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.With two months to spare, it became obvious that the car would not be ready in time.Better safe than sorry, I asked the mechanics not to hurry and to make sure the car isdone right.The standards for mechanical work on high-speed cars must be significantlyhigher than it is for road-going and autocross vehicles, for safety.The standards shouldbe comparable to those in aviation.Hurrying is a recognized no-no in aviation, and Iapplied the same logic to the car work.As I write, I have an ultimate goal of running itin SCC and NORC in '01 and '02.I had already committed to run the '00 NORC, so I slapped a roll cage in my '98 Mallett435 and went on down.This is another fabulous vehicle, but I hadn't intended to run itin high-speed events until the last minute.It was quite a hustle to get the required safetygear properly installed in time.In hindsight, I don't regret the decision.The car reallycame to life at NORC and I've run it in several high-speed events since then.63 Our flight plan called for holding speeds up to 165 for minutes at a time.As part ofplanning, we did a survey and calibration run of the course at legal, highway speeds.Onthe survey run, we noticed several bumpy spots.Driving over them at 70 mph, theywere not frightening.But, we had to figure out what to expect at 165.So, right there inthe middle of nowhere, we whipped out some envelopes, turned them over, pulledmulticolour pens from our pocket protectors, and started scribbling.Geek racing at itsbest.Let us take a moment to review the goals and methods of the "back-of-the-envelope"(BOE) style of analysis introduced in Part 3 of this series.Frequently, one simply needsa ballpark estimate or a trend.These are often much easier to get than are detailed,precise answers.In fact, they are often easy enough that they can be literally scribbledout on the backs of envelopes in the field.And that's the key point: we needed a roughidea of how the violence of the bumps varies with speed, and we needed it right thenand there in the field.Another benefit of the BOE style is that it can give one a quick plausibility check onnumerical data back at the lab.Thoroughgoing engineering analysis usually entailsdozens of interlocking equations solved on a computer resulting in tables, plots, andcharts.The intuition gets lost in the complexity.It's sometimes impossible to say, justby looking at a table or chart, whether the results are correct.On the other hand, to getour BOEs, we often make very gross approximations, such as treating the car as a rigidbody; or ignoring its track width, that is, treating it as infinitely thin; or ignoring thesuspension altogether; or even treating the whole car as a point mass, that is, as if all itsmass were concentrated at a single point.Even so, the results are often not wildly offthe numerical data, and the discrepancies can usually be explained via non-quantitativearguments.If the BOE and numerical results are wildly different, then some detectivework is indicated: one or both of them is probably wrong.BOE is really a semi-quantitative oracle to the physics.These articles are about thephysics of racing as opposed to the engineering of racing.We're primarily interested inthe fundamental, theoretical reasons for the behaviour of racing cars.The trends andballpark estimates we get from BOEs often do the job.Of course, this doesn't mean wewon't get into more detailed treatments and computer simulation.It's just that we willalways be focusing on the physics.All that said, as usual for BOE, we start with a simplistic model we can solve easily.Think of a bump in the road as a pair of matched triangles, one leading and one trailing.Let the width of each triangle be w and the height be h.Suppose a car approaches thebump with horizontal speed v.To assess the violence of the bump, let's ask whatvertical acceleration the car will experience? If we assume a simplistic model of the caras a rigid body, we get an instantaneous, infinite acceleration right at the instant the carcontacts the rising edge.We get further infinite, vertical accelerations at the two othercusps of bump the geometry.However, we know that the tyres and suspension willsmooth out these sudden impulses.Calculating the effects of tyre and suspension flex istoo time-consuming to do in the field even if we had data and computers on hand.64 However, we can get a useful approximation by assuming that the acceleration isdistributed over the entire bump.If the bump is shallow (h « w) and the car is fast, then the horizontal speed doesn'tchange very much and the car goes up the leading edge of the bump in time t = w / v.Inthat time, the car goes upward a distance h, thereby acquiring a vertical speed ofvy = h / t = vh / w.Since it acquires that velocity, very roughly, in time t, we can estimatethe vertical acceleration to beUh oh.BOE says that the severity of a bump goes up as the square of the speed.Abump you can feel at 50 mph is going to be sixteen times worse at 200 mph and willmost definitely get your attention.The little whoopdeedoos we were noticing at 70 mphwould feel (165/70)2 = 5.5 times worse at our planned speed: definitely something toanticipate on-course before we hit them.This BOE also says that the nastiness variesinversely as the width.The wider the bump, the less nasty, linearly.This is plausible.Now, let's refine the analysis a little.Conservation of energy dictates that the horizontalspeed of the car must change.In our simplified, two-dimensional BOE, the velocityvector, , consists of two components, horizontal speed, vx, and vertical speed, vy.Thesequantities obey the equationwhether on the flat or on the bump, that is, no matter what the inclination of the road.We've presupposed, here, that vertical always means "in the direction of Earth'sgravitation." If we do not change the kinetic energy of the moving car, then ½ mv2 staysconstant, therefore v2 stays constant.On the leading-edge ramp of the bump,remembering trigonometry,Define, as shorthand, , yielding vx = vw / r, vy = vh / r.Using the sameapproximation as above, we assume that we acquire a vertical velocity of vy in timet = w / vx = wr / vw = r / v, for a vertical acceleration ofThis still varies as the square of the speed, we just take a little more time to go over thebump.The only difference to the prior formula, v2h / w, is the appearance of h2 in thedenominator.Consider the case of a high, narrow bump.This case was not covered by our first BOE,which assumed that h « w.Now, with a high bump, h2 » w2 and , meaning thatthe severity of the bump will go down linearly with increasing height.Within theconfines of our model, this makes sense, because a higher bump gives the car a greatervertical distance in which to suffer its increased vertical velocity, but this doesn't seemintuitively correct
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