Proportional reasoning with motion | AP Physics 1 | Khan Academy

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. We have to think about how much fuel we need, how much oxygen we need, how much time we'll be spending over there, and how much radiation exposure humans will be having. There are so 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. You can call it Mission A, Mission B, Mission C, and so on and so forth. Then we try to create mathematical models for them and compare the models to find the trade-offs. Eventually, we finalize one of them.

So the goal of this video is to get a glimpse of how to create mathematical models and compare them. 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. We'll assume that it's traveling with a constant acceleration, and lastly, we'll assume that the initial velocity is zero.

For the first case, let's assume, let's call this Mission A, that the acceleration is 10 m/s², which is very close to the acceleration due to gravity on Earth. So the astronauts over there would be feeling pretty much at home, and the spacecraft will travel some distance—let's call it Delta X—in some time. But now, let's create a second plan, a second mission, in which again it's constant acceleration, initial velocity is zero, and it's going to travel the same Delta X but in half the time.

Now our goal is to compare some of the mission parameters. The first thing you could compare is the average velocity. I mean, think about it, right? If we want to go from here to here in half the time compared to over here, then clearly, the average velocity over here would be higher than over here. That makes intuitive sense, but how much higher? We want to compare that.

So to do that, we want to build a mathematical model for the average velocity and then compare them. We can try using the model for average velocity, which is Delta X / T. Let'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. 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. We'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.

Since we know the relationship between these two and the relationship between these two, we can now use this equation—use this model—to find the relationship between the average velocities. This is going to 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, and that's why it's important to pick the right models. But you don't have to worry too much if you pick a wrong model; we'll figure it out and scratch it and use another model. Trial and error—no big deal.

Okay, anyways, let's go ahead with this. Let's write down the average velocity for Mission B. So we could say the average velocity for Mission B would be Delta X B by T B, and the average velocity for Mission A would be Delta X A by T A. We know Delta X B and Delta X A are the same. It's just Delta X, so let's just call them Delta X.

Now to compare them, we can just divide the two equations. Now we simplify, so Delta X divides out, and so we get that equals 1 / T B / 1 / T A, which is T A / T B. Now we know what T B is: the time for this is half the time for A. We can plug that in, so if you get half T A, the T A divides out, and you end up with two.

So look, this ratio ends up becoming 2. 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.

But now the interesting question is: could we have just looked at this equation and without doing the math figured 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/T," or it's inversely proportional to T. What 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 the reciprocal of that, half. If T triples, the average velocity would be the reciprocal of that, 1/3. In our case, the T became half, so the average velocity would be the reciprocal of half, which is 2. So immediately, just by looking at this, we could have said, "Hey, if T becomes half, the average velocity would double." Powerful, right?

All right, now let's see if we can compare something else. Let'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 acceleration. One of such models is this. This is the model for things that are moving with constant acceleration, right?

Again, let's look at stuff that we already know, and then see if this is going to be useful for us. Okay, so what do we know? Well, first of all, since we have X_0, which is the initial position, we can just set that to zero. Then the X, which is the final position, we can just call that as Delta X. So we know those two. Okay, that's the same for both of them.

We also know the initial velocity, which is V_0. That is zero, and we know the time over here needs to be half the time over there. Finally, we know the acceleration of the first one. So these are the things that are given to us. 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, and this goes to zero.

So I'll get Delta X = 1/2 a t². Since I want to compare accelerations, I will rearrange this to get the acceleration. So acceleration becomes 2 Delta X / t². 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 are the same. And do I know the relationship between time? Yes, 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. So for B, it'll be 2 * Delta X B / T B², and for A, it'll be 2 * Delta X A / T A². But we know Delta X B and Delta X A are exactly the same, so we can just get rid of A and B over there.

Then we can divide the two just like before and then simplify. So the two Delta X and two Delta X can divide out, leaving us with 1/T B² / 1/T A², and then we can rearrange this to get T A² / T B². Again, we can finally plug in what T B is. T B is half T A. If we plug that in, we get half T A the whole squared. Finally, T² divides out, leaving us with 1/1/4, which is 4.

So from this, I now know that the acceleration in Mission B would be four times that of the acceleration in Mission A. And so there we have it: we need four times more acceleration here compared to over here. Four times more is 4 * 10, which is 40. Then we could say that, hey, 40 m/s² is too much of an acceleration for humans to withstand. That's not going to be possible because we'll be traveling for a long time.

So we could say, "No, we can't do that even 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/T² or inversely proportional to T². This means whatever factor T changes by, your acceleration will change by the reciprocal of the square.

So if T were to double, acceleration would be the reciprocal of half squared—1/4. If T were to triple, acceleration would change by the reciprocal square, 1/9. But in our case, T became half, so acceleration would be reciprocal to square 4. There you have it; this is a very powerful way of doing this.

Since we're having so much fun over here, we should try one more plan, Mission C. Here, we'll keep the same acceleration as before but will allow the spacecraft to travel for 50% more time than before. So same acceleration but 50% more time, it's going to travel farther. Now the question is how much farther?

So we know what to do by now: think about a mathematical model. Since we're dealing with positions and accelerations, let's use the same model as before. Write 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 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 X to be zero. We also know the initial velocity is zero, so we know that. We also know the accelerations are the same in this time, so acceleration K_A and K_C is the same.

We are allowing for 50% more time, which means the time here is 50% more compared to time A, which means 1.5 * T_A. So T_C must be 1.5 * T_A. Again, we can plug in zeros to simplify. This goes to zero, this goes to zero, and that gives us a simplified model: X, which is the final position 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. Do 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 * T_A.

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? So 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 T².

Whatever factor T changes by, X will change by the square of that factor. You can see that's directly proportional this time, right? So if T doubles, X will change by a factor of 2². If T triples, X will change by a factor of 3². In our case, T has become 1.5 times more, so X will change by a factor of 1.5².

So we can immediately write X_C. The final position over here has to be 1.5² * X_A, the final position over here. If you simplify, 1.5² is 2.25, and so we immediately get the answer. So X_C would be 2.25 * X_A, which is amazing if you think about it, right?

Just by 50% more time, you're allowing it to travel more than twice the distance compared to the first one. By the way, we can check the math, and I'll not go through all the steps over here; I'll just show you all the steps. You can pause the video, and you can just see if you divide it and do it the long way, you get the same answer.

But anyways, look at what this means. This means that, 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 1.5 years, it will travel 2.25 billion kilometers. That'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. You can divide them out, and you can figure out, and you can compare whatever you want to compare.

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 both, and then figuring out the proportionality relationship—the right proportionality relationship. If 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 of physics.