Brake applications and the remaining velocity Hans Humenberger University of Vienna, Faculty of mathematics

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1 Hans Humenberger: rake applications and the remaining elocity 67 rake applications and the remaining elocity Hans Humenberger Uniersity of Vienna, Faculty of mathematics Abstract It is ery common when teaching the topic acceleration and deceleration that the formula s = is aimed at. Here s denotes the braking distance, the initial elocity b and b the so called braking deceleration. One can see in the formula that depends quadratically on, that means: with double initial elocity one needs four times the braking distance. This is an interesting fact and it should warn us against driing too fast.there is the danger that students learn this formula just like other formulas ( it ist just another formula ). I think there is a way to deal with this topic in a more striking and exciting way: Do not stop at this point (formula for s ) but think of the so called remaining elocities. Unfortunately in most cases dealing with the topic brake applications stops when arriing at the formula for the braking distance s =. ut one can b face the enormous consequences of too high speed in traffic much better if one thinks of remaining elocities. Therefore in the following pages there are presented three consecutie worksheets, the first two aiming at the third: remaining elocities. Dealing with the first worksheet A students should be able to establish autonomously the formula for the braking time t =, the function for the b elocity in terms of the elapsed time t () = bt, and the formula for the braking distance s =. This worksheet has nearly no prerequisites, it can b be gien to students of grade 9. If students know already these formulas, one can start with worksheet maybe this is the case when you deal with these items in later grades (1, 11, 1). The second one () deals with the elocity in terms of the distance coered. esides other tasks students should come to the associated formula s () = bs. After haing established these formulas in worksheet C students should be faced with the enormous consequences of driing too fast in a more striking way than just say: If you double your elocity, the braking distance will grow by factor 4. s

2 68 Hans Humenberger: rake applications and the remaining elocity A) raking distance 1) There are two common units to measure elocities: km/h and m/s. a) What are the adantages of the one or the other unit? b) Take some typical car elocities in km/h and conert them in m/s; also the other way round: km/h m/s c) Establish a formula for both calculations. m/s km/h ) When braking with constant pressure the elocity gets less steadily in time: In Austria car brakes must hae a minimum power of braking : Per second the elocity has to get less at least by 5 m/s. This amount is called braking deceleration b (here: b = 5 m/s/s = 5 m/s²) a) We assume that during a steady brake application the elocity of a car gets less by 5 m/s eery second (i. e. b = 5 m/s/s = 5 m/s²). How long does it take to stop ( braking [m/s] time t ) when the initial elocity t is gien? [s] b) Can you gie a general short formula to calculate the braking time t using the initial elocity and the braking deceleration b? 3) When you neither brake nor accelerate your motion the elocity will stay at the leel. When braking steadily the elocity will decrease steadily. In the figure to the right both graphs show the elocity in terms of the time. a) Find the formula for the function t () for t t in the case of braking steadily. What meaning does the constant slope of t () hae? b) What is the distance coered during the time t in case of (1) not braking and () braking respectiely? What is the relation between the two distances? Can you gie a general formula for the braking distance s in terms of the initial elocity and the braking deceleration b? [Hint: In case of constant elocity the distance coered in a time interal is gien by the area under the -t-graph, explain why! Try to argue for the same fact also in case of non constant elocity: cut the time interal into many small interals; in those you can assume approx. constant elocities. Continue this reasoning with own words.]

3 Hans Humenberger: rake applications and the remaining elocity 69 c) Choose two reasonable braking 18 m/s 35 m/s 8 km/h decelerations and s 5 m 5 m 75 m complete the table in each case. What happens to s if i) ii) 3? Possible Solutions: b) t = ; be careful of the units: if you want t in seconds you must use b m/s for and m/s² for b. 3a) t () = bt ; the (negatie) slope of the function corresponds to the braking deceleration b. 3b) The distance coered ( braking distance ) is the area of the grey s 1 triangle: s = t =. Of course the used units for and b hae to b be compatible (e.g. m/s and m/s²)! easoning that the area under the -t-graph can be interpreted as the distance coered although the elocity is not constant: We assume the elocity as constant in ery small periods of time: then the small rectangles are of course distances and their sum approximates the area of the triangle. Therefore it is clear that the area under the -t-graph can be interpreted as the distance coered. 3c) e careful of the units; i) : s 4 ii) 3 : s 9 s s With such a worksheet students should be able to establish the formulas t t =, t () = bt and s = = by themseles. This worksheet b b has nearly no prerequisites, it can be gien to students of grade 9. With the knowledge of the formula for s it is not difficult to answer also questions like What happens to the braking distance when you increase the elocity by about 4%?.

4 7 Hans Humenberger: rake applications and the remaining elocity So far it is rather standard and quite common. Often the fact the braking distance grows quadratically in terms of the elocity ( s = ) is learned b just like other formulas, without being aware of the enormous consequences. ut there is a good way to elucidate the striking consequences of the relation between elocity and braking process (safety in traffic)! In the way described aboe the focus lay on the braking distance s. In 3a) we also had the elocity in terms of the time elapsed ( t () = bt ) but it is more striking to deal with the elocity in terms of the distance coered. This will especially be seen in the worksheet C) at the end. Students (and also people who know the formula s = ery well) could b be asked to estimate elocities: A car is driing at a speed of 3 (5, 7) km/h; suddenly the drier sees a child coming on the street (e.g. fetching a ball without haing a look on the street) and can stop just in front of the child. Which remaining elocity would the car hae at this position, if it had an initial elocity of only a bit higher, say 5 (7, 1) km/h instead? It is surely surprising, striking and exciting that the remaining elocity of the faster car would hae been about 4 (5, 7) km/h. In the first case een more than the initial elocity of the slower car (4 > 3), in cases and 3: as if the car with the lower initial elocity had not braked at all: 5, 7 km/h! The corresponding explanations follow in worksheet C. I think these facts are more astonishing and go much deeper than the mere formula s = and the mere knowledge double elocity causes four b times braking distance. Therefore we hae in mind worksheet C as a central focus. For preparing it the following worksheet wants to show a way how students can hae an autonomous approach to the formula s () = bs, which is a prerequisite for C. Here s () denotes the elocity in terms of the distance coered s (b is the braking deceleration and the initial elocity). If students know already s () = bs it is possible to start immediately with C, but a repetition and getting familiar with the situation in the sense of may be appropriate also in this case.

5 Hans Humenberger: rake applications and the remaining elocity 71 Worksheets should make possible that the students can discoer and understand phenomena by themseles. In the best case they also should be able to gie proper reasoning. If the students are in grade 11 and hae already dealt with accelerated motions they should be able to repeat the reasoning for the formula maybe in a shorter way than it is done in worksheet. ut in grade 9 (or een in 1) students may happen not to know the basic facts about accelerated or decelerated motions. Especially in this case students should not only be gien the formula by the teacher or from the internet; they themseles should make first guesses and experiences (see below: a and b). As we all know: Teaching and learning mathematics is not only about facts, formulas and algorithms, it should be seen more and more as a process! ) Velocity of the car in terms of the distance coered A car is driing at a speed of = 7 km/h. Suddenly the drier sees some sort of barrier and starts braking so that the elocity steadily reduces eery second about 1 km/h. We know from worksheet A that the graph of the function t () that shows the elocity at the moments of time during the brake application looks like following: 1) Find out the braking time t and the braking distance s in the aboe case (formulas from A). ) How will the elocity of the car deelop in terms of the distance coered? a) Sketch the graph of the deelopment ( free-hand drawing ) of the elocity s () as you think it will be without calculating (s denotes the distance coered).

6 7 Hans Humenberger: rake applications and the remaining elocity b) Assume a reasonable aerage speed in eery second and calculate the distances coered in these seconds. With this information you can draw the graph of s () more accurately still without formula for the associated function. c) Let s consider a brake application from elocity stop; what is the meaning of b? What is the meaning of in a brake application from b elocity ( < ) stop? What is the meaning of the difference s = b b? y soling this equation for you get a formula for the elocity (initially ) in terms of the distance coered s: = ( s) explain why! d) Plot the graph of the function s ( ) of from part c) aboe with a computer compare with the free-hand sketch of a) and with the approximate method in b). e) y what fraction is the initial elocity reduced after ¾ of the braking distance is coered? f) When the initial elocity has been reduced by 75%, what percentage of the total braking distance has been coered? Is elocity reduced primarily at the beginning or primarily at the end of the braking distance? Possible Solutions and intended actiities 1) e careful of the used units! It is better to switch to m and s instead of km and h, because the elocity reduction is meant per second. 7 km/h 19.4 m/s; 1 km/h,8 m/s; Using the formulas from worksheet A one gets: t = = 7s and s = 67. m. b b In a) students should sketch the graph of s (); there are three possibilities as shown in the figure: linear, cured downwards, cured upwards. They should do it without calculating or reasoning; in b) it turns out

7 Hans Humenberger: rake applications and the remaining elocity 73 which one is the right shape of the cure. Students can check by themseles whether they were right or wrong. b) After 1 s the elocity is 6 km/h 16.7 m/s; a reasonable aerage elocity within this first second therefore is 65 km/h 18 m/s so that the distance coered in this second is approx. 18 m. Analogously one can do this estimation in the following seconds: time [s] elocity [km/h] elocity [m/s] total distance [m] When you draw the graph of (s) approximately with the last two rows of the aboe table you get the points (;19.4), K,(68;) : Here one can see that the graph is not linear but cured upwards. c) Meaning of b : braking distance with initial elocity Meaning of b : braking distance with initial elocity <.

8 74 Hans Humenberger: rake applications and the remaining elocity Meaning of the difference s = b b : needed distance for reducing the elocity from. Soling for yields 7 d) = 19.4 m/s; 3.6 b.8 m/s²; graph of s () = bs: = () s = bs. e) After ¾ of the braking distance only half of the initial elocity is reduced: 3 3 Inserting s = instead of s in 4 4 b 3 1 s = 4 s () = bsyields: f) For reducing 75% of the initial elocity nearly the whole braking distance is necessary, exactly 93.75% (or 15/16 ) of it: Let x be the necessary fraction of s ; then we hae = bx = 1 x x= 4 b 4 16 That means the main part of the elocity is reduced at the end of the braking distance! This can also be seen in the graph of the function s () = bs (square root function). This phenomenon is particularly important and should be thought of thoroughly in brake applications of eery day traffic! The main part of the elocity is reduced at the end of the braking distance! esides the formula s () = bs this is also a central message of worksheet! For proper understanding I think one should rather focus on the shape of the graph of s () = bs (square root function, parabola) and not so much on concrete calculations.

9 Hans Humenberger: rake applications and the remaining elocity 75 C) emaining elocity ersion 1 A car is driing at a speed of 3 km/h in a zone with a speed-limit of 3 km/h (as common in many inner cities). Another car oertakes it, traelling at a speed of 5 km/h. At the moment that the two cars are side-by-side a child crosses the road without looking. (e.g. to fetch a ball). The cars hae equally strong brakes and the two driers react equally quickly. The car traelling at 3 km/h just comes to stop in front of the child. What remaining elocity does the other car hae at this position? 1) a) Estimate this remaining elocity. b) Estimate this remaining elocity also in cases of other initial elocities: instead of (3km/h, 5 km/h), use (5km/h, 7 km/h) or (7km/h, 1 km/h). ) In the case aboe the remaining elocity was more than 4 km/h; can this be true? Such a crash elocity is in most cases lethal for a child. Assume that both brake applications begin at the same position. Plot the graphs of both elocity functions (elocities of the two cars in terms of the distance coered) in one plot and read off the remaining elocity (do the same also for the other alues mentioned in 1)). Can you plot the graphs of the elocity functions so that the elocities are measured in km/h and the distances in m (for better comprehension)? 3) In reality the brake applications will begin at the same time but not at the same position as we assumed in ) explain why! This makes things worse concerning the remaining elocity the aboe alue (read off the graph, 4 km/h) therefore is just an unrealistic minimum! A maximum alue of the remaining elocity in the aboe case would be 5 km/h (why?). 4) Find the unrealistic minimum and the maximum alue of the remaining elocity in the general case (depending on the initial elocities, and the braking deceleration b). What do you notice concerning the minimum alue and b?

10 76 Hans Humenberger: rake applications and the remaining elocity 5) Now take into account the different reaction distances. denotes the slower initial elocity, the faster one. The graphs of the elocities then are like shown here. Explain the graphs! What is the meaning of s and s? What is the meaning of s + s and s + s? Gie formulas for them. Why are the break points ( ertices ) on the same straight (broken) line through the point ( )? Calculate the remaining elocity in the aboe case ( = 3 km/h, = 5km/h ) assuming a reasonable reaction time in two cases: (1) both cars hae ery weak brakes: b = 3m/s () both cars hae strong brakes: b = 9m/s. 6) Let s assume a reaction time of t = 1s and general alues for the initial elocities (, ). For what alues of the braking deceleration b does the faster car hae its full initial elocity as remaining elocity? Are these alues for b realistic or unrealistic ones (can full initial elocity be true) in the aboe situations: -) = 3 km/h, = 5 km/h -) = 5 km/h, = 7 km/h -) = 7 km/h, = 1 km/h

11 Hans Humenberger: rake applications and the remaining elocity 77 emarks to Worksheet C Its main focus is: Not making own experiments or doing modelling in a classical way ( modelling cycle ), but by dealing with graphs, functions, and formulas ( s () = bs) the students should realize the dramatic phenomena concerning the remaining elocity. Prerequisites: Students know: 1) What does the braking deceleration b mean? ) Formula for the braking distance: s /( ) = b 3) Formula for the elocity in terms of the distance coered s: s () = bs The aim of the tasks (dealing with graphs and functions) is that students use their knowledge in order to see, understand, and explain the underlying dramatic phenomena. I am aware that the actiities are rather based on plotting and interpreting graphs, treating formulas appropriately (and not so much modelling ) but the underlying message of the problem in this context is ery striking and illustratie. So it still has a lot to do with reality and is ery important in traffic education! The estimations of the remaining elocities usually are by far too low. The preious and following thoughts can maybe open ones eyes. Often the fact the braking distance grows quadratically in terms of the elocity ( s = ) is learned just like other formulas, without being aware of the b enormous consequences taking into account the remaining elocities. These phenomena are a good opportunity to work together with physics, road safety education or driing schools and to show: mathematics can help to explain and to understand many phenomena mathematics is useful

12 78 Hans Humenberger: rake applications and the remaining elocity Possible Solutions: ) Many students will think that a remaining elocity of more than 4 km/h is highly exaggerated. ut a short glance on the graphs of the elocity functions Fig. 1 s () = bs and ( s) = bs [ = 3 km/h 8.3 m/s, = 5 km/h 13.9 m/s; students hae to make an assumption on the brake deceleration b, e.g. b = 5 m/s²] makes clear that the remaining elocity of the faster car really can be quite near to its initial elocity (this as a priori probably not clear!). In the first step one can work only graphically with the plots of the functions s () and ( s). Since the students hae to make an assumption on b it is likely that they also will try other alues of b (if not the teacher should encourage them to do so). When trying other alues of b one will notice that the alue of the remaining elocity rem (approx. taken from the graphs) seems to be always the same (independent of the alue of b ), approx. 11 m/s or 4 km/h respectiely. This means the gien alue more than 4 km/h can surely be true. Here is a plot of the situation concerning all four gien initial elocities 3 km/h, 5 km/h, 7 km/h, and 1 km/h (again b = 5 m/s²; for better comprehension the elocities in km/h): The other remaining elocities (read off the graphs approx.) (5 km/h, 7 km/h): remaining elocity 5 km/h (7 km/h, 1 km/h): remaining elocity 7 km/h

13 Hans Humenberger: rake applications and the remaining elocity 79 I think this is a striking and exciting result! The remaining elocity of the faster car is een higher than the initial elocity of the slower car! At the position of the child it is faster than the slower car would be completely without braking! Such graphs concerning the elocities in terms of s and approx. alues of the remaining elocities (yet without calculation) show the dramatic phenomena and consequences of high speed traffic better and more instructiely than formulas like s = /( ) b. 3) Although for both cars the reaction time is said to be equal, the faster car begins its brake application somewhere ahead because it has a longer reaction distance. Therefore in reality the remaining elocity is een higher than the alue read off in graphs like here (in such graphs we assume that the cars begin braking at the same position). ecause the initial elocity of the faster car is 5 km/h the remaining elocity can t be higher! Therefore in reality the possible interal for the remaining elocity in this case is (4km/h; 5kmh]. 4) There are two initial elocities: > ; for the unrealistic minimum alue of the remaining elocity we are looking for the function alue of ( s) at s = : rem = ( s ) = = b = b b b Now we hae the mathematical explanation of the fact seen aboe: This unrealistic minimum remaining elocity is completely independent of the braking deceleration b. The maximum in the general case is of course gien by the higher initial elocity. Therefore the domain in which the remaining elocity lies in the general case is the interal ( ;. With = 5 km/h, = 7 km/h: rem (49 km/h; 7 km/h]. With = 7 km/h, = 1 km/h: rem (71km/h; 1 km/h].

14 8 Hans Humenberger: rake applications and the remaining elocity 5) Due to the equal reaction time and different reaction distances the elocity begins to decrease at different positions. s and s are the reaction distances, s + s and s + s are the oerall stopping distances. The break points (ertices) of the graphs lie on the same straight (broken) line through the point ( ) because of the equal reaction time in both cases. Formulas: s = t, s = t, s + s = t +, b b + = + s s t emark: We do not use piecewise defined functions and we don t consider the horizontal shift algebraically (in the term of the function) because students will probably also not deal with the problem in this way when they work autonomously (unless the teacher tells them to do so). This way would be appropriate, if a general formula for the remaining elocity is aimed at. + bt ( for ) s + s > s esult: rem = for s + s s We assume a reaction time t = 1s ( = 3 km/h, = 5 km/h). (1) b = 3m/s : s + s 19,9 m ; from this we hae to subtract the longer reaction distance compute s 13,9 m. For the remaining elocity we hae to ( s) = bs at s 19,9 13,9 = 6m: rem (6) 1,5 m/s = 45km/h. () b = 9m/s : s + s 1, m ; the reaction distance s 13,9 m of the faster car is here een longer than the oerall stopping distance s + s 1,m of the slower car; so it is clear that the remaining elocity in this case is the full initial elocity of the faster car: 5 km/h (as if it would not brake at all!).

15 Hans Humenberger: rake applications and the remaining elocity 81 6) Full initial elocity as remaining elocity happens if the oerall stopping distance of the slower car is not longer than the reaction distance of the faster car: t + t {. Taking t 1s 1443 b = and soling for b yields: b ( ) s+ s s -) = 3 km/h, = 5 km/h: b 6.5 m/s realistic! -) = 5 km/h, = 7 km/h: b m/s unrealistic -) = 7 km/h, = 1 km/h: b.69 m/s unrealistic raking decelerations: asphalt, concrete (dry): b m/s² asphalt (wet): b m/s² That means when the two cars hae initial elocities of 3 km/h and 5 km/h respectiely it is well possible that the faster car has not yet started to brake (and still has its full elocity of 5 km/h) at that point the slower one comes to stop! C) emaining elocity ersion A car is driing at a speed of 3 km/h in a zone with a speed-limit of 3 km/h (as common in many inner cities). Another car oertakes it, traelling at a speed of 5 km/h. At the moment that the two cars are side-by-side a child crosses the road without looking. (e.g. to fetch a ball). The cars hae equally strong brakes and the two driers react equally quickly. The car traelling at 3 km/h just comes to stop in front of the child. What remaining elocity does the other car hae at this position? 1) a) Estimate this remaining elocity. b) Estimate this remaining elocity also in cases of other initial elocities: instead of (3km/h, 5 km/h), use (5km/h, 7 km/h) or (7km/h, 1 km/h). ) Find out a reasonable alue for the remaining elocity not only by estimating. 3) How can one find a reasonable answer for the remaining elocity in the general case: initial elocities like in 1) or with ariables:,?

16 8 Hans Humenberger: rake applications and the remaining elocity emark: This ersion is much more open ended. The concrete questions of ersion 1 lead somehow step by step to a certain result. In ersion 1 students also work autonomously but are at the same time led by the structure of the gien tasks. Here in ersion students hae to look for a way for themseles to come to an answer. Of course this needs not to be the same as in ersion 1! It depends on many circumstances (teacher, students, how are students used to ery open ended problems,...) which one is the better in a specific teaching situation. To summarize: Although worksheet C is not ery easy to cope with and although students hae to work not so much in modelling and but rather on working with the formula s () = bs, plotting and interpreting graphs and the like, we think that this worksheet has a lot to do with reality: Fundamental and important phenomena concerning the remaining elocity can be discoered and seen in a ery striking way. So mathematics can help to understand the enormous consequences hidden behind too high elocities. These consequences can be seen more clearly when one deals also with remaining elocities and not only with the formula s =. b Address of the author: hans.humenberger@uniie.ac.at Hans HUMENEGE, Faculty of mathematics, Uniersity of Vienna, Nordbergstraße 15, A 19 Vienna.