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# The case of huffing Harry

There are many excellent reasons not to sniff the fumes of aerosol cleaners, as what happened to Harry illustrates. Harry and friends were zipping along North in the right hand turn lane when he passed out. They were entering a curved ramp protected by curbs on both sides that directed traffic into the east-bound lane. Harry's BMW (3,659 lbs with three adult occupants), jumped the curb curving ahead and crossed the median, crossed the east-bound lane, jumped the curb and median going airborne between the east- and west-bound lanes, and hit an Explorer (3,961 lbs with a man, his wife, and a 3 year old) stopped in the west-bound left turn lane. The BMW hit the Explorer about 10 inches above its center of mass (CM) causing it to roll on its side.

In reconstructing accidents, actual situations rarely fall into textbook categories. Among other things, a complete analysis would look at the speed required and energy dissipated in jumping the two curbs, the effects of braking, a trajectory analysis, the energy dissipated in deforming both vehicles, and an analysis of the motion of both vehicles after impact. This case is further complicated by the fact that the impact is eccentric, i.e. it is directed above the CM of the Explorer. Although a complete analysis is not within the scope of this article, this factor deserves further discussion.

When the impact force of one vehicle (A) on another is eccentric, it influences two aspects of the motion of vehicle B. First, according to Newton's laws, the impact forces "F" produce a linear acceleration "a" of the CM:

a = F/m

where "m" is the mass. Second, the sum of the moments "M" produced about the CM by the impact forces results in an angular acceleration "a"

a = M/I

where "I" is the moment of inertia. These relations formulated as impulse/momentum relationships can be used to get an estimate of the speed of A just prior to impact (VA).

To obtain a minimum value for VA, we will assume the case of impending rollover and work backwards to determine the conditions prior to impact. Knowing the CG height (26.5 inches) and tread width (58.4 inches) of vehicle B we can calculate the potential energy change APE between its position prior to impact and in impending rollover. This is the amount of kinetic energy (KEB) of B that must be available after impact for rollover to occur. Assuming B is rolling without slipping, we set KEB = APE, and solve for the angular velocity just after impact.

Assuming plastic impact and conservation of linear and angular momentum, with a little more hand waving we obtain a relationship between uB and VA prior to impact, giving

VA = 16 mph

Due to the relative position of the two vehicles, it doesn't take much speed to make B roll. Remember that this is a minimum estimate.

Note that if the CG of B is higher, it is less likely to roll over in this situation, one reason why SUVs and pickups provide some safety advantages. As I am writing this, the news is breaking regarding California Governor Davis signing the bill that some think will eliminate SUVs and pickups in the state. Without getting into politics, there are implications for safety that should be considered when making such laws. Environmental issues in the bill should not trump all others.