Modeling motion subject to drag forces

1. Modeling motion subject to drag forces PHYS 361 Spring, 2011

2. physics • Goal is to predict the motion of an object • position vs. time ... x(t) • velocity vs. time ... v(t) • Several forces may be acting on this object • Connect motion to forces using Newton’s laws • Obtain differential equation(s): “Equations of Motion” Solution is trivial if Fnet is constant. Most interesting forces, such as those involved in riding a bicycle, are not constant.

3. deriving a useful Equation of Motion We want a differential equation of the form But Newton’s 2nd law does not, at first glance, have this form: Of course, this equation is interesting (i.e. worthy of a computational colution) only if the force is not constant. It could be a function of time, position, or even velocity. Let’s consider a situation where force depends on velocity and time. How could we rewrite Newton’s 2nd Law in the desired form?

4. Forces that depend on velocity A cyclist’s power output is more typically constant than applied force. Power is defined as Drag force: viscous and inertial viscous drag: Stokes Law. Valid for small v. pushing air out of the way valid for larger v inertial drag:

5. Equation of motion for a cyclist Assume inertial drag is much larger than viscous drag

6. Euler method Our differential equation: remember: b2 is a constant Euler method for obtaining a finite difference equation: Substituting into our equation, we can solve for vi+1