June 2021 by Aydin Ural

“The shortest stopping distance of a vehicle is observed when crashed to a stationary object – only then you realize your speed was just too high” (racers’ proverb)

In early ‘90s, Turkish Traffic Police Department released a tentative study showing some information signs to be placed at the sides of highways, to prevent people from speeding. There were three signs designed resembling comic strips in the old newspapers, telling a short story in three frames. And they would be placed 100m apart along the side of the road so the driver would see the three signs in sequence. As can be expected from state-selected ‘artists’ those times, drawings were significantly worse than mediocre. All three were from the view at the center of the back seat of the car. The first was showing the hands of the driver at the steering wheel, a good straight two lane (one for each direction) road beckoning at the front, with beautiful trees on both sides. A very sharp eye could see the nose of an old truck coming out of the trees towards the right side of the road in the distance. Underneath, in block capitals with an ugly font was the writing: “What Is Your Speed?” In the second frame, the car was stopped millimeters away from hitting a truck that had taken the complete road (both lanes) transversely, to turn to the oncoming direction. The writing underneath the drawing of the near-miss situation was: “90 km/h – You Are Safe”. In the last drawing, the car had crashed, windshield was broken like a single-glazed window shattered with a football from the street, the driver face down on the bonnet, with blood all around! The writing underneath said: “110km/h – You Are Dead!” 

Of course, there were (at that time) sensible people at the office and these signs were disapproved at some level; probably because of the graphic depiction of death and poor art. They were absolutely not suitable for public roads with a lot of children in the cars looking out of the windows asking about anything interesting or not interesting they see. 

For me, as a mechanical engineer specialized in automotive engineering, it was stupid – at first. I said, “At 90km/h, we don’t even dent our bumper, yet at 110km/h, we die, eh? Imbeciles!” Of course, the mind immediately calculates the difference as 20km/h and determines this as the crashing speed and knowledge tells him that even a least safe crappy small car of those times would endure such a crash without killing any of its occupants if everyone was wearing seat belts, maybe some wounded in the worst case. 

Than, right after that ‘at first’ second, the engineer in the soul kicks in – whispering “are you sure that would be the impact speed? Wouldn’t you need to analyze?” And I did. Actually, you don’t need to be an engineer, not even a university graduate, to solve the problem once it’s laid out; high school physics knowledge is good enough. We should know speed is the change in distance per unit time and acceleration is the change in speed per unit time, that’s all (if you can’t remember even that much, you may skip to another reading material now).

The important variable, the fixed variable that would become the boundary condition in this braking situation is distance. So we must calculate the distance available first (the distance the car being driven at 90km/h could be marginally stopped). 

In any braking process, there are three stages: 1) reaction time between seeing the obstacle on the road necessitating heavy braking and starting to press the brake pedal (the distance travelled during that time period is crucial), 2) starting to press the brake pedal until the maximum braking power possible is reached (now, that (the distance travelled during that period) is difficult to calculate with high school physics, as not only the speed but the acceleration (negative acceleration, deceleration) is continuously changing during that period, from zero to maximum available). For practical purposes, let’s assume our driver presses the brake pedal fully promptly following the reaction time and brings the car to the maximum deceleration in no time, 3) the time and distance travelled with the maximum deceleration until the car stops (or, the speed is dropped to a required value, for general purposes). So, we must assume a reaction time and a maximum deceleration level. Well, reaction times differ substantially from one person to another, in a range of 1-2 sec., but for an average driver, let’s assume 1.2 sec. (For a race driver, it is much less than 0.5 sec). The maximum braking (deceleration) level varies with the make/model of the vehicle (braking systems employed in different cars differ substantially), tires, road surface, load on the vehicle and… the driver. This last item may surprise many people, thinking how on earth a driver makes a difference of braking performance at the same car, on the same road; but it’s a fact, because braking is not like pushing or pulling a switch, it’s one of the features of the art of driving. Modern sports cars may reach well above 1.0g in braking under good hands or feet, but for the sake of an average car with an average driver in early ‘90s let’s assume 0.6g (it’s typical to give acceleration figures in g’s (earth’s gravitational acceleration, 9.81meter/second square). Now, with those assumptions, our driver while moving at 90km/h (25m/s) will travel 25×1.2= 30m during the reaction time of 1.2 sec and then to bring speed from 90km/h to 0km/h with a deceleration of 0.6g, will need (25-0)/(0.6×9.81)= 4.25sec. As the speed will be shed linearly during that constant deceleration, we may accurately assume average speed during that period will be half of the summation of the initial and final speeds. ((25+0)/2)*4.25= 51m. 

Now that we have the distance available (30+51=81m), we come to the intriguing consequence of what would the crashing speed of the same car with the same driver be, if it were being driven at 110km/h (30.55m/sec). Same reaction time in this case gives 36.7m and that means only 81.0-36.7= 44.3m is left to decelerate! And the car should decelerate from 110km/h this time! ((30.55+V)/2)x((30.55-V)/(0.6×9.81))=44.3 is the equation; and this is actually the fourth equation of motion in an inverted form (for deceleration), final speed is square root of (initial speed square minus 2 times acceleration times distance), in our example, 20.3m/sec or….73 km/h! 

God forbid, even the safest cars of today may not protect you in such a collision. Euro NCAP tests are performed at 50km/h; and the impact force changes proportional with the square of the speed.. 

So, the guys who came up with the idea of those very poorly illustrated info signs were actually right. Were the drawings more cute and less pornographic, could they have a green light? I don’t know. But being an individualist, I didn’t leave the calculations there. I had a very simple mathematical model (a formula!), why wouldn’t I take the liberty of playing with the assumptions now? What if, the driver was a very good one, with high skills to extract 0.8g braking power from a decent car he was driving and good scanning and fast responses to cut down reaction time to 0.7sec? First of all, the distance covered during reaction time would be 30.55×0.7= 21.4m hence the distance left to decelerate would be 81-21.4= 59.6m. And the crashing speed with 0.8g would be… slightly less than zero! (the car would come to halt well before 59.6m; how much before can also be calculated but I won’t go into that).

If you play with the formula, you’ll see that reaction time is as important as the braking power: 0.1sec faster reaction time is equivalent to 0.04g more braking deceleration extracted out of the vehicle (equivalence would slightly change at different initial speed conditions).

That’s why a good rider should always practice to develop his abilities to extract the highest possible braking (deceleration) from his/her bike (at various speeds, at various surfaces) AND should practice to sharpen his/her scanning skills (continuously catching/identifying/assessing risk factors) to lower the reaction time. Ironically, the lower your reaction time gets (thru better and better scanning), the less likely you’ll ever need it (nor maximum braking) – as you become so good at anticipating anything that’d force you to alter your speed abruptly, you take precautions beforehand starting to very slightly lower your speed, to reposition, to be prepared to take evasive maneuvers… And yes, this example is very odd in not taking into account any evasive maneuver possibility, showing hard braking as the only option, yet there are such odd cases in reality, cases in which a good rider anticipates these situations and sets a speed to leave a braking distance well above his/her best possible braking ability, putting a good safety margin – through setting a lower speed.

Braking vs Braking

In the previous article, we had omitted to go into the stage in which acceleration (or, deceleration) was changing – as it would be a very short period of time in that example so the effect could be omitted. Because, we were to extract the maximum braking power and extract it immediately. In daily riding, as we always use part-throttle, we also partially use the brakes. We never need the full braking force (ideally). Yet, the time from starting to squeeze the brake lever until the required braking force is reached is not that short in these instances. And this period deserves to be analyzed – not in mathematical detail (not this time!), not even as time; but conceptually.

Acceleration is not something that disturbs us humans. That said, there are of course limits: to be able to withstand 9g of a fighter jet maneuver, one must be fiercely fit and extremely well-trained. Just as the drivers who withstand 4g braking or 3g cornering forces of modern F1 cars, developing those accelerations much more abruptly. 

That “abruptly “ is the keyword we’d try to elaborate in this article. 

As speed is the change of distance with time and acceleration is the change of speed with time, jerk is the change of acceleration with time. And jerk is not something we humans like. We may withstand high levels of acceleration, even 6g, for a very short time (0.2 sec for normal people) after all, effect of acceleration is a static force as it’s applied to our mass; but the jerk in the motion is a discomfort to us. Therefore jerk should be as short and as linear as possible. 

Jerk cannot be avoided! If we increase (or decrease) the speed, we have to have acceleration, the speed cannot change without (positive or negative) acceleration; yet, acceleration itself cannot start from zero and reach a certain level without a change in acceleration, there has to be jerk (well, actually, jerk cannot go from zero to a certain value without a further derivative with respect to time either, and that (change of jerk with time) is called snap, but as this shouldn’t be an engineering essay more than it already has been, let’s forget about snap – for, after all, things don’t stop there either, snap has a derivative as well).

During acceleration, we usually control jerk much better than we do while braking – delivery of the engine power is easier to master; because it’s internally mastered: most modern engines employ highly sophisticated engine mappings for various part-throttle response levels, they’re in a way perfectly “tamed”. An immensely skilled rider was required to master the ‘70s sports bikes with 100hp engines, but a below average rider can more-or-less master the acceleration of a 200hp engined bike of today. This is how tamed today’s engines are. 

We also practice acceleration much more than we do braking, as we love it so much and don’t like to brake that much. But this should change: mastering braking with practice, both for extracting the maximum braking and finding out how to best modulate the brakes for smoothest possible riding are joyous acts that should be devised, both for safety and not to accuse the pillion rider for hitting our helmet. But more than anything else, to enjoy our riding more – excessive and/or lengthy periods of jerk are discomforting for us riders too, besides, they deteriorate the smoothness. 

There are many techniques available: the easiest and most practical is, setting a jerk that would bring us to the required deceleration level very quickly and then holding this deceleration level. This is easier said than done of course, the greatest difficulty is misjudging the level of necessary deceleration and getting into the position of being have to brake harder just before the stopping point. Holding a slightly stronger deceleration level than what you feel enough at first sight is a good counter measure, releasing some braking power towards the end is not discomforting. On the contrary, comforting if done expertly, with experience.

There are a couple of engineering marvels you may visit to “learn” how it’s done perfectly. One is the elevator. But I’m not talking about the cheap elevator of a 4-story suburban block. You should find a classy hotel or business center with 40+ stories. The lift is incredibly fast, yet you don’t feel any discomfort while acceleration nor deceleration. It doesn’t spend twice the time it travels 20 floors to be able stop gently either, it stops swiftly – leaving you in awe. In some of them, they use a slight level of jerk then a continuous slight acceleration, so while going up you feel you weigh a little more on your legs. But some don’t even have that, you feel as if you’re beamed up. The other example is roller coasters. They are truly marvels of engineering. It’s difficult to “learn” from them as they are incredibly complicated, but you really get astonished how such levels of speed and acceleration can be reached with such minimal levels and periods of jerk. 

After all, part of the joy of riding is always finding new levels of advancing, right?