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Re: Truck stopping distances?



Second shot...there will be many more.

And now a third.

I expect that we may have an example of different scaling dimensions
at work here.

The kinetic energy that must be dissipated as thermal energy when
the vehicle stops scales as the mass of the vehicle, which scales
approximately as the cube of the linear size L of the vehicle.

The maximum tolerable temperature rise in the braking surface is
probably fixed by the material properties of the braking materials
which is probably approximately independent of vehicle size to a
crude approximation.

The rate of dissipation at the braking surface that causes a given
temperature rise is governed by the thermal conductivity and
geometry of the brakes. If we assume that the ultimate thermal sink
that dissipated energy spreads to is the huge metallic mass of the
rotors (or drums) wheels, axles, etc. of the vehicle, and if we
can for all practical purposes consider it as an infinite sink, then
the problem of the temperature rise at a surface where thermal energy
is deposited at the surface and conducting through a conductor to
infinity approximately obeys the law that the temperature rise is
proportional to the energy dissipation rate per unit area of the
surface where the energy is being initially deposited. (Crudely, we
have a 1-d conduction problem where the heat flow moves perpendicular
to the braking surfaces to (effective) infinity.)

Since the total dissipation rate is the product of the disspation
rate per unit area times the area of the braking surfaces we see that
the maximum total power dissipation rate is crudely proportional to
the area of the braking surfaces.

The minimum stopping time scales as the quotient of the initial
kinetic energy to be dissipated divided by the maximum power
dissipation rate.

The geometry of the vehicle shape approximately requires that the
braking surface area scales as the second power of the linear size L
of the vehicle. This means that the minimum stopping time is
effectively a quotient a factor that is proportional to L^3 divided by
a factor that is proportional to L^2. This makes the stopping time
scale proportional to L itself. Thus, it would take big vehicles
longer to stop than small ones *other things being equal*.

David Bowman
David_Bowman@georgetowncolleg.edu