Copyright ©2011 by Paul Niquette All rights reserved.
Featured in the Historic Car Crash puzzle is a photograph taken by a Speed Graphic camera like this one, which dates back to the 1930s. Look closely and you will see a solenoid mounted horizontally below the lens. It was designed to be activated by a speedgun unit for synchronizing the shutter with a flash attachment (not shown in the photograph).
For the car-crash photograph, the solenoid was activated by an electronic timer, which was built in the laboratory of ITTE. The design of that timing device is quite a story in itself, inasmuch as transistors, let alone microprocessors, were not commercially available in 1954.
A little parsing music, please: Notice that the Historic Car Crash puzzle does not seek a prediction of collision forces and energy dissipation. ITTE's research, of course, would give paramount importance to those topics, but the puzzle merely calls for a time interval. According to Engineered Automobile Crashes a timer tripped the shutter at a precise interval after the car contacted the barrier in order to capture the instant of "maximum deceleration and crush-in volume." That phraseology, by the way, was specified by Derwyn Severy, the project engineer. Both of those measurements might pertain to the car itself. But what about the second impact and the collision forces that cause injuries to people?
For the Historic Car Crash puzzle, let us reprise the 1954 analysis by the research assistant...
Sophisticated solvers will recognize the equation d2x/dt2 = - (k / m) x as the most familiar homogeneous second-order linear, constant coefficient ordinary differential equation, which is best known for its power to describe simple harmonic motion (animation).
The research assistant's mathematical model, in effect, had a 1937 Plymouth 'vibrating' linearly back and forth in obedience to x = Xmax-crush cos[(k / m)1/2 t], such that the period is given by the following expression: 2 / ( k / m)1/2. Well, of course, the vibration ceased after only one-quarter of its period, with x = Xmax-crush, as the car's structure stopped its "harmonic motion." Accordingly, Tmax-crush = ( / 2) (m / k)1/2.
Here may be a surprise for some people:
Tmax-crush is independent of Xmax-crush and
therefore independent of the speed of the vehicle V!
The exclamation point would have been an undeserved
indulgence at this point in the analysis, inasmuch as
the research assistant did not yet have an estimate
for Tmax-crush. Whatever that turns out to be,
though, it will be almost constant for a range of
crashing speeds and vehicle damage characterized by Xmax-crush.
Back in 1954...
Now, with respect to the second
impact and the fact that Tmax-decel > Tmax-crush,
ITTE's research focussed on the right front seat,
which in those days was commonly called "the death
seat." That funereal phrase gave recognition to the
vulnerability of passengers seated thereon. A
post-crash photograph in the article showed
one hapless anthropometric dummy that suffered
horrendous consequences that day while riding in the
death seat. Let us continue
with a review of the 1954 analysis by ITTE's
And the rest is history. Thus, the solution to the Historic Car Crash puzzle can be...
More than a half-century later, sophisticated solvers will surely want to reconsider -- all right, second-guess -- the three assumptions above...
 Values of m and k are surely not constant over the range 0 < x < Xmax-crush. Inasmuch as the harmonic period and therefore Tmax-crush is proportional to (m / k)1/2, one must expect that...
Thus these two effects seem to offset each other, and thus did think ITTE's research assistant.
 Of course, something less than 100% of the kinetic energy of the vehicle was absorbed by deformation of that old Plymouth. For one thing, the back end was lifted up in an obvious partial conversion of kinetic to potential energy. Most likely, Ymax-crush acts in reciprocity with Xmax-crush, but it is hardly clear, even today, what effect energy shifts would have on Tmax-crush.
 Sophisticated solvers of the Historic
Car Crash puzzle undoubtedly doubt that a
perfectly linear average (f =Fmax-crush / 2) would
prevail throughout the range, 0 < x
< Xmax-crush. Today, with a simple spreadsheet,
people can lash up a piece-wise model to explore
non-linear elasticities for a crashing
automobile. However, many will be astonished to
see how little Tmax-crush changes with various shaped curves
postulated for k-versus-x.