A car, travelling at a constant speed is about to crash into another,
stationary car.

The moving car has a "crumple zone" which is designed to take as much
energy out of the collision as possible.

At impact, if the crumple zone yields too easily, the stationary car
will be thrown forward with great force. This will also be the case if
the crumple zone is too hard.

For any given set of conditions, (i.e. mass of stationary car, mass
and speed of moving car) how soft or how hard would the crumple zone
need to be, (in terms of joules required to crumple?) to minimize the
effects on the STATIONARY car (in terms of momentum?)?

Could you derive an equation(s) to describe or model this system?
(please without calculus?)  ;-(
I'd like to be able to try out different values/scenarios.


Assumptions:

Friction can be ignored.

The crumple zone is the only part of the system that deforms.

I'm assuming the length of the crumple zone isn't important (just the
energy required to crush it), but this may not be the case?

The answer could assume that the "crumpling" action is linear but
maybe a "compressing spring" action would give different or better
results?