 Let us turn our attention now to considering extreme cases. This is best shown through an example. At JFK International Airport, Austin and Pat are curious about what would happen if they raced to the end of a walkway in back. One on the floor on the other on the walkway. Given that they are equally fast on the ground at constant speed, they think it should not matter who runs where because the race must end in a tie. The advantage one gains in one direction will be cancelled when going in the opposite direction. Is the reasoning correct? If not, who should win? Let us look at three cases, each extreme in its own right. What happens if the belt moves faster than they move? Well, this is what I call the blooper case because the person will go very fast in one direction but when he comes back, he won't even be able to get on the belt. He will just fall off. What happens if the belt moves as fast as they move? Well, this is what I call the treadmill case. He will go fast in one direction but then he will just stay stationary. He won't be able to move on his way back. And what happens if the belt is slower than Austin and Pat? Consider the extreme case where the belt does not move. That is when the race ends in a tie. That is in the first case, the one on the floor wins. In the second case, the one on the floor wins. And in the third case, we are not certain but it seems like also the one on the floor should win. Let us use some algebra. Let S be Austin's and Pat's speed and W be the speed of the walkway. Let capital D be the total distance back and forth. We want to consider the case where S is greater than W. We also know that the velocity is equal to the distance over time for a constant velocity or constant speed. Let T sub W be the time on the walkway and T sub F be the time on the floor. We want to compare TW and TF. T sub W can be broken down into two cases. Going in the same direction as the walkway and going against or going in the opposite direction as the walkway. When we are going in the same direction, we add the speeds. When we are going in the opposite directions, we subtract them. Keep in mind that we are considering the case where S is greater than W. If we add those two rational expressions, this is what we get, which is equal to this. We must compare that to the time on the floor, which is simply the total distance over the constant speed. How do those two quantities compare? Well, this is just the same as comparing D times S squared and D times S squared minus W squared. Since S squared, again, keep in mind that S is greater than W. Since S squared is always greater than S squared minus W squared, we have that this quantity is greater, this quantity is greater. So the time on the walkway is always greater than the time on the floor, so the person on the floor always wins. Thank you.