So, while I was reading “Genghis: Birth of an Empire” I noticed that there are quite a few instances when the book describes a man being knocked down by an arrow. As someone who reads and watches a lot of fantasy and historical fiction, this was far from the first time I’ve seen such a thing, but as a physics person I wondered whether an arrow would really hit someone with enough force to knock them down.
Now, I’m not disputing that getting hit by an arrow might cause someone to lose their balance and fall off of a horse, or off of a castle wall or the like. The question here is how hard the arrow actually would hit. For anyone who remembers their high school physics, that means we’re talking about a momentum problem. We need to figure out how much momentum a typical arrow would have, and figure out what that amount of momentum would do when transferred to a human. Specifically, this type of problem is an “inelastic collision”, where two bodies collide and then stick together. If the arrow somehow bounced off the person, that would be an elastic collision, but we won’t worry about that.
We are going to assume that our hapless victim is standing still and the problem starts with an arrow speeding toward them. To find the arrow’s momentum, we need to know its mass and velocity. For an estimate of the arrow’s mass I will go with 65 g based on this source. The velocity depends on the draw weight of the bow that fired the arrow, but generally I am seeing arrow velocities of around 200 feet per second (61 m/s) for longbows. Momentum is just mass x velocity, so the momentum of a 65 g arrow going 61 m/s is 3.965 kg*m/s.
I don’t know about you, but that doesn’t mean very much to me, so let’s figure out how fast some more familiar objects would have to be going to have the same momentum, starting with a baseball. A standard baseball has a mass of ~145 g. So a baseball with the same momentum as our speeding arrow would be going 27.3 m/s, or 61 miles per hour. We can do the same calculation with a soccer ball, which has a typical mass of 450 g. A soccer ball with the same momentum as our speeding arrow would have to be going 8.8 m/s, or ~20 miles per hour.
A 60 mile per hour baseball or a 20 mile per hour soccer ball don’t really have enough oomph to knock a person over unless they’re already off balance. So, a typical arrow would not knock a man down, or stop him in midair, or the like. He might fall down because he suddenly has an arrow sticking in him, but it’s the damage done by the arrow, not its momentum that would make him fall.