Michael Press wrote:
> > a = kv^2
> > -1 = k(37)^2, so k = -1/37^2
> > so dv/dt = (-1/37^2)v^2
> > Separate the variables and take the antiderivative of both sides
> > v = 37^2/(t + 37)
> > s = 37^2 ln[t + 37] + c
> This model leads only to solutions with an infinity in
> finite time. In this case at t = -37. Do we not have a
> model without infinities?
Models are allowed to have infinities. It's only the real world
that doesn't have infinities (except in the case of things that are
actually infinite). In this case, assuming a = kv^2 is valid if the
only force is typical subsonic air drag. However, when you
extrapolate the equation backwards (negative time), the object's
velocity exceeds the speed of sound in air. At this point, long
before infinity, the model has ceased to be accurate;
supersonic and transonic drag behave differently. A bigger
issue is whether extrapolating backwards is meaningful at all.
It's a little clearer to write the equations symbolically.
a = kv^2 (k is negative)
v0 = 37 ft/sec, v1 = 29 ft/sec, t0 = 0.
dv/dt = kv^2, so dv / kv^2 = dt.
Integrating, t1-t0 = -1/k * (1/v1 - 1/v0), which gives t1 = 10.2 sec.
velocity: v(t) = v0 / (1 - v0*k*t).
distance: s(t) = -1/k * ln(1 - v0*k*t)
Extrapolating backwards to negative t, the velocity goes to
infinity at t = 1/(v0*k). This would mean that if the a = kv^2
force law always applied, an object at arbitrarily high velocity
could be decelerated in a finite time. There's two issues
here. One is that the force law may not always apply outside
the subsonic regime. The other is that you can only trigger
this mathematical singularity by starting with an object with
infinite velocity. If your initial conditions specify an object with
any finite velocity, its _future_ behavior governed by this
deceleration equation is always non-singular.
Because fluid drag is a dissipative process, the equations of
motion are not time reversible, so extrapolating backwards in
time is not guaranteed to give a sensible result.
(In the case of the bicycle rider, obviously some energy input
was required to accelerate to the initial speed, and while
pedaling the force law is different.)