Proportional reasoning with motion

0:00- [Lecturer] NASA is researching how to send humans to Mars by as early as 2030. Now, this is a complex mission, because you're traveling for millions of kilometers, and this will involve a lot of things, like you'll think about how much fuel you need, how much oxygen we need, and then how much time we'll be spending over there, and how much radiation exposure humans will be having.

0:20So many different things. So how do we figure out what's the best approach? Well, what we usually do is try to come up with different plans. I can call that as Mission A, Mission B, Mission C, and so on and so forth. And then we try to create mathematical models for them, and then compare the models to find the trade-offs.

0:39And then, eventually, we finalize on one of them. And so the goal of this video is to get a glimpse of how to create mathematical models and compare them. Now, of course, we'll not do it for such a complex mission. We'll simplify it. So in our case, we'll assume that the spacecraft is traveling in a straight line, because that's simple, and we'll assume that it's traveling with a constant acceleration.

1:04And lastly, we'll assume that the initial velocity is zero. So for the first case, let's assume, let's call this Mission A, let's assume that the acceleration is 10 meters per second square, which is very close to the acceleration due to gravity on Earth. So the astronauts over there would be feeling pretty much at home.

1:22And the spacecraft will travel some distance, let's call it delta X, in some time. But now, let's create a second plan, second mission, in which, again, it's constant acceleration, initial velocity is zero. It's gonna travel the same delta X, but in half the time, okay?

1:39Now, our goal is to compare some of the mission parameters. The first thing you would compare is the average velocity. I mean, think about it, right? If we want to go from here to here in half the time as over here, then clearly, the average velocity over here would be higher than over here.

1:53That makes intuitive sense. But how much higher? We want to compare that. So to do that, we wanna build a mathematical model for the average velocity and then compare them. So let's do that. We can try using the model for average velocity, which is delta X divided by t.

2:08Let's see if this model is useful for us. Well, first of all, we need to compare average velocity. So it's good that that's there, okay? What about delta X? We know the relationship between them, they're exactly the same. Delta X for Mission B is the same as delta X for Mission A.

2:21We're considering the same delta X value, so that's good. And what about time? We also know the relationship between them. We know the time over here has to be half the time over here. So time for B is half the time for A. So since we know the relationship for this between these two and the relationship for this between these two, we can now use this equation, use this model to find the relationship between the average velocity between the two.

2:44So this is gonna be a useful model. Now, imagine if you had used a model for average velocity, which involved acceleration, then that wouldn't be useful because we don't know the relationship between the accelerations, right? We don't know what the acceleration over here is.

2:57And so that's why it's important to pick the right models, but we don't have to worry too much. If you do pick a wrong model, we'll figure it out and we'll scratch it, and we'll use another model. Trial and error, no big deal, okay? Anyways, let's go ahead with this.

3:08Let's write down the average velocity for Mission B. So we could say the average velocity for Mission B would be delta XB devided by tB. And the average velocity for Mission A would be delta XA divided by tA. We know delta XB and delta XA are the same, it's just delta X, so let's just call them as delta X.

3:27And now to compare them, we can just divide the two equation. And now we simplify. The delta X divides out. And so we get that equals one over tB divide by one over tA, which is tA divided by tB. And now we know what tB is. The time for this is half the time for A, we can plug that in.

3:47So if you get 1/2 tA, the tA divides out, and you end up with two. So look, this ratio ends up becoming two. So we can rearrange now, and write that the average velocity in Mission B will be twice as much as the average velocity in Mission A. So to cover the same delta X in half the time, we need twice the average velocity, which is not super obvious, because we're dealing with accelerated motion over here.

4:13But now the interesting question is, could we have just looked at this equation and without doing the math, figure this out? And the answer is yes. If you look at this model carefully, because delta X is the same for both, we can look at this and we can say, hey, average velocity is proportional to 1 over t, or it's inversely proportional to t.

4:33What this means is that if t changes by some factor, your average velocity will change by the reciprocal of that factor. So if t becomes double, your average velocity will be reciprocal of that, 1/2. If t triples, the average velocity reciprocal of that, one by third.

4:49In our case, the t became 1/2. So the average velocity would be reciprocal of 1/2, which is two. And so immediately, just by looking at this, we could have said, hey, if the t becomes 1/2, the average velocity would double. Powerful, right? All right, now let's see if we can compare something else.

5:07Let's compare their accelerations. What's the acceleration in Mission B compared to Mission A? And again, for that, we need to use a model, this time a mathematical model that involves an acceleration. And one of such models is this. This is the model for things that are moving with constant acceleration, right?

5:24Again, let's look at stuff that we already know, and then see if this is gonna be useful for us, okay? So what do we know? Well, first of all, since we have X that operates in the initial position, which is over here, we can just set that to zero. And then the X, which is the final position, we can just call that as delta X.

5:41So we know those two, okay? That's the same for both of them. We also know the initial velocity, which is V0, that is 0. And we know the time over here needs to be half the time over there. And finally, we know the acceleration of the first one. So these other things that are given to us.

5:59So the first thing we can do is plug in wherever zeros are to make our model slightly simpler. So if you do that, this goes to zero, this becomes delta X, this goes to zero. So you'll look at delta X equals 1/2 at square. And since I wanna compare accelerations, I will rearrange this to get the acceleration.

6:15So acceleration becomes two delta X divided by t square. And again, let's see if this model is useful for us. I want to compare acceleration, so that's great. I need accelerations. Do I know the relationship between delta X? Yes, they're the same. And do I know the relationship between time?

6:31Yes, I do. So I can use this to find the relationship between the accelerations. It'll be a great idea to pause the video and see if you can try this on your own. Very similar to what we did earlier. All right, let's do this. So we can write down the accelerations for both the missions.

6:45So for B it'll be two times delta XB by tB square, and for A will be two delta XA by tA square. But we know delta XB and delta XA are exactly the same, so we can just get rid of A and B over there. And we can divide the two just like before. And then simplify.

7:02So the two delta X and two delta X can divide out. That leaving us with one over tB square, divide by one over tA square. And then we can rearrange this to get tA square divided by tB square. And again, we can finally plug in what tB is, tB is 1/2 tA. If we plug that in over here, we get 1/2 tA, the whole squared.

7:24Finally, tA square divides out, leaving us with one divided by 1 over 4, which is 4. So from this, I now know that the acceleration in Mission B would be four times as the acceleration in Mission A. And so there we have it. We need four times more acceleration here, compared to over here.

7:42Four times more is 4 times 10, that is 40. And then we could say that, hey, 40 meters per second squared is too much of an acceleration for humans to withstand. That's not gonna be possible, 'cause we'll be traveling for a long time. And so we could say that no, we can't do that.

7:55Even if we reach in half the time, that's not possible. But anyways, just like before, could we have figured this out without doing the whole math? And the answer is again, yes. Again, if you look at the model carefully, we see that delta X is the same for both of them, which means acceleration will be proportional to 1 over t squared, or inversely proportional to t squared.

8:16This means whatever factor t changes by, acceleration will change by the reciprocal of the square. So if t were to double, acceleration would be the reciprocal 1/2 square, 1/4. If t were to triple, acceleration would change by one over three reciprocal square, 1/9.

8:35But in our case, t became half. So acceleration would be reciprocal two square, four. And there you have it. This is a very powerful way of doing this. Since we are having so much fun over here, we should try one more plan, Mission C. Here, we'll keep the same acceleration as before, but we'll allow the spacecraft will travel for 50% more time than before.

8:57So same acceleration, but 50% more time. It's gonna travel farther, right? And the question is how much farther? So we know what to do by now. Think about our mathematical model. So again, since we're dealing with positions and accelerations, let's use the same model as before.

9:15Write down what we know. And at any moment, feel free to pause and try this on your own, okay? But let's write down what we already know. We know, I mean, we can set our X0 to be zero, the initial position to be zero, and the final position would just be X. So we can just set X0 to be zero.

9:30We also know the V0, initial velocity, is zero. So we know that. We also know the accelerations are the same in this time. So acceleration case A and case C is the same. And we are allowing for 50% more time. So we know that the time here is 50% more, compared to time A, which means 1.5 times A.

9:48So tC must be 1.5 times tA. And again, we can plug in zeros to simplify. So this goes to zero, this goes to zero. And that gives us a simplified model, X, which is the final position which we want. That is the model. So let's see if it's useful, okay? We want X, we want to compare that, so it's good that's there.

10:10Do we know the relationship between the accelerations? Yes, the same accelerations. Do we know the relationship between the time? Yes, we do. So we know the relationship between time, 1.5 times tA. And so we can figure this out. Again, we can do the math, but we also know how to do this in a shortcut now, right?

10:26So we can look at this model and we can say that, hey, since the acceleration is the same, since A is the same, X is directly proportional to t squared. So now whatever factor t changes by, X will change by the square of that factor. You can see that's directly proportional this time, right?

10:46So if t doubles, X will be, X will change by a factor of two square, if t triples, X will change by a factor of three square. In our case, t has become 1.5 times more. So X will change by a factor of 1.5 squared. So we can immediately write XC, the final position over here, okay, that has to be 1.5 times square XA, the final position over here.

11:10And if you simplify, 1.5 square is 2.25. And so we immediately get the answer. So XC would be 2.25 times XA, which is amazing if you think about it, right? Just by 50% more time, it'll travel more than twice the distance, compared to the first one. And by the way, we can check the math, and I'll not go through all the steps over here.

11:33I'll just show you all the steps. You can pause the video and you can just see, if you divide it and, you know, do it the long way, you get the same answer. But anyways, look at what this means. This means that, you know, in the first case, if we allowed it to travel for one year, and let's say the distance it traveled was a billion kilometers, in the second case, if we allow it 50% more time, which is 1 1/2 years, it will travel 2.25 billion kilometers.

11:58That's awesome. So, long story short, by using mathematical models, we can see how changing one variable affects the other variable. This is super useful in comparing scenarios. What's important is that you don't need actual values, you just need the models and you can divide them out, and you can figure out, and you can compare, you know, whatever you want to compare.

12:20And another way is you can do it without dividing, you can do it without doing the math, by just looking at the model, seeing what variables are the same for them both, and then figuring out the proportionality relationship, the right proportionality relationship.

12:34If you know the factor by which one variable changes, we can figure out the factor by which the other variable would change as well. This is not just useful for kinematics, it's useful for all physics.