Shortly after last week's post, I got a followup from my brother Nate: I like your latest blog post about how you feel the need for speed-mowing. However, I think you did not follow your curve far enough. Can you make a follow-up post where you show when travel-speed becomes a problem because the blades don’t cut all the grass?
What Nate is suggesting is that as the speed of the mower across the grass increases, the speed of the blade relative to the grass during part of its rotation will slow down, since it's moving in the opposite direction:
We can write an expression for the ends of the blades using a parametric equation:
where v is the forward speed of the mower, and ω is the angular velocity of the blades. The manual for the mower gives the blade speed as 2800 RPM, so we can look at how the coverage changes as we approach the speed of the tips, which works out to about 200 MPH. I tried a couple different steps in the range, along with the mower's default maximum, 3 MPH:
Around 60 MPH, we can see some gaps appearing, and if we went for the full 200 MPH, we'd be missing about half the width of the mower. On the other hand, I could finish the lawn in 17.3 seconds, so maybe there's something to Nate's idea! In fact, he's far from the first person to consider supercharged lawnmowers, though that example only went up to ~27 MPH. Call a physicist when you really want to exceed the limits of reality!
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