This past thanksgiving break, I spent a couple hours each day cleaning my yard in preparation for the holidays. Oaks surround my property, so a lot of leaves and downed branches accumulate on the lawn. Instead of raking, we prefer to mulch the leaves as doing so saves space in the contractor’s bag, and we’re charged to dump the leaves on a per-bag rate. As I’ve found out over the years, mulching the yard can be dangerous business. We try to separate the sticks from the leaves before pushing the mower over a pile, but our sorting is far from perfect. From time to time, a wooden projectile will fly from the mower, endangering whoever is standing in its way.
But what is the velocity of the projected stick? If we imagine that the stick is being caught and launched at the end of the mower’s blade, we can treat the blade as a throwing arm. The average lawn mower blade rotates at 3000rpm. Using ω = ΔΘ/Δt, we find that ω = 300 Θ/s for the spinning blade. We can find the linear velocity of the blade’s tip using v = ωr: v = 150 m/s (150 m/s = 335.54 mph). Using conservation of momentum, we can determine the velocity of the blade and stick after inelastic collision, mbladevblade + mstickvstick = (mblade + mstick)v’. We can approximate the blade’s mass to be .65kg, while the mass of the stick is .050kg. We find that v’ = 140 m/s. We can imagine that soon after collision, the stick is flung from the blade at this velocity. Assuming the blade’s acceleration is zero after the collision, the stick is shot at a velocity of 140 m/s.