updated notebooks 1-4

README.md

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 # Computational Mechanics 03- Initial Value Problems
+## Learning to frame engineering equations as numerical methods
 
-![Warning: Under
-Construction!!](https://publicdomainvectors.org/photos/under-construction_geek_man_01.png)
+Welcome to Computational Mechanics Module #3! In this module we will explore
+some more data analysis, find better ways to solve differential equations, and
+learn how to solve engineering problems with Python. 
+
+[01_Catch_Motion](./notebooks/01_Catch_Motion.ipynb)
+
+* Work with images and videos in Python using `imageio`.
+* Get interactive figures using the `%matplotlib notebook` command.
+* Capture mouse clicks with Matplotlib's `mpl_connect()`.
+* Observed acceleration of falling bodies is less than $9.8\rm{m/s}^2$.
+* Capture mouse clicks on several video frames using widgets!
+* Projectile motion is like falling under gravity, plus a horizontal velocity.
+* Save our hard work as a numpy .npz file __Check the Problems for loading it back into your session__
+* Compute numerical derivatives using differences via array slicing.
+* Real data shows free-fall acceleration decreases in magnitude from  $9.8\rm{m/s}^2$.
+
+[02_Step_Future](./notebooks/02_Step_Future.ipynb)
+
+* Integrating an equation of motion numerically.
+* Drawing multiple plots in one figure,
+* Solving initial-value problems numerically
+* Using Euler's method.
+* Euler's method is a first-order method.
+* Freefall with air resistance is a more realistic model.
+
+[03_Get_Oscillations](./notebooks/03_Get_Oscillations.ipynb)
+
+* vector form of the spring-mass differential equation
+* Euler's method produces unphysical amplitude growth in oscillatory systems
+* the Euler-Cromer method fixes the amplitude growth (while still being first
+* order)
+* Euler-Cromer does show a phase lag after a long simulation
+* a convergence plot confirms the first-order accuracy of Euler's method
+* a convergence plot shows that modified Euler's method, using the derivatives
+* evaluated at the midpoint of the time interval, is a second-order method
+* How to create an implicit integration method
+* The difference between _implicit_ and _explicit_ integration
+* The difference between stable and unstable methods
+
+[04_Getting_to_the_root](./notebooks/04_Getting_to_the_root.ipynb)
+
+* How to find the 0 of a function, aka root-finding
+* The difference between a bracketing and an open methods for finding roots
+* Two bracketing methods: incremental search and bisection methods
+* Two open methods: Newton-Raphson and modified secant methods
+* How to measure relative error
+* How to compare root-finding methods
+* How to frame an engineering problem as a root-finding problem
+* Solve an initial value problem with missing initial conditions (the shooting
+* method)
+* _Bonus: In the Problems you'll consider stability of bracketing and open  methods._