Physics with Richard Bleil
Let’s do some mathematics today. Recently, I’ve been on a gun buying kick. Don’t ask me why. Honestly, I used to make fun of people like me, with gun collections. I never understood why anybody would need more than one gun. Now I get it. It’s just fun. Every gun is different, with its own characteristic. Some guns are practical for hunting or self-defense, but I cannot imagine hunting and so only defend myself against paper targets in my gun club. But I’ll tell you what, those targets aren’t going to get me!
And, as a quick aside, let me say that I have no problem with hunting. I understand the ecological service hunters do, I know where meat comes from, and I certainly like it when a hunter gives me game meat. Trust me, it’s fine. I just don’t think that I have it in me to look down a barrel and actually pull a trigger. And, no, don’t show me pictures of cute and furry animals hoping to turn me vegetarian. It won’t work. Trust me, those animals would kill me if they needed to for food or self-defense if they could. Death is a part of life, and if the hunter is using the meat, then I’m good with it.
Recently I bought one of the nastiest looking guns that I’ve ever seen. Technically it’s a handgun because the barrel is so short, but believe me, I would never use it as such. I was curious, and decided to look at its muzzle velocity, certain it would be the highest in my collection, but it is not. My AR-15 has the highest muzzle velocity in my collection at 3,300 feet per second. I’m very curious about how far I could shoot this gun, so let’s figure it out, shall we?
Under still air (no wind), and ignoring wind resistance (air friction), the maximum distance one can fire is if the gun is fired at an angle of forty-five degrees. You could shoot in a general direction, but of course you would not be able to aim at any particular target, but that’s not the point of this exercise. At forty-five degrees, half of the muzzle velocity is vertical, and half is horizontal. That means that, horizontally, the velocity of the bullet is (3300/2=)1,650 feet per second. This is also the initial vertical velocity.
Acceleration due to gravity is 32 feet per second squared. That means, starting at zero, a body in free fall will be traveling thirty-two feet per second after one second. In two seconds, it will be sixty-four feet per second, then ninety-six in the third second and so forth. The formula of the velocity for a constantly accelerating body is DV=at, where DV is change in velocity. So how long will it take for the vertical velocity to reach zero feet per second. Then we have ((0-1650)=32t, solving for time gives t=)-51.5625 seconds. This is negative because the round would be decelerating (slowing down) on its ascent.
The formula for distance of a constantly accelerating body is d=(1/2)a*t*t. If the bullet is in the air for 51.5625 seconds, the maximum height it will reach is ((1/2)*32*51.5625*51.5625=)42,539 feet. There are 5,280 feet in one mile, so this is about eight miles. The bullet will reach, indeed, eight miles high. To put this into some kind of perspective, commercial jets typically cruise at about 35,000 ft, so an AR-15 round could hit a commercial jet (although it would be nearly impossible to aim to do so intentionally).
Traveling horizontally, though, it won’t just travel for 51 seconds. If it reaches its maximum velocity in fifty-one seconds, it will take another fifty-one to fall back to the ground. The bullet will be in the air for about one hundred and two seconds. Traveling at 1,650 feet per second, that means that the bullet will travel 84,000 feet (with rounding), or a total of about sixteen miles.
My AR-15 could shoot something sixteen miles away (although, again, I couldn’t aim at a target; it would be a random projectile). From where I live, I could literally shoot over the area in Omaha called Boys Town and almost to Waterloo, NE (the next town over). Or I could shoot over Council Bluffs, and well into Iowa. Manhattan Island is about thirteen and a half miles long. You could literally shoot from one end of Manhattan and miss the other side. By the way, the island is only about two and a half miles wide.
This really means nothing. Just a fascinating calculation. This is why I enjoy physics. There are so many fascinating things you can do with it.
Bleil's Banter 