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 back-of-the-envelope 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, kinetic energy, into the energy of heat
in the brakes. Converting energy from useful forms (motion, electrical, chemical, *etc.*)
to heat is generally called dissipating the 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 not by 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
back-of-the-envelope, 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! |