What do you do when you mow the lawn? I like to listen to podcasts and think of new blog ideas. This one has been in my head for a long time. What is the best way to mow the lawn? Should I just go back and forth, or should I make a box-shaped loop? These questions must be answered.

Video Data ———-

I need some data. How fast does a lawn mower move? How long does a turn take? How fast do you go when pulling the lawn mower backwards? Without these answers, really the only thing I could do to optimize the lawn mowing would be to make sure I don't go over the same spot twice.

Video time. Yes, I made a video. If you really, really, really want to see it - fine. Why would you even watch that? Oh, I guess I should also say something about the lawn mower. This is one of those "self propelled" models. You still have to push - so it doesn't really propel itself, but it does make a difference. Oh, if you look at the video, the cones are 1 meter apart.

Here is my first plot. This shows the horizontal position of the lawn mower. The first thing to look at is the acceleration.

This shows an acceleration of about 0.6 m/s2 for about 1 second. After that, the velocity seems fairly constant.

This shows an approximate speed of about 1.67 m/s. The acceleration during the stopping part of the motion is quite a bit larger than the starting acceleration. It seems to be somewhere around 4 m/s2.

In the video, I did several things. In the first case, I went a few meters and then turned the mower around. In the next case, I went a few meters and then just pulled the mower backwards without turning it around. So, are the forward and backward speeds significantly different? Since I went back and forth several times, I can make a histogram showing the distributions of speeds for both forward motion and backwards motion.

I know it isn't as much data as I would like, but I can only mow the same spot so many times before I get bored. Also, there are more "forwards" than "backwards" because whenever I was turning around I would go forward twice. Anyway, the speeds seem to be clearly different with an average forward speed of 1.607 m/s and a backwards average of 1.255 m/s.

How long does it take to turn around? Looking at the data (which I won't plot since that gets boring), I get an average turn around time of 2.213 seconds.

What about a right angle turn? How long does this take? I get an average time of 1.326 seconds. Ok - I think that is all the data I need.

Finally, what about the time to stop and go backwards if I am just pulling the mower back and not turning it around? That would be about 0.893 seconds.

Modeling a Lawn Mow ——————-

Let me now start with a simple square lawn that is 30 meters by 30 meters. I think I need one more piece of information, the cutting width. My particular mower has a 22 inch blade. This gives and approximate cutting width of about 0.52 meters (I cut off some of the width for a little bit of overlap).

Here is my first cutting strategy, back and forth.

How long does this lawn take to mow? For this case, it isn't too difficult to figure out. If the width of the mowed path is s, then the mower would need to make L/s cuts to finish the lawn. If the number of paths doesn't fit perfectly, round up. The total mowing time would then be:

The number of turn around times is 1 less than the number of rows. If I have 8 rows, I will have to turn around 7 times. So, back to my 30m square yard (with no trees or anything in the middle). Using my values from the video, this lawn would take 20.14 minutes to mow.

What about the other common mowing pattern - the spiraling square? (I just made that name up)

First, how many of these square patterns will be needed? This is similar to the back and forth case, but there will be half as many "squares" as rows in the previous case (I think). That would put N at L/(2s). For each square, there will be 4 right angle turns. The thing that changes is the length of each side for the successive squares. If the first square is L x L, the next square will be (L - 2s) x (L - 2s).

But how would you calculate this? One way is a with a super-simple python program. Like this:

In the loop, the program calculates the time for mowing in a square including 4 right angle turns. It then decrease the size of the square and repeats the time calculation. Using this method, I get a mowing time of 21.14 minutes. Just slightly longer than the back and forth method. Why does it take longer? Although a right angle turn is quicker than a full turn around, there are more of them. And yes, I know sometimes this square pattern is better for mowing since you can push all the grass clippings into the center (if you are into collecting grass clippings).

What about a non-square yard? I think (but not completely sure) that for non-square yards, the spiral-square method will still be at a disadvantage. Think about two square yards that are placed side by side. For both methods, you would still have the same number of turns as the square case. Perhaps the only way the square-spiral method could win would be if it was a non-square yard and you use the back and forth method in the direction of the small side of the yard.

What About Backing Up? ———————-

Suppose you get to a small part of your yard. Maybe it is that small section on the other side of your driveway. What is the best strategy to mow this part? Should you do a short run, turn around and then do it again? Or is it quicker to do one side and then pull the lawn mower backwards? The answer probably depends on the length of the yard area to cut.

Let me first consider the case for turning around to make the cut. If I consider one row to include one turn around, then the total time for this one row would be:

But really, to compare to the pull back method I need to look at two rows. This time would just be twice the value for one row. Now, what about the pull back case. Here is the time for two rows:

Perhaps my notation is a bit confusing. Here, I am using Δt~pull back~ to represent the time to go from pushing the mower forward to pulling it backwards. The Δt~turn around~ is the time to turn the mower around. The velocity with the f subscript is the moving forward velocity and the b subscript is the pulling back velocity.

Using the velocities and turn around times from above, here is a plot of the time for the two rows as a function of the length of the row.

So here you go. If the length is less than about 5 meters, you would save time by just pulling the lawn mower back. If the row is longer than 5 meters, your best bet is to turn around. Perhaps I should change that to "my best bet is to turn around" since you will have different speeds and turn around times with your own lawn mower. Probably.

Of course there are more things to explore (always true). What if it is an irregularly shaped lawn? Are there any other lawn mowing strategies that would be more efficient?