Over the summer, I got to spend a lot of time with my son, Arthur. I enjoyed this immensely, as there is nothing quite like being able to watch your child grow up. Lately, he has been learning to walk and is now able to take several steps at a time. In particular he is at the point where he can walk short distances, but he can still crawl faster than he can walk. He enjoys standing, so he likes to stop by some object that he can brace himself. However, he seldom stays in one place long as he loses focus on what he’s currently doing and wants to make his way over to some other toy, me, his mother or anything else that grabs his attention.
While this is all interesting enough to me, what I wanted to focus on was the fact that some times when he moves form one place to another he walks, and other times he crawls. However, he very rarely does both in the same trip. This normally only occurs when he loses balance and falls, and doesn’t seem to be intentional. Of course I found myself wondering why, so I thought about it.
To begin with I thought, what is he trying to do? Well, he is trying to get from one place to another. His movement, while very interesting to me, was just a means to get to the next object of interest, so he seemed to want to accomplish this task as quickly as possible. That is, he wanted to travel some distance while minimizing time spent. Aha! We have an optimization problem to solve!
Of course, I now want to model and solve this problem. While I was tempted to get out a stop watch and try to determine Arthur’s speed crawling and walking, I decided that guessing numbers was enough since I don’t feel comfortable experimenting on my son. So, let’s suppose that he can currently walk 1 foot per second and crawl 2 feet per second. In order to transition from standing to crawling position, it takes him approximately 1 second to get down and 1 second to get back up. Now we can set up some equations. Note that distance traveled by walking, dw, is given by dw=tw where tw is time spent walking. The distance traveled crawling is dc =2(tc-2), since 2 seconds is spent going down and getting back up. If he both walked and crawled we would see the total distance is dt=dw+dc=tw+2(tc-2). Noting that the total time is t=tw+tc, we know that the total time is given by t=d if there is no crawling and t=d/2+2+tw if there is any crawling.
Well, we can see that if he both walks and crawl that any time spent walking will actually add to the total time spent getting to where he wants to go. It makes sense then to only crawl or only walk. It is quite surprising that even a one year old seems to intuitively sense this. Now, when should he walk and when should he crawl? Since he crawls faster than he walks, he should crawl if he can make up for the extra 2 seconds getting up and down. Therefore, he should crawl if he has to go further than 4 feet and walk if he has to get less than 4 feet.
I don’t think he quite has the exact distance figured out for when he should walk or crawl, but he does walk shorter distances and crawl longer. Also, it seems that his crawl speed is about capped out, but his walking speed is getting faster. If we change the numbers above this would mean the distance where it’s beneficial to crawl would be get longer, and this seems to be happening as well. While, eventually, you will walk faster than you crawl and should therefore just walk all the time, it has been fun for me to watch him as he develops.