I want to take time in this chapter to introduce you to calculations of a type that you have probably never encountered in a math class. There are equations or sets of equations that may easily be written down, but which upon inspection have no known solutions. We say that such systems have no [b]analytic solutions[/b] - which is a fancy way of saying that none of the analytic functions like sines, cosines, exponentials, polynomials, nor combinations of such functions will satisfy the equations.[br][br]When this happens - and it happens a lot in nature - we are left with only one choice. We need to solve the equations using [b]numerical methods[/b]. Numerical sounds like it involves numbers. That's for a good reason. Solving equations this way requires us to assign numeric values to all parameters and will only give us a numeric solution rather than a generalized one in terms of x or t or any other variables.

A good place to start this discussion is to ask when numerical methods are necessary. If you think back to our studies of kinematics, you will recall that we had to integrate acceleration to find change of the velocity, or [math]\Delta\vec{v}=\int \vec{a}\;dt.[/math] Similarly we integrated velocity to find displacement.  If the integral has no known solution, you need numerical methods.  Really it goes deeper than this, but we'll get to that soon enough.  Regarding the integrals, there are cases where solutions may exist but you yourself just don't recall how to solve the integral, or perhaps a known method exists but requires trig substitution followed by integration by parts several times. In such cases, if you really just want an answer for a specific scenario rather than a generalizable solution, use numerical methods and save the time and effort. [br][br]Another way of putting it is that all systems are able to be solved numerically, but only the easier ones are able to be solved analytically. The only thing lost to a numerical solution is that you get results in a case by case fashion rather than a generalizable solution. [br][br]As a concrete example, with just a little work doing kinematics, it can be shown that a drag-free projectile subject to earth's constant gravity will fly to a maximum distance on level ground when it's launched at a 45 degree angle. This problem can be done on paper, and a single analytic equation may be written that will predict the range of such a projectile given an initial speed [math]v_0[/math] and a launch angle [math]\theta.[/math] [br][br]In the numerical version of such a problem you'd have to run the calculation again and again with different launch angles to deduce the angle that gives the maximum range. Then you'd probably want to change the launch speed and see if the results change, just to be sure that such interaction between velocity and angle terms doesn't exist. While this necessity to repeat the calculation over and over seems like a major shortcoming, it turns out since computers are so fast at doing such calculations (usually), we can sample all launch angles in a split second. So it's really not much of a problem.

The most common examples this semester that require numerical methods are problems involving air drag. Kick a soccer ball across a field, or hit a golf ball and the force of air drag becomes a major consideration. Ride a bike down a hill or even on level ground and air drag is the major force resisting forward progress. The resistance felt by trains, by cars, by planes, by boats and other vehicles is usually air drag or fluid resistance in general. Furthermore, vehicles with wheels also have rolling resistance which grows with speed, but at a different growth rate than the air drag. [br][br]In later semesters you'll see that any sufficiently interesting differential equations tend to require numerical solutions - for instance those related to atomic structure and quantum mechanics.

The forces on a system until now have been either constants or time dependent forces that were relatively easily integrable. But things get really interesting when forces depend on multiple other factors besides time, or when they have more complex time dependence. Here are some examples of all sorts of functional dependencies including time dependence:[br][list][*]A rocket's thrust to a good approximation depends only on time. If you wanted to be very careful, however, it also depends on the atmospheric pressure surrounding the rocket, which depends on altitude. We won't model rockets to that extreme, but it will still need to be time-dependent to act realistically.[/*][*]The force of air drag depends on speed squared and the direction of the velocity vector.[/*][*]Small bodes moving through the same air, or slow motion through more viscous fluids tend to depend only on speed linearly and on the direction of travel.[/*][*]The force that a car engine can produce to propel the car down the road depends in a complicated fashion on engine speed. It first rises with engine speed and then falls rather rapidly as the maximum engine speed is approached. This is generally called a torque curve, but becomes an engine speed-dependent force function when the "rubber hits the road".[/*][*]Gravitational and electrostatic forces, as you already know, depend functionally on distance and location with respect to surrounding masses and charges.[/*][*]The force of a spring that causes it to vibrate depends linearly on displacement from equilibrium.[/*][*]Molecular bonds which cause materials in general to vibrate also look just like spring forces unless the bonds are stressed too far. In such cases, much more complicated functions related to displacement from equilibrium must be used.[/*][*]The force with which a human can push against an object due to skeletal muscle contraction drops off proportionally to the speed of the motion according to something called [url=https://en.wikipedia.org/wiki/Muscle_contraction#/media/File:Muscle_Force_Velocity_relationship.png]Hill's muscle model[/url].[/*][/list]While it might seem like an impossible task or just way too sophisticated a calculation at this point in your education to take some of these forces into account, we will have occasion to deal with most of them at some point during our semesters together, and several of them in this chapter.  It's really not a big deal.