Modeling motion subject to drag forces PHYS 361 Spring, 2011

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 F net is constant. Most interesting forces, such as those involved in riding a bicycle, are not constant.

deriving a useful Equation of Motion We want a differential equation of the form But Newton’s 2 nd 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 2 nd Law in the desired form?

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

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

Euler method Our differential equation: Euler method for obtaining a finite difference equation: Substituting into our equation, we can solve for v i+1 remember: b 2 is a constant