In a current episode of Mythbusters, Brian and Jon (the brand new MythBusters) needed to see what occurs if you shoot an arrow straight up into the air. It is going to clearly come again down—however would it not nonetheless be shifting quick sufficient to kill you? In case you are a MythBuster, the best choice is to truly shoot an arrow straight up into the air and measure its velocity on affect. However for regular folks, it may be higher to simply calculate the ultimate arrow velocity. Additionally one be aware—I am at the moment a science marketing consultant for the present.
OK, let’s begin with a fundamental physics downside. How briskly would an object be shifting in the event you dropped it from a top of 300 ft (91.four meters)? When an object is in free fall, there’s solely the gravitational pressure appearing on it. Since each the acceleration and the gravitational pressure rely upon the mass, all objects fall with the identical acceleration (no matter mass) with a fee of g = 9.eight m/s2. In fact, that is solely true in the event you ignore air resistance—however we’ll get to that quickly.
If the item begins at a vertical postion y = h and ends at y = zero meters with a beginning velocity of zero m/s, then the ultimate velocity could be discovered with the next equation. In order for you extra particulars on kinematics, here is a resource.
Utilizing the beginning top of 91.four meters, I get a ultimate velocity of 42.three meters per second (which might be 94.6 mph). So proper there you possibly can see that this can be a pretty good estimation of the movement of the arrow. However that evaluation leaves off one vital side—air resistance. As an object strikes by the air, there’s a pressure pushing again in the wrong way of the item’s velocity. The magnitude of this air resistance pressure will increase with velocity and in addition is dependent upon the air and the scale of the item. This air resistance pressure could be written as the next equation:
The v, after all, is the item’s velocity, C is a coefficient that is dependent upon the form of the item, A is the cross sectional space, and ρ is the air density.
Air resistance is okay and the whole lot—but it surely provides an enormous downside to the falling arrow. Because the arrow falls, it will increase in velocity. Because the velocity will increase, the drag pressure will increase. Which means that a falling arrow has a non-constant pressure and a non-constant acceleration. The traditional introductory physics stage kinematic equations (just like the one used above) solely work for fixed acceleration. This implies we want a trick to unravel this falling arrow downside. Sure, we primarily must cheat. But it surely’s probably not dishonest if it really works.
The strategy we’ll use known as the Euler method. The fundamental thought is to take the issue of the falling arrow and break it into many smaller issues. As an alternative of letting the arrow fall all the best way to the bottom, I’ll simply let it fall for a really brief time period—let’s go along with zero.01 seconds. Throughout this small time interval, the falling object’s velocity does not change very a lot (as a result of it is such a short while). Which means that the air resistance pressure additionally does not change very a lot. The truth is, we are able to approximate this pressure as being fixed throughout this interval. Now we’re again to what we are able to deal with—movement with fixed forces.
However we made an enormous sacrifice. Selecting a short while interval permits us to calculate stuff, however now I’ve an entire bunch of issues to unravel. If I wish to mannequin the movement of an object over an extended time interval of 1 second, I must do 100 calculations (utilizing small steps of zero.01 seconds). This many calculations would just about suck. I do not wish to try this many, so I wont—I’ll make a pc do it.
Don’t fret, you do not have to be a code monkey to get this factor to work, however you do have to put in writing a pc program. Truthfully, it is not as unhealthy as you assume—and anyway, I feel everyone should be able to program a computer (not less than just a little bit). Right here is the fundamental (very fundamental) plan for the pc program. Throughout every small time step (of zero.01 seconds), I’m going to calculate the entire pressure on the item. Subsequent I’ll use that whole pressure to calculate the acceleration and the speed of the item on the finish of this time interval. Lastly, I’ll use this ultimate velocity to calculate the place of the item on the finish of the interval. Then I simply have the pc repeat these calculations till the item will get to the bottom.
Let’s get to it. Right here is the code (written in python). Go forward and run the code (click on the play button) and see what occurs.
There are a number of extra feedback I ought to make. Whenever you see a quantity register a line of code, the whole lot after that image is handled as a remark and the pc ignores it (the feedback are only for people). I attempted to place sufficient feedback so to see what is going on on—however extra importantly, I level out which issues within the code you may wish to change. Sure, you must certainly change one thing after which run it once more and see what occurs. No, you possibly can’t completely break something. However attempt altering issues just like the mass of the arrow or the drag coefficient of the arrow. It is enjoyable.
Oh, however what about that “terminal velocity” calculation within the code? As an object falls, it will increase in velocity which causes the air resistance pressure to additionally enhance. At a while, the air resistance will probably be nearly as giant because the gravitational pressure and the entire pressure on the item will probably be near zero Newtons. With zero internet pressure, you’ve zero acceleration and the item simply strikes down at a relentless velocity—we name this terminal velocity. Why is it vital? Effectively, if the ultimate velocity may be very small in comparison with the terminal velocity you possibly can principally ignore the air resistance pressure and skip the Euler methodology all collectively. It is simply one thing else you possibly can play with.
Now for one thing just a little completely different. What if I wish to mannequin the movement of the arrow going each up and down? On prime of that, I’m going to incorporate an object that begins with the identical velocity however has no air resistance (only for comparability). This is what that appears like. Keep in mind, you possibly can click on the “pencil” icon to view and edit the code.
Keep in mind, I am simply estimating a few of these values for the air resistance. Actually, the toughest parameter to estimate is the drag coefficient (C)—however nonetheless, the entire numerical mannequin appears to be like ok for an approximation.
However ultimately, this numerical mannequin (that is what we name a calculation like this) principally agrees with the ultimate arrow velocity that the MythBusters measured utilizing a digital camera. Oh, in case you wish to check it out in actual life—do not. In addition they discovered that this arrow velocity could be doubtlessly deadly.