# Throwing Spear – Part 1

Posted on January 4, 2010

Instead of writing another huge post, I’m going to cut this one into several pieces which will be posted throughout the week.  This may turn out to be a better format in general.

IN THIS POST: I’ll introduce the purpose and scope of the current discussion.  I’ll also introduce some concepts that will help with the parts to follow.

I’ve been introduced to cord-making and flintknapping and now I want to put the two together.  I plan to make a throwing spear.  But how?  There are many characteristics of a spear that can be varied.  What would work best, assuming that I will be the thrower?  How would paleolithic people have figured it out?  Probably, at first, by exhaustive trial and error.  Actually, I shouldn’t say “exhaustive”.  I’ll give them a little more credit and say “heuristic”.  When enough possibilities have been tried, discernible patterns would emerge and humans would be good at choosing next possibilities to try which are more likely to be an improvement than not.

I don’t have the time for trial and error.  But what I do have is modern scientific knowledge.  In this case, I’m talking about Newtonian physics.  Since both of those tactics will sooner or later approach an optimal result, I’m going to “cheat” and hope that reasoning with Newtonian mechanics will get me a similar result as generations of trial and error would.

So … a quick “review” of Newtonian mechanics…  Newtonian mechanics can really be summarized using just one simple equation.  Oops!  I said the E word.   No, don’t go!  It’s not that scary.  I promise.  That one equation is $F = ma$, read “force equals mass times acceleration”.  The rest is just properly defining $F$, $m$, and $a$.  $m$ is the easiest.  In this model, all material objects have a property called “mass”.  This is easily related to common experience because more massive objects are also heavier.

Defining acceleration is only a little more tricky.  At any given moment in time, a material object has a position.  Its position can be described as its three-dimensional coordinates in space (x, y, z).  Its velocity is the rate of change (also called the time derivative) of its position (in each of its three coordinates).  That’s different than speed because it tells you not only how fast the object is moving, but also the direction in which it’s moving.  If you in turn take the rate of change of the velocity, then you get acceleration.  Roughly speaking, that’s how fast the object is speeding up – but like velocity, there is also a direction component.  For that reason, acceleration sometimes describes how fast the object is slowing down or even the way its trajectory is curving.

Now we come to force.  Intuitively, force describes how strongly an object is being pushed or pulled and in what direction.  There are many things that exercise forces on objects.  The ones we’ll be concerned with here are gravity, friction, the normal force (when one object is directly pushing on another), and air resistance.  In the equation $F = ma$, $F$ stands for the sum of all the forces acting upon the object in question.  As an example, if you put a cup down on a table, the force of gravity “tries” to accelerate it downward.  But the table exerts a contact force (normal force and probably a little friction) on the cup which is exactly opposite the force of gravity on the cup.  So the sum of the forces on the cup is zero.  Solving $F = ma$ for $a$ with $F = 0$ shows us that the acceleration of the cup will indeed be zero, regardless of the cup’s mass – it’ll stay put!

So far, what I’ve been calling “objects” have positions which change over time, but not orientations.  A spear, of course, can both translate (travel) and rotate.  So how do we describe that?  It turns out that the materials around us are actually made up of lots of little “objects” called particles.  In a solid object, such as a spear, forces between the particles (chemical bonds) hold them in an essentially rigid configuration.  Now that we’re talking about a rigid body of particles, it makes perfect sense to discuss orientation.  Although the individual particles might not have orientations in this model, the entire constellation of particles can certainly rotate through space.

IN THE NEXT POST: I’ll talk about torques (rotational forces) and begin applying these physics concepts to the problem of the ideal spear.  I’ll also discuss what a spear actually is and what different kinds of spears there are.