# Cycling Force Equation

### Building the Cycling Force Equation

The Cycling Force Equation is built by identifying all of the forces acting on the CyclistCycle, their functional description, and their collective directions of application to the cycle. Unless we are doing complex scenario modeling, architectural trade offs, or research analytics, this equation can be easily written as a one-dimensional along the direction of motion.

### Writing Cycling Force Equation

We have the formulas for the individual forces, and now all we need to do is add them up. While this may look involved, it turns out the first three terms, forward force, rolling resistance, and gravity, are constant. We can therefore group them into a single constant force term so when writing the equation out, we can add them together and reduce the three terms to one. This leaves Aerodynamic Drag as the only other term, and it is the one that involves the most work. ### Solving the Cycling Force Equation

Despite our simplifications, the presence of v² in the drag term turns this into a Nonlinear Differential Equation, and making the direct solution mathematically challenging. However, we have an alternative if we can accept not having the actual expression for v(t) and x(t) but instead the graphs, we have a second approach which is called Incremental Numerical. This latter approach is intuitive and devoid of the mathematics of the direct approach.

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