,390 --> 00:11:37,230 But we know the opposite side is 10; we know the hypotenuse here is 31.18 126 00:11:37,250 --> 00:11:40,320 So why not use sine? Sine is opposite over hypotenuse 127 00:11:40,510 --> 00:11:51,780 So we know the sin of theta is going to be equal to 10/31.18 128 00:11:51,800 --> 00:11:54,200 Or if you want solve for theta 129 00:11:54,860 --> 00:11:58,790 you take the arcsine or the inverse sine of both sides 130 00:11:58,820 --> 00:12:12,900 Theta is equal to the inverse sine, or arcsin of 10 / 31.18 131 00:12:12,930 --> 00:12:16,390 Once again get the calculator out to figure out that value 132 00:12:16,400 --> 00:12:20,340 I'll take the inverse sine--this is the same thing as arcsin 133 00:12:20,340 --> 00:12:25,080 This says, give me the angle. When I take its sine, I get this value 134 00:12:25,350 --> 00:12:33,850 So the inverse sine of 10 divided by our previous answer 31.18 135 00:12:33,850 --> 00:12:35,810 is equal to-- 136 00:12:36,160 --> 00:12:40,460 This says, give me the angle whose sine is 10/31.18 137 00:12:55,300 --> 00:12:59,750 Here we've constructed a vector. We took its vertical component and its horizontal component 138 00:12:59,850 --> 00:13:02,790 and we're able to figure out the total vectors 139 00:13:02,900 --> 00:13:07,670 This projectile in this situation, its total placement--just to make it clear 140 00:13:07,820 --> 00:13:15,510 Its path will look something like this 141 00:13:15,550 --> 00:13:18,150 And we've just calculated its total displacement 142 00:13:24,630 --> 00:13:27,660 And I realize that when I started this problem, I asked you 143 00:13:27,720 --> 00:13:30,750 I think I was asking, how far along the platform 144 00:13:30,940 --> 00:13:33,980 We figured out its total horizontal displacement 145 00:13:34,130 --> 00:13:38,740 So if you want to know how far along the platform, the platform starts 2 m to the right 146 00:13:38,890 --> 00:00:00,000 so it's really 27.53 m along the platform is where it lands