1 00:00:03,450 --> 00:00:07,270 One of the most common motions we see in our everyday lives 2 00:00:07,270 --> 00:00:10,820 is the path of an object moving thrown 3 00:00:10,820 --> 00:00:12,870 and moving through space. 4 00:00:12,870 --> 00:00:15,220 Now, this type of motion has a very famous name 5 00:00:15,220 --> 00:00:17,160 called projectile motion. 6 00:00:17,160 --> 00:00:21,600 And when we look at it, let's introduce a coordinate system. 7 00:00:21,600 --> 00:00:24,270 i hat and j hat. 8 00:00:24,270 --> 00:00:28,080 Here's our y plus y-axis and our plus x-axis. 9 00:00:28,080 --> 00:00:31,150 Then, in order to understand the kinematics of this motion, 10 00:00:31,150 --> 00:00:33,380 we'd like to apply Newton's second law. 11 00:00:33,380 --> 00:00:36,670 So separately we'll draw our object. 12 00:00:36,670 --> 00:00:39,990 It has gravitational force acting downward. 13 00:00:39,990 --> 00:00:43,290 Remember our unit vectors in the j hat direction. 14 00:00:43,290 --> 00:00:47,890 And so when we write our equations of motion, 15 00:00:47,890 --> 00:00:48,880 f equals ma. 16 00:00:48,880 --> 00:00:50,800 We have two different directions. 17 00:00:50,800 --> 00:00:54,070 In the j hat direction, we have the gravitational force 18 00:00:54,070 --> 00:00:56,100 downward minus mg. 19 00:00:56,100 --> 00:01:02,710 And remember here that g is our positive quantity, 9.8 meters 20 00:01:02,710 --> 00:01:07,380 per second, and we have m a y. 21 00:01:07,380 --> 00:01:11,160 Newton's second law equates these two different things. 22 00:01:11,160 --> 00:01:13,980 And that's-- we'll write it like that. 23 00:01:13,980 --> 00:01:16,140 So those two quantities are equal. 24 00:01:16,140 --> 00:01:20,730 And so our conclusion is that the acceleration 25 00:01:20,730 --> 00:01:26,090 is equal to minus g in the y direction. 26 00:01:26,090 --> 00:01:28,920 Now, likewise, keep in mind that we 27 00:01:28,920 --> 00:01:31,330 have our horizontal direction as well. 28 00:01:31,330 --> 00:01:33,979 And notice that we're now assuming that there's 29 00:01:33,979 --> 00:01:35,440 no horizontal forces. 30 00:01:35,440 --> 00:01:38,630 In the real world, there can be all types of air resistances. 31 00:01:38,630 --> 00:01:40,860 But here, for the simplicity of this model, 32 00:01:40,860 --> 00:01:42,670 we have no horizontal forces. 33 00:01:42,670 --> 00:01:46,160 So we have 0 equals m a x. 34 00:01:46,160 --> 00:01:50,820 And so we have our separate equation, a x equals 0. 35 00:01:50,820 --> 00:01:54,080 Now, both of these equations are equations that we've already 36 00:01:54,080 --> 00:01:55,950 been working with in kinematics. 37 00:01:55,950 --> 00:01:58,759 Here we have a constant acceleration, 38 00:01:58,759 --> 00:02:00,500 and here we have zero acceleration. 39 00:02:00,500 --> 00:02:04,150 So using the results that we had earlier from integration, 40 00:02:04,150 --> 00:02:07,020 we can write down the equations of motion 41 00:02:07,020 --> 00:02:09,610 in the y and the x directions. 42 00:02:09,610 --> 00:02:12,020 First, we'll write the velocity. 43 00:02:12,020 --> 00:02:17,579 Vy as a function of time is equal to some initial value. 44 00:02:17,579 --> 00:02:21,329 And because the acceleration is negative, minus gt. 45 00:02:21,329 --> 00:02:25,540 You can test your integration technique to see that. 46 00:02:25,540 --> 00:02:28,490 And the position as a function of time, 47 00:02:28,490 --> 00:02:31,880 remember, is just some constant value. 48 00:02:31,880 --> 00:02:34,390 This is our constant y nought. 49 00:02:34,390 --> 00:02:42,340 And we have plus vy nought t minus 1/2 gt squared again 50 00:02:42,340 --> 00:02:43,980 doing integration. 51 00:02:43,980 --> 00:02:47,620 Now, the horizontal equation of motion, 52 00:02:47,620 --> 00:02:51,510 the x(t), because there is no acceleration 53 00:02:51,510 --> 00:02:54,170 in the horizontal direction, this 54 00:02:54,170 --> 00:02:59,130 is a constant value given by the initial value of the component 55 00:02:59,130 --> 00:03:01,400 of the velocity in the x direction 56 00:03:01,400 --> 00:03:07,260 and the position x(t) is then equal to some initial position 57 00:03:07,260 --> 00:03:10,190 plus Vx nought t. 58 00:03:10,190 --> 00:03:12,930 And so those are our functions of time 59 00:03:12,930 --> 00:03:15,160 for the components of the velocity 60 00:03:15,160 --> 00:03:17,680 and the components of position. 61 00:03:17,680 --> 00:03:21,780 Now, for our particular example, it's much easier. 62 00:03:21,780 --> 00:03:24,560 Here we have X nought equal to 0. 63 00:03:24,560 --> 00:03:29,180 Notice that our object is starting at the origin. 64 00:03:29,180 --> 00:03:33,579 And so that tells us that x as a function of time 65 00:03:33,579 --> 00:03:35,150 is just Vx nought t. 66 00:03:35,150 --> 00:03:37,910 I'm dropping the parentheses t. 67 00:03:37,910 --> 00:03:39,620 Remember, that's not a product. 68 00:03:39,620 --> 00:03:41,970 That's just a function of time. 69 00:03:41,970 --> 00:03:44,210 And given this fact, that tells us 70 00:03:44,210 --> 00:03:48,522 that t is x divided by Vx nought, 71 00:03:48,522 --> 00:03:51,079 and now I can take this value of t 72 00:03:51,079 --> 00:03:53,850 and put it into our vertical equation 73 00:03:53,850 --> 00:04:02,230 and I get V y is some initial value plus V y nought x over Vx 74 00:04:02,230 --> 00:04:07,520 nought, substituting for t, minus 1/2 gx 75 00:04:07,520 --> 00:04:12,450 squared over V x nought squared. 76 00:04:12,450 --> 00:04:14,500 When we described projectile motion, 77 00:04:14,500 --> 00:04:19,720 we had an equation describing y as a function of t, 78 00:04:19,720 --> 00:04:22,140 an equation describing x as a function 79 00:04:22,140 --> 00:04:24,430 of t-- the horizontal motion and the vertical motion-- 80 00:04:24,430 --> 00:04:27,130 and here we have a separate equation, 81 00:04:27,130 --> 00:04:30,960 which is describing the y as a function of x. 82 00:04:30,960 --> 00:04:34,159 Notice there is no time involved in this equation. 83 00:04:34,159 --> 00:04:35,980 Now, let's try to look at graphically 84 00:04:35,980 --> 00:04:37,480 what we're seeing here. 85 00:04:37,480 --> 00:04:42,280 So, for instance, when we look at a plot of the motion of y 86 00:04:42,280 --> 00:04:46,540 versus x, we can see the vertical component going 87 00:04:46,540 --> 00:04:50,850 up and down and we can see the horizontal component moving 88 00:04:50,850 --> 00:04:56,100 to the right while the object is following this trajectory. 89 00:04:56,100 --> 00:04:58,530 And this is a parabolic trajectory 90 00:04:58,530 --> 00:05:01,870 of y as a function of x. 91 00:05:01,870 --> 00:05:06,790 Now, if we wanted to actually plot out y as a function of t, 92 00:05:06,790 --> 00:05:09,180 then here we can look at the simulation 93 00:05:09,180 --> 00:05:11,830 where we're just looking at the y component 94 00:05:11,830 --> 00:05:15,970 and you can see that that motion too is a parabola. 95 00:05:15,970 --> 00:05:18,700 However, what's crucial to look at is 96 00:05:18,700 --> 00:05:25,750 that it's y as a function of t and not y as a function of x. 97 00:05:25,750 --> 00:05:28,060 Finally, we can look at the horizontal motion 98 00:05:28,060 --> 00:05:32,010 and again recall that when we looked at the trajectory as y 99 00:05:32,010 --> 00:05:34,980 as a function of x, you can see the horizontal motion 100 00:05:34,980 --> 00:05:35,730 component. 101 00:05:35,730 --> 00:05:39,600 Let's plot that separately, x as a function of t. 102 00:05:39,600 --> 00:05:41,460 And when we make that plot, you can 103 00:05:41,460 --> 00:05:45,610 see that this is just a linear equation in time 104 00:05:45,610 --> 00:05:47,550 where there is some initial condition, 105 00:05:47,550 --> 00:05:52,510 x nought, and so that's the plot of x as a function of t. 106 00:05:52,510 --> 00:05:57,470 And so we see three separate representations of this motion. 107 00:05:57,470 --> 00:06:05,370 y as a function of x, y as a function of t, and x 108 00:06:05,370 --> 00:06:07,223 as a function of t.