RESTATEMENT OF PROBLEM

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RESTATEMENT OF PROBLEM

This paper describes how and where to position a pile of cardboard boxes that will cushion thefall of a stunt motorcyclist who will jump over an elephant. We will discuss the flight path of thestuntman given several different initial velocities, and a range of maximum heights and distancestraveled. We will also give directions for where to start the box pile for a given initial velocity,as well as how wide and long the pile should be. Finally, after comparing prices andcompression test values on different types of boxes, we will choose a specific box type and willuse the McKee formula for compression strength to determine how much force each box cantake. We will use this information to determine how deep the pile of boxes should be.

Assumptions

- The motorcyclist will start his approach 2.8 m from the end of the ramp.- The motorcyclist will jump off a 45-degree ramp that is 1.4 m long, 2 m wide, and 2 m

off the ground.- The motorcyclist is able to leave the ramp within the middle 0.5 m (see Figure 7).- The motorcyclist is able to leave the ramp within 1.5 m/s of his target initial velocity.- Motorcycle will be either a sport bike (average mass 199.58 kg) or an off-road bike

(average mass 131.54 kg).- Motorcyclist will have a mass of 72.58 kg.- Jumps will be of minimum height 4.5 m and of maximum height 8.5 m.- Elephant will not move during jump.- There is negligible wind.- Motorcyclist will not let go of bike during jump.- Motorcycle will stay upright.- Motorcycle tires slow down to a negligible velocity before impact on boxes.- Boxes will not catch fire.- Structural integrity of the cardboard boxes is affected by the environment.- Area on which box structure will be built is level- A motorcycle and its rider can withstand hitting the ground at a speed of 3.8 m/s (9.8

mph)- The motorcycle will decelerate at a constant rate as it falls through the box structure- The maximum deceleration rate a rider can take is 9.8 m/s2 (that is, one G)

Variables

- Initial velocity will be within 1.5 m/s of either 13 m/s, 16 m/s, or 19 m/s- Mass of cycle plus stuntman will be either 272.16 kg (sport bike) or 204.12 kg (off-road

bike). See Figures 1 and 2 for determined center of mass. [2], [5], [9], [10], [16], [26]

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Figure 1. Center of mass for Honda off-road motorcycle and rider

Figure 2. Center of mass for Honda sport motorcycle and rider

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ANALYSIS OF PROBLEM

Flight path

In this section of the analysis, we will describe the flight path of the motorcycle, from themoment the bike leaves the ramp. We will momentarily consider this a two-dimensionalproblem and derive parametric equations relating distance traveled and height. (See the sectionbelow, Landing area, for changing the problem from 2-D to 3-D.) To make the problem morerealistic, we included the force of drag in our analysis. This section will be divided into twoparts: deriving the kinematics equations, then using them to choose target initial velocities andaltitudes of the motorcyclist, as well as an estimated range of accuracy of these values.

Part I: Deriving the Equations

There are two forces acting on the moving body. The gravitational force causes a downwardacceleration of < 0, -g >, where g = 9.8 m/s2. The magnitude of the force due to drag is given bythe equation: D=0.5CρAv2, where D is the force due to drag, C is the coefficient of friction, ρ isthe density of the air, A is the effective area of the object, and v is the norm of the velocity. Thedirection of the force D is opposite in direction to the moving object. [14] The acceleration dueto D will be considered carefully.

From [16], the coefficient of friction of a motorcycle with a rider was given to be 6.2 ft2, with theeffective area figured in, i.e., CA=6.2 ft2. Since our motorcycle is airborne, we estimated theeffective area to be doubled. Using ρ = 1.2 kg/m2 from [14], and converting CA to metric, wecalculated D = .691 v2. Then we derived the following equations for acceleration, with m =mass, ax = dvx/dt = the horizontal component of acceleration, and ay = dvy/dt = the verticalcomponent of acceleration.

dvx/dt = (-.691*(vx2 + vy

2)1/2* vx)/m

dvy/dt = -g + (-.691*(vx2 + vy

2)1/2* vy)/m

Unfortunately, there is not a very easy way to solve this system of differential equations.Because of the time constraint, we decided it would be an acceptable estimate to assume that themagnitude of the force due to drag is proportional to the norm of velocity. That is, in our case,we assumed D=b*v, where b = .691. [21] The result of this assumption is the followingequations:

vx(t) = vx0e-bt

vy(t) = vy0e-bt + g/b*(1- e-bt)

x(t) = x0 + vx0/b*(1- e-bt)

y(t) = y0 + g/b*t + (vy0/b – g/b2)*(1- e-bt)

Part II: Using the Equations

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We used these equations to choose target initial velocities and altitudes of the motorcyclist, aswell as an estimated range of accuracy of these values. Because of our assumptions, the onlyvariable upon which we need to find is the initial velocity. By [1], the average height of theAfrican elephant is 11-12 ft. (See Figure 3.) Using an Excel spreadsheet, we decided to choosetarget velocities of 13, 16 and 19 m/s. Using the 1.5 m/s range of accuracy, the values we choosegave us the correct corresponding heights we desired in the assumptions. See Figures 4, 5, and 6for the three cases.

Figure 3. African elephant, which averages 11-12 ft. in height.

Figure 4. Motorcycle path with target initial velocity of 13 m/s, with an upper bound of 14.5 m/sand a lower bound of 11.5 m/s

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Landing area

We determined that to ensure simplicity of the construction of the box pile, we should haveidentical landing areas for all cases of initial velocity. Thus to determine the landing area, wedetermined the maximum width and length of the box pile needed for the highest possible initialvelocity, 20.5 m/s because this case yields the largest landing area.

Width of landing area:

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In our assumptions, we stated that the motorcyclist would began his approach to the take-offramp 2.8 m before the ramp’s edge, where the ramp is 1.4 m long, 2 m wide, and 2 m highstraight surface at a 45-degree angle. We also assumed that the motorcyclist would be able totake off from the ramp within the middle 0.5 m. Using trigonometry, we determined that thehorizontal error of the motorcyclist using the starting point 2.8 m away from the edge of the rampwould be 10.1 degrees (see Figure 7). Furthermore, observing a total distance traveled of 17.47m in the case of the highest possible initial velocity, 20.5 m/s (see Figure 6), we determined thatthis 10.1 degree error in width would translate into a width of 3.11 m that would need to becovered by boxes. Assuming that the motorcyclist landed on the extreme end of this 3.11 m, wedetermined that he would need another 1 m to his side in order to land safely. This gave us atotal box pile width of 5.11 m (see Figure 7).

Length of landing area:

We again used the information from Figure 6 to determine the necessary length of the box pile.Because 19 m/s was the highest target velocity, accounting for a 1.5 m/s error in speed either wayyielded a larger difference in total distance traveled than in the other target velocities. We notefrom Figure 6 that the total distance traveled if the initial velocity was 17.5 m/s is 14.25 m, whilethe total distance traveled if the initial velocity was 20.5 m/s is 17.47 m. This yields a differenceof 3.22 m which must be covered by boxes. Assuming that the motorcyclist landed on theextreme end of this 3.22 m, we determined that he would need another 1.5 m either in front ofhim or behind him in order to land safely. This accounts for the fact that our kinematic equationsmodel the path of the center of mass of the bike. This figure accounts for the length of themotorcycle and the placement of the center of mass. (See Figure 1 and 2). This gave us a totalbox pile length of 6.22 m (see Figure 7).

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Figure 7. Necessary length and width of box pile given a take-off error of 10.1 degrees and amaximum distance traveled of 17.47 m.

Landing on boxes

To determine the height of the box structure needed how high the pile of boxes would need to beto decelerate the heaviest combined weight (272.16 kg) to safe levels, we used the McKeeformula to estimate the initial box compression strength (ICS). [7], [26], [27] This formula hasbeen found to determine the initial box compression strength within 15% of actual values:

ICS = 2.03*(ECT)0.746*(DMD*DCD)0.5*P0.492

In this formula, ECT is the edge crush test value of the cardboard in lb/in, DMD is the boardflexural stiffness in the machine direction in lb/in, DCD is the board flexural stiffness in the crossdirection in lb/in, and P is the box perimeter (2*interior length + 2*interior width). Wecalculated the ICS of a box with perimeter 72 in. (as was determined to be the most cost efficient

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size in the “Design” section), an ECT of 32 lb/in, and a height of 3/16” (which were determinedin [7] to be the approximate values for such a box) using an estimation formula [6]:

ICS = 5.0(ECT)(P*h)0.5 = 5.0(32)(72*3/16)0.5 = 587.9 lb = 2615 N

However, this force value is not accurate under any normal environmental circumstances,including factors such as heat and humidity. To more accurately predict the force in an actualimplementation, we must divide the above force value by a “safety factor” of 8.0 [17]. Thisyields a force of 327 N per box.

We next determined the necessary height of the box structure. We assumed that the motorcyclewill decelerate at a constant rate as it falls through the structure, that the stuntman andmotorcycle can withstand hitting the ground at a rate of 3.8 m/s (8.5 mph), and that the maximumdeceleration rate a rider can take is 9.8 m/s2 (that is, one G). (Note: we do not know that this is alikely deceleration rate – we simply used it as an upper bound. The actual deceleration rate couldbe determined when our system is tested.) Using the equations d=½*at2 and t = (vf – v0)/a, , wederived the equation

0 = ½*(vf – v0)2/a – d

where v0 is the velocity at which the motorcycle hits the box structure and d is the height of thebox structure. Plugging in the values 9.8 m/s2 for a and 3.8 m/s for vf, we obtained:

0 = ½*(3.8– v0)2/9.8 – d

Because both v0 and d are dependent on time, we approximated the solution to this equationusing our Excel spreadsheet by calculating the value for the left-hand side of the equation when dranged from 0.5m to 2.5 m and determined that it was closest to 0 when d = 1.5 m and v0 = 9.16m/s.

Again referring to our Excel spreadsheet when d = 1.5 m, we see that a1.5m = 4.63 m/s2. Thususing the equation F = m*a, we have that the sport motorcycle (m = 272.16 kg) strikes the boxstructure with a force of 1260 N, and that the off-road motorcycle (m = 204.12 kg) strikes thebox structure with a force of 945 N. Using our determination above that each box can withstanda force of 327 N, we find that the sport motorcycle requires four layers of boxes in the structureand that the off-road motorcycle requires three layers of boxes.

DESIGN

As was determined in the Landing area section, the box pile must be at least 5.11 m wide by6.22 m long. We also showed in the Landing on boxes section that for a sport bike, the structuremust be at least four boxes thick with a minimum height of 1.5 m (4.9 ft). (We do not considerthe case for the off-road bike when determining how many boxes to order as it requires the samebox area but only needs the structure to be three boxes thick.)

We first researched whether we should use single wall, double wall, or triple wall boxes in someor all of the structure. We found that double wall boxes and triple wall boxes, which are a

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combination of B and C flutes, were nearly twice as expensive as more traditional single wallboxes, which are made entirely of C flute [13], [20], and also yielded lower ICS rates than singlewall boxes. Consequently, we decided to build our entire structure out of single wall boxes.

Obtaining a price estimate of the total number of boxes needed proved difficult since the majorityof box companies did not post their prices. We found one company which did post prices fordifferent types and sizes of boxes per box, so it is very likely that the boxes could be purchasedin bulk elsewhere at a reduced price. However, we did determine from www.mrboxonline.comthat the cheapest option would be to purchase 672 18”x18”x16” boxes at $1.17 each for a totalprice of $786.24 and a box pile height of 1.63 m, or 5’4” (see Figure 6). [20]

Dimensions Cost perbox

Total boxes needed Total price Box pile height

1)

13”x13”x16” $0.92 1216 $1118.72 5’4”

2)

16”x12”x10” $0.61 1700 $1037 5’4”

3)

15.25”x11.5”x7.125”

$0.58 2520 $1461.60 5’1”

4)

18”x18”x16” $1.17 672 $786.24 5’4”

Figure 8. Number of boxes needed and corresponding total prices from mrboxonline.com [20]

We recommend building a rectangular prism with dimensions approximately 6.22 m by 5.11 mby 1.5 m by stacking 3 or 4 layers of boxes depending on the combined weight of the motorcycleand motorcyclist. The boxes should be arranged adjacently and upright. We also determinedthat the boxes should be single-layer, C-flute because these are the most common andeconomical and yield the best compression strength.

While we only accounted for the price of the 672 boxes in our above price calculations and weonly accounted for using solid, unaltered boxes in our calculations for the design of a safe boxstructure, we believe that several modifications to the boxes would improve the effectiveness ofour design. Historically, movie stunt cushioning boxes often have their corners cut to allow themto compress more evenly and that flattened boxes are placed between box layers to spread theforce more evenly among each layer. Finally, the box structure is traditionally secured at thebase with rope and at the top with a tarp. [4], [29] We recommend performing thesemodifications during implementation.

Structure location:

Figures 9, 10, and 11 display the recommended relative locations of the ramp, elephant, andboxes given target initial velocities of 13 m/s, 16 m/s, and 19 m/s, respectively. The positionpath of the motorcycle is also displayed to show the clearance over an elephant of height 3.35 m(11 ft.). Notice that the larger initial velocities produce conditions conducive to obtaining a goodcamera shot of the stunt.

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Figure 9. Recommended relative locations of the ramp, elephant, and box structure given targetinitial velocity of 13 m/s

Figure 10. Recommended relative locations of the ramp, elephant, and box structure giventarget initial velocity of 16 m/s

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Figure 11. Recommended relative locations of the ramp, elephant, and box structure giventarget initial velocity of 19 m/s

TESTING

To test the proposed box structure, we believe it would be best to build a box structure comprisedof:

1) The same size of boxes, because so little is known about how boxes withstand this type ofstructural stress. Additionally, we would not have to add in the error in scaling the datato fit another size of box.

2) The same height of boxes, for similar reasons as in (1)3) Same extra components:

a. Same tarp securing boxesb. Same layers of flat cardboard in between layers of boxes

4) Using a crane to drop a given weight of dimensions similar to a motorcycle: using onlythe maximum weight of 272.16 kg if money constraints permit only one destruction ofboxes

5) Different length and width dimensionsa. We can more easily drop a large weight than launch it; thus, we will be more

accurate and will not need to account for the safety margin we considered in theDesign section

b. Smaller length and width dimensions will save money

Analysis of the test:

We will know with what force, acceleration, mass, and velocity the weight hit. Not only can weobserve how the boxes react, but we can also use a digital video camera to save the data for amore careful visual analysis. Further, we can use the visual data coupled with the timing dataprovided by the digital video camera to perform a quantitative analysis on the test. Mostimportantly, we can calculate the deceleration of the object to ensure that the human being cansurvive that amount of G’s. (Note: as stunt coordinators for a move, we have access to high-techvideo cameras. This would not add significantly to the cost of the testing.)

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STRENGTHS AND WEAKNESSES

Strengths:

- The analysis takes drag constant into account, which vastly improves the accuracy of theplacement of the box pile

- Design is very easy to build, since its a rectangular shape with virtually the sameassembly (plus or minus the top layer of boxes) for different initial velocities andcombined weights of the stuntman and motorcycle

- $768.24 is a very reasonable price estimate, in comparison with other types and sizes ofboxes

- Allows for a 10.1 degree side-to-side error in take-off from the ramp- Allows the motorcyclist to hit the box pile if within 1.5 m/s of the target velocity- Allows for two different kinds of motorcycles with very different combined weights

Weaknesses:

- “Corrugated board cannot be considered an engineering material due to the fact that itsstrength characteristics are not predictable within its normal range of use.” [15]

- We made assumptions of what a professional motorcyclist or stuntman is capable ofbased on our personal experience with mountain biking and watching [17]. It could bethat their skills are much better than we anticipated; thus the area of the boxes and thetotal price could be reduced.

- Our equations from the Analysis of Problem section would have been more accurate if wehad an accurate drag coefficient for our specific motorcycles that was calculatedexperimentally

- We only allow for three target velocities, but the movie director might want a muchdifferent target velocity

- Our two combined weights of 272.16 kg for the stuntman and a sport bike and 204.12 kgfor the stuntman and an off-road bike are rather crude averages, and it might have beenmore appropriate to have data for a bigger variety of combined weights. However, it isunlikely that anything heavier than a sport bike would be used in a movie, and thus a 272kg upper bound on the combined weight is reasonable. Furthermore, it is only thisheavier case that we need to consider.

- Using the edge crush test (ECT) and the corresponding initial box compression strength(ICS) rates is not entirely appropriate in this problem. From individual trials of jumpingon boxes, edges are extremely strong in comparison with the center of a panel.Consequently, the motorcycle will not completely crush a box before hitting the nextlayer, as is suggested by our model. If we had more explanatory resources for themeaning of the tests, we could probably use the data more appropriately.

- ICS rate does not take into account the height of the box and has been shown to only beaccurate to about 15% [6]

- Statistical methods should have been used to determine a reasonable variance in speed foreach target velocity and the likelihood of departing the ramp within a certain distancefrom its center.

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- More accurate position and velocity equations describing the flight of the motorcyclecould have been obtained by solving the system of differential equations using the factthat force is directly proportional to the square of the velocity.

- Limits in structural data for boxes limits our choices for box. For example, cutting thecorners off of the boxes would lower the structural integrity of the box – thus enabling usto use more box layers; consequently, the deceleration process would be much morecomfortable. However, pending doing strength experiments on the boxes ourselves, wehave no way of reasonably quantitatively predicting the strength of this setup.

- We found no information on the maximum velocity at which a person could safely hit theground. In our calculations, we assumed that a person could tolerate landing at 3.8 m/s(8.5 mph), but this value could be either too low or too high.

- In our calculation of the necessary height of the box structure, we assumed a maximumdeceleration rate upon hitting the box structure of 9.8 m/s2. This value might be muchhigher than necessary, in which case the required height of the box structure would belower.

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BIBLIOGRAPHY

[1] The Amazing World of Elephants. (2001). Retrieved February 7, 2003 from http://wildlywise.com/ele_text.htm

[2] American Honda Motor Co., Inc. (2002) Retrieved February 7, 2003 from http://www.honda.com

[3] Baldini, N.C. et al. (Eds.) (2002). Paper; Packaging; Flexible Barrier Materials; Business Imaging Products (Vol. 15). Annual Book of ASTM Standards; Section 15: General Products, Chemical Specialties, and End Use Products . West Conshohocken, PA: ASTM International.

[4] Baxter, J. (1974). Stunts. Garden City, NY: Doubleday.

[5] Bike Stand Drills & Bike Training Practice. (2002, March 4). Retrieved February 7, 2003, from Medsker Racing College website:http://www.racingsmarter.com/bike_stand_training_simulator.htm

[6] Buschow, J.K.H. et al. (Eds.) (2001). Packaging: Corrugated Paperboard. In Encyclopedia ofMaterials: Science and Technology (pp. 6637-6641, Vol. 7). Kidlington, Oxford: ElsevierScience.

[7] Buschow, J.K.H. et al. (Eds.) (2001). Paper Products: Container Board. In Encyclopedia of Materials: Science and Technology (pp. 6696-6701, Vol. 7). Kidlington, Oxford: ElsevierScience.

[8] Corrugated Paper Products: Product Information. Retrieved February 7, 2003 from: http://www.shipmaster.com/main.html

[9] D’Acquisto, L.J. Anatomical Kinesiology: The Center of Gravity and Stability. Retrieved February 7, 2003, from Central Washington University website:http://www.cwu.edu/~acquisto/balance.htm

[10] Foale, T. (1997). Balancing Act. Retrieved February 7, 2003 from http://www.ctv.es/USERS/softtech/motos/Articles/Balance/BALANCE.htm

[11] Gerrity Corrugated: Products and Services. Retrieved February 7, 2003, from http://www.gerrity.com/products.htm

[12] Granger, R.A. (1985). Fluid Dynamics. New York: CBS College Publishing.

[13] Great Little Box Company. Retrieved February 7, 2003 from http://www.greatlittlebox.com/cat_b_1.htm

[14] Halliday, D., Resnick, R., & Walker, J. (2001). Fundamentals of Physics (6th ed.). New York: John Wiley & Sons, Inc.

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[15] HP Compression Test Method. (1996). Retrieved February 7, 2003 from http://packaging.hp.com/testing/sectab.htm

[16] Hoerner, S.F. (1951). Aerodynamic Drag. Dayton, Ohio: The Otterbein Press.

[17] Hendricks, B. (Writer/Director). (2003). ESPN’s Ultimate X: The Movie. [Motion picture]. United States: Buena Vista Home Entertainment, Inc.

[18] Hough, D. (2003, February). Riding Skills: Coming Unglued. Retrieved February 7, 2003 from http://www.soundrider.com/archive/safety-skills/coming_unglued.htm

[19] Konopasek, M. & Ferguson, R. Solving Differential Equations with Runge-Kutta. Retrieved from February 7, 2003. http://www.uts.com/arc/diffeq.html

[20] Mr. Box Online (2003) Retrieved February 7, 2003, from http://www.mrboxonline.com

[21] Projectile Motion. Retrieved February 7, 2003 from Suny Brockport Department of Physics website: http://www.brockport.edu/physics/index.html

[22] Scibor-Rylski, A.J., (1984). Road Vehicle Aerodynamics (2nd ed.). New York: John Wiley & Sons, Inc.

[23] Skytran Safety Information (2003). Retrieved February 7, 2003 from http://www.skytran.net/09Safety/01aSftyINTRO.htm

[24] Sovran, G., Moral, T. & Mason, Jr., W.T. (Eds.) (1978). Symposium on Aerodynamic DragMechanics of Bluff Bodies and Road Vehicles, General Motors Research Laboratories,1976. New York: Plenum Press.

[25] Stacking Strength (1998). Retrieved February 7, 2003 from http://www.topseng.com/T_Stack.pdf

[26] TAPPI Testing Lab. Retrieved February 7, 2003 from http://www.mysticsheets.com/tappi.htm

[27] Tests for Properties of Corrugated Boxes. (1998). Retrieved February 7, 2003 from http://www.fcbm.org/activities/tests_for_corr_media.htm

[28] Tuluie, Rob. Wrenching with Rob. Retrieved February 7, 2003 from http://www.motorcycle.com/mo/mcnuts/chassis.html

[29] Wise, A. & Ware, D. (1973). Stunting in the Cinema. New York: St. Martin’s Press.

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RESTATEMENT OF PROBLEM