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The Physics of Racing, Part 11: Braking
Brian Beckman, physicist and member of No Bucks Racing Club
P.O. Box 662 Burbank, CA 91503
©Copyright 1991
I was recently helping to crew Mark Thornton's effort at the Silver State Grand Prix in
Nevada. Mark had built a beautiful car with a theoretical top speed of over 200 miles per
hour for the 92 mile time trial from Lund to Hiko. Mark had no experience driving at these
speeds and asked me as a physicist if I could predict what braking at 200 mph would be
like. This month I report on the backoftheenvelope calculations on braking I did there
in the field.
There are a couple of ways of looking at this problem. Brakes work by converting the
energy of motion, kineticenergy, into the energy of heat in the brakes.
Converting energy from useful forms (motion, electrical, chemical, etc.) to heat
is generally called dissipatingthe energy, because there is no easy way to get it
back from heat. If we assume that brakes dissipate energy at a constant rate, then we can
immediately conclude that it takes four times as much time to stop from 200 mph as from
100 mph. The reason is that kinetic energy goes up as the square of the speed. Going at
twice the speed means you have four times the kinetic energy because . The exact formula for kinetic energy is , where is the mass of an object and is its speed. This was useful to Mark because braking
from 100 mph was within the range of familiar driving experience.
That's pretty simple, but is it right? Do brakes dissipate energy at a constant rate?
My guess as a physicist is `probably not.' The efficiency of the braking process,
dissipation, will depend on details of the friction interaction between the brake pads and
disks. That interaction is likely to vary with temperature. Most brake pads are formulated
to grip harder when hot, but only up to a point. Brake fade occurs when the pads and
rotors are overheated. If you continue braking, heating the system even more, the brake
fluid will eventually boil and there will be no braking at all. Brake fluid has the
function of transmitting the pressure of your foot on the pedal to the brake pads by
hydrostatics. If the fluid boils, then the pressure of your foot on the pedal goes into
crushing little bubbles of gaseous brake fluid in the brake lines rather than into
crushing the pads against the disks. Hence, no brakes.
We now arrive at the second way of looking at this problem. Let us assume that we have
good brakes, so that the braking process is limited notby the interaction between
the pads and disks but by the interaction between the tires and the ground. In other
words, let us assume that our brakes are better than our tires. To keep things simple and
backoftheenvelope, assume that our tires will give us a constant deceleration of The time required for braking from speed can be calculated from: which simply follows from the definition of constant
acceleration. Given the time for braking, we can calculate the distance , again from the definitions of
acceleration and velocity:
Remembering to be careful about converting miles per hour to feet per second, we arrive at
the numbers in Table 1.
We can immediately see from this table (and, indeed, from the formulas) that it is the distance,
not the time, that varies as the square of the starting speed v. The braking time only
goes up linearly with speed, that is, in simple proportion.
The numbers in the table are in the ballpark of the braking figures one reads in
published tests of high performance cars, so I am inclined to believe that the second way
of looking at the problem is the right way. In other words, the assumption that the brakes
are better than the tires, so long as they are not overheated, is probably right, and the
assumption that brakes dissipate energy at a constant rate is probably wrong because it
leads to the conclusion that braking takes more time than it actually does.
My final advice to Mark was to leave lots of room. You can see from the table
that stopping from 210 mph takes well over a quarter mile of very hard, precise, threshold
braking at 1!
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