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