## Numerical Methods - A very brief introduction

### Analytic vs. Numeric

$$\frac{dv}{dt} = a = \frac{F}{m}$$

In the case of free fall, this becomes: $$\dot{v} = g$$

#### More Analytic Solutions:

If $F = -bv$, then we can solve for velocity and obtain: $$v(t)= v_0 e^{\frac{t}{\tau}}$$

But, what about when $$\int dv = \int \left(\textrm{Something Nasty} \right) dt$$

### Numerical Methods

Re-writing our original definition of acceleration, we can say that: $$\frac{\Delta v}{\Delta t} = a$$

Let's say we know a value for $v_i$ $$v_f - v_i = a \Delta t$$ Then we can just solve this to get a value for $v_f$ at some $\Delta t$.

#### Numerical Integration

Let's start with $v_i = 0$. Then we can see that $$v_f = a \Delta t$$ That the area of the first rectangle in the previous graph. Now we can use that $v_f$ in the original equation as the $v_i$, and we will obtain: $$v_f = v_i + a \Delta t = a \Delta t + a \Delta t$$ ... and so on.