# Bullet time

I decided to apply a little physics to my KitchenAid mixer misadventure, in which I claim I was nearly killed by a hunk of metal propelled past my head by spinning mixer blades. When that projectile whizzed past my head, how much danger was I really in compared to, say, a bullet from a sniper’s rifle?

It comes down to a calculation of the kinetic energy of the projectile. Fortunately it’s very easy to approximate by making a few assumptions and by ignoring the effects of air resistance and the spray of cake batter.

The projectile was the mixer’s own removable endcap. As soon as it vibrated loose, fell into the bowl, and was struck by the spinning mixer blades, it was on a ballistic trajectory, arcing up, past my head, and then down behind me. Let’s assume that the highest point of the trajectory was about level with the top of my head, roughly 1.75 meters off the ground, and that this height was attained just as the projectile was passing me, meaning that once it did pass me, it had already started down.

We can decompose the motion of the projectile into a pair of vectors: the one pointing straight down to the floor and the one perpendicular to it, pointing horizontally past my head. The speed in that direction was constant until the endcap hit the floor. The speed in the floorward direction was increasing due to gravity.

Since we’ve assumed the endcap reached its apex as it passed my head, we know that its downward velocity at that moment was zero. We also know that the acceleration in that direction (due to gravity) is 9.8 meters per second per second. Finally we know from high school physics that:

distance = initial-velocity×time + acceleration×time2/2

and since we know initial-velocity is 0, we can rearrange this to say:

time = √distance/acceleration

And since we know “distance” is 1.75 meters and “acceleration” is 9.8 m/s2, we know that it took about 0.4 seconds for the endcap to fall from the height of my head, regardless of its motion in the horizontal direction.

In that 0.4 seconds I estimate (based on where I later found the endcap) that it covered a horizontal distance of 3 meters, giving it a speed of 7.5 meters per second.

I haven’t weighed the endcap but I’m going to guess it’s about 0.25 kilograms (around half a pound). Again, high school physics tells us that kinetic energy is:

mass×velocity2/2

which means the endcap, if aimed just a bit differently, would have struck me with about 7 joules of energy.

How bad would that have been, compared to a bullet? Apparently even the wimpiest guns deliver hundreds of joules to their targets, so we’re not looking at a shearing-off-the-top-of-my-head scenario here. On the other hand, I was there, and I’m not exaggerating when I say that hunk of metal could have dealt me a grievous injury at the very least. If that was what seven joules looks like, I have a whole new appreciation for the stopping power of a bullet.