bu er zones in basketball - university of new...

Needs and current stardards Biomechanics of human walking and running Study design Measurement and data analysis

Buffer Zones in Basketball

Dan LiuJoint work with: Ceyda Mumcu, Gil Fried

University of New Haven

January 29, 2017


Injuries caused by inadequent buffer zones

• Lebron James caused a concussion to a spectator.

• Fatal accident: In 1997, 8th grade Lamar Pope went head first intothe gym wall less than 5 feet from the end line as playingbasketball.

• In 2001, 14 year old Katie Patrick fell backwards and hit her headon the unpadded metal wall less than 4 feet from the end line. Shesustained a traumatic brain injury.

”... the lack of a sufficient buffer zone was alleged to have been theprimary causes of injury in 67% of basketball lawsuits.”Todd L. Seidler, Basketball buffer zones: accidents waiting to happen, HighSchool Today, Oct 2011.

http://espn.go.com/nba/story/_/id/14392300/lebron-james-cleveland-cavaliers-bowls-ellie-harvey-wife-golfer-jason-day-stands/


Buffer zone is the space between the activity area and any obstructionssuch as walls, benches and equipment.

Figure: National Federation of State High School Association (NFHS) courtand field diagram guide.


Current standards and regulations

• National Federation of State High School Associations BasketballRules .. at least 3 feet (preferably 10 feet)

• National Collegiate Athletic Association (NCCA) Mens andWomens Basketball Rules Committees and the Amateur AthleticUnion (AAU) recommend ”... at least 10 feet of open spacebetween the boundary lines and the seating.”.

• American Alliance for Health, Physical Education, Recreation andDance (AAHPERD) recommends at least 10 feet clear spacebeyond the end lines with a very minimum of 6 feet with full wallpadding.

• Is the space sufficient? if not, then what should be the minimumbuffer zones?


Physics behind of walking

Normal force

Sta-c fric-on

gration can be chosen so that the time averages of a and vare zero, and the time average of z is either zero or its mea-sured value.7The results of such a calculation are shown in Fig. 7. For

these calculations, the observed force wave forms were ap-proximated and slightly smoothed by fitting a sixth-orderpolynomial. The total ground force is a maximum when theCM is at its lowest point and is a minimum when the CM isat its highest point. The amplitude of the vertical displace-ment is typically about 4 cm peak to peak. A similar calcu-lation based on the horizontal force wave form indicates thatthe horizontal speed is a maximum when the CM is at ithighest point and the speed is a minimum when the CM is atits lowest point.A surprising result is that z is almost exactly sinusoidal,

despite the fact that a is distinctly nonsinusoidal and theforce wave form is asymmetric at high walking speeds. Sincethe a wave form is periodic, it can be modeled as a Fourierseries containing a strong fundamental component andweaker harmonics. When this is integrated twice, the higherharmonic components are effectively filtered out. Con-versely, the force and acceleration wave forms are very sen-sitive to small variations in the vertical displacement. Forexample, suppose that the vertical height of the CM, when itis near its maximum position, is given by z�1�t2/2�t4/24. This represents the first three terms in the seriesexpansion of cos(t). Then z is a maximum, and a��2�t2/2 passes through a minimum, at t�0. However, if z�1�t2/2�t4/24 then a��2�t2/2 is a maximum at t�0.During the time interval �0.3�t�0.3, the two z wave formsare almost indistinguishable when plotted on a scale from 0to 1 m since they are equal at t�0 and differ by only 0.7 mmat t�0.3 s. In both cases, z is a maximum at t�0, a isnegative, and the ground reaction force F is less than Mg.However, it is unlikely that the two z wave forms could bedistinguished experimentally by direct measurement, e.g., ofall body segments, in an attempt to locate the position of theCM. It is known that the CM of a person is located withinthe pelvis, but it is not known to within one mm.The dip in F in Fig. 6 is due to the fact that the CM is

raised to its maximum height as the CM passes over thestationary foot on the ground. At this stage of the walkingcycle, the CM describes an arc of radius r�1 m for adultsand the ground force is given by F�Mg�Mv2/r . Conse-quently, if v is greater than about 3.2 ms�1, the ground forcedrops to zero and the walker becomes airborne, or breaksinto a running stride.2,3 For young children, r is smaller andthey start running at a lower speed. Consequently, a childneeds to run to keep up with a fast walking adult.A simplified model of walking is obtained by regarding

the leg on the ground as an inverted pendulum, and the leg inthe air as a pendulum suspended at the hip. An interestingdemonstration is to hold the top of a meter stick close to thehip, set it oscillating as a pendulum, and then walk in stepwith the pendulum. It is easy to walk faster or slower thanthe pendulum, but walking in step results in a comfortablewalking speed and presumably helps to minimize the effortrequired. In general, the stride length increases with walkingspeed, but the cadence �i.e., step frequency� increases onlyslightly as the walking speed increases.8,9A sideways oscillation is also observed when walking, at a

frequency that is half the frequency of the vertical oscillationand slightly out of phase at low walking speeds.4,9 At highwalking speeds, the sideways oscillation is in phase with thevertical oscillation. Consequently, if one observes a walkerfrom behind, the CM traces out a �-shaped Lissajous figureat low walking speeds, or a U-shaped figure at high walkingspeeds, with a vertical amplitude of about 4 cm and a hori-zontal amplitude of about 4 cm. This would also make aninteresting and amusing lecture demonstration or video pre-sentation.

V. RUNNING WAVE FORMS

Running differs from walking in that both feet are in theair for a significant part of the running cycle, there is noperiod when both feet are on the ground, and the feet spenda shorter fraction of the time on the ground, as well as ashorter time on the ground each stride. In walking, the feetare off the ground for about 40% of the time, so both feet areon the ground for a short period each cycle. In running, eachfoot is off the ground for about 70% of the time at a runningspeed of 5 ms�1, increasing to about 80% of the time at 9ms�1. In running, the peak vertical force on each foot, whileit is on the ground, is typically about 2Mg when running atlow speed, increasing to about 3Mg when sprinting at a highspeed. The average vertical ground force, for a complete run-ning cycle is Mg since the average vertical acceleration iszero.10Measurements of the stride length, L, of a runner at top

speed indicate that L�2.4 m, where L is the distance be-tween the landing point of the left foot and the subsequentlanding point of the right foot.8,13 This distance is larger thatthe distance traveled in flight, since the CM is well in frontof the back foot when the runner becomes airborne, and iswell behind the front foot when the front foot lands. Theflight distance is typically about 0.57L , as indicated sche-matically in Fig. 5.The vertical component of the ground force acting on one

foot when running at a slow pace �or jogging�, measured onthe homemade force plate, is shown in Fig. 8. As the footlands on the ground, there is an initial spike due to the heelstriking the ground, followed by a further increase in theground force owing to the rapid deceleration of the CM in

Fig. 7. The vertical component of the ground force F(L�R), obtained byadding the force due to the left foot F(L) and the force due to the right footF(R). These results were used to calculate the acceleration a �ms�2�, thevelocity v �ms�1�, and the vertical displacement z �m�.

307 307Am. J. Phys., Vol. 67, No. 4, April 1999 Rod Cross

Figure: F(R) in the right graph is the normal force on right foot, which ismeasured by force plate. [1]

−−→GRF =

−→N +

−→fs

where GRF is Ground Reaction Force.Physics model of walking - simple pendulum

• Speed limit of walking depends on the leg length Lleg .vmax =

√gLleg .

[1] Rod Cross, Standing, walking, running, and jumping on a force plate, Am. J.Phys. 67, 304 (1999).


Running

the vertical direction. The ground force then decreases rap-idly as the runner leaps off the ground. There is no interme-diate dip in the force, below Mg, as there is in a walkingstep. The CM sinks to its lowest point while one foot is onthe ground, since the knee bends significantly to absorb theinitial impact and to prepare the runner for the leap off theground. At no stage does the leg extend fully while the footis on the ground. The CM rises to its maximum height whenthe feet are off the ground, not when one foot is on theground. Consequently, the ground force wave form containsa single maximum that occurs when the vertical height of theCM is a minimum.The horizontal component of the ground force for running,

in the direction of gait, is also shown in Fig. 8. This was notmeasured but it is well documented in the literature. Theaverage horizontal force is close to zero. The total horizontalforce, including wind resistance, must average to zero whenrunning at a constant average speed. The peak horizontalforce is about six times smaller than the peak vertical force,and is zero when the vertical force component is a maxi-mum, indicating that the line of action of the resultant forcepasses near or through the CM, and that the line of action ofthe resultant force makes a maximum angle of about 25°with the vertical. The horizontal force increases approxi-mately linearly with running speed,8 and is typically about0.5Mg at a speed of 6 ms�1. The average horizontal force ispositive when accelerating from a starting position, and thisis achieved by leaning forward so that the feet spend moretime behind the CM than in front.Analytical solutions for the vertical displacement and ve-

locity of the CM can be obtained if the vertical force waveform is approximated as a series of half-sine pulses of theform F�F0 sin(�t), where F0 is the maximum force, ��/� , and � is the duration of each impact on the ground. Ifone foot first contacts the ground at t�0, and the other footfirst contacts the ground at t�T , then the acceleration of theCM in the vertical direction, for a runner of mass M, is givenby a�(F0 /M )sin(�t)�g in the interval 0�t�� and a��g in the interval ��t�T . Since the average accelerationover the period 0�t�T is zero, F0 is given by F0�(�T/2�)Mg . The ratio T/� is typically about 2 in running,so F0�3Mg . The vertical impulse, �0

�F dt�MgT de-creases as the running speed increases since T decreases,typically from T�0.37 s at 4 ms�1 to about 0.25 s at 10ms�1. Wave forms computed under this approximation areshown in Fig. 9. These wave forms are the same as those fora perfectly elastic bouncing ball of mass M that bounces atintervals of T and remains in contact with the ground for atime � each bounce. The contact time depends on the spring

constant of the ball. The effective spring constant of a runneris about 2�104 N/m, similar to that of a tennis ball, butincreases with running speed since the ground contact timedecreases as the speed increases.11An estimate of the fractional change in speed at each step

can be obtained from the results shown in Fig. 8. For ex-ample, a peak horizontal force of about 200 N is observedwhen jogging at 3 m/s. The force lasts for about 50 ms ineach direction, giving an impulse of about 5 Ns in each di-rection. Acting on a 70 kg jogger, the center of mass speeddecreases by 0.07 m/s when the front foot pushes forwardand increases by about 0.07 m/s when the back foot pushesbackward. In this way, the center of mass speed is held con-stant to within 3%, even though the feet vary in speed by�100%.

VI. CONCLUDING REMARKS

It has been shown that an inexpensive force plate can beconstructed to provide vertical component ground force datathat is similar in quality to commercial versions. Such a platecan be used in a teaching environment to demonstrate rela-tionships between force, acceleration, velocity, and displace-ment in a manner that relates directly to a student’s dailyexperiences, and in a manner that is both entertaining andinformative. The plate can also be used for quantitative ex-periments in walking, running, and jumping, as well as otheractivities such as lifting weights, climbing stairs, etc. Anexperiment that may interest students is to test whether theirhighly priced sneakers have any effect on the heel strikewave form. For research work, two force plates are required,one for each foot, and sensors to detect the horizontal com-ponents of the ground force should also be incorporated.However, many universities already contain fully equippedbiomechanics laboratories, and it is unlikely that ‘‘home-made’’ equipment would have a significant impact in thisfield. A possible improvement over commercial force plateswould be to include more than four piezos between the platesin order to raise the vibration frequency and hence increasethe useful frequency response. A small version designed tomeasure impacts of duration about 1 ms was recently de-scribed by the author.14

Fig. 8. The wave form observed when jogging. The dotted line representsthe horizontal component of the ground force, in the direction of the gait.

Fig. 9. A series of half-sine wave forms used to approximate the groundforce when running at a speed of 6 m/s, and the corresponding values of a,v , and z for a runner of mass 80 kg.


lowering of the CM, in preparation for the jump. The im-pulse is simply related to the height of the jump, which wasabout 3 cm in this case.

IV. WALKING WAVE FORMS

Walking or running is not a topic that is usually studied ina physics course, but it is ideally suited for a class of life-science or sports science students. It illustrates some inter-esting aspects of elementary mechanics in a way that shouldappeal to all physics students. For example, when walking orrunning at constant speed, no horizontal force is required, atleast in a time-average sense. Why then do we keep pushingbackward on the ground to maintain speed, and why do wenot keep accelerating with every stride? The answer is wellknown in the biomechanics field, but it is probably less wellknown by physicists. Wind resistance is not sufficient to sup-ply the retarding force required to keep the speed constant.Top sprinters push backward on the ground with a forcecomparable to Mg. If wind resistance were the only retardingforce, and if their legs could move fast enough, sprinterswould reach a terminal velocity comparable to the freefallspeed of a person jumping from a plane.The main retarding force in both walking and running

arises from the fact that the front foot pushes forwards on theground, resulting in an impulse that is equal and opposite theimpulse generated when the back foot pushes backwards.This is shown schematically in Fig. 5. The instantaneouswalking speed therefore fluctuates, but if the average speedremains constant, then the average horizontal force remainszero. The retarding force can therefore be attributed to africtional force, between the front foot and the ground, thatprevents the front foot sliding forward along the ground.6This result is surprising since it means that top sprinters, aswell as slow walkers, must spend about half their time push-

ing forward on the ground in order to slow down. It is nec-essary to slow down, by bringing the front foot to rest, sothat the back foot can catch up to and then pass in front ofthe CM. While the front foot is at rest, the back foot travelsat about double the speed of the CM. The maximum speed ofa runner is therefore limited to about half the speed at whichthe back foot can be relocated to the front.The vertical component of the force acting on one foot

when walking on the force plate is shown in Fig. 6. Alsoshown is the horizontal component in the direction of thegait. The latter component was not measured with the home-made force plate, but the wave form is well documented inthe biomechanics literature.1–4,7–13 The ratio of the verticalto the horizontal component indicates that the line of actionof the ground force acts through a point that is close to theCM at all times. In this way, the torque about the CM re-mains small and the walker or runner can maintain a goodbalance.The vertical force wave form is interesting since it has two

distinct peaks where F�Mg and a dip in the middle whereF�Mg . Both peaks are similar in amplitude when walkingat a slow or medium pace, but the second peak is smallerthan the first when walking at a fast pace. The force risesfrom zero as weight is transferred from the back to the frontfoot, and returns to zero when weight is transferred back tothe other foot at the end of the stride. The force wave formcan be interpreted in terms of the vertical motion of the CM,provided one adds the force on both feet to obtain the result-ant force, as shown in Fig. 7. It is reasonable to assume thatthe left and right wave forms are identical for most people.With the aid of two force plates, it is found that the timing ofthe ground forces on the left and right feet are such that thefirst force peak observed on each foot coincides with thetime at which the back foot lifts off the ground, and thesecond force peak coincides with the time at which the otherfoot first lands on the ground. The vertical acceleration, a, isobtained by subtracting Mg from the force wave form. Inte-gration of a yields the vertical velocity, v , and integration ofv yields the vertical displacement, z. The constants of inte-

Fig. 4. The wave form observed when stepping onto and then jumping offthe force plate.

Fig. 5. A schematic diagram showing the ground forces when running, andthe vertical displacement of the CM �dashed line�. Also shown are typicaldistances and times when running at 10 ms�1.

Fig. 6. The wave forms observed when walking �a� at a slow pace and �b� afast pace. The dotted line represents the horizontal component of the groundforce, in the direction of the gait.


Figure: The left figure shows the normal force when the running speed is 6m/s. [1]

• Stride length (m)

• Stride frequency (Hz)

• Duty factor


Duty factor

Duty factor ”D” is the fraction of the stride period at which one foot isin contact with the ground.

• D ≈ 0.3 as the speed of 5m/s; D ≈ 0.2 as the speed of 9m/s. [1]

•D = 0.394Fr−0.174 (1)

where Froude number, Fr = v2/(gLleg ). [2]

[2] Alexander, R. McN. and Jayes, A. S. (1983). A dynamic similarity hypothesis forthe gaits of quadrupedal mammals. J. Zool., Lond. 201, 135152.


Study design

What is the minimum first stride length x1 as athletes slow down fromtheir highest speeds in court?

• The contact time calculated from the stride before slowing down =The contact time calculated from the stride after slowing down

l

v0D0 =

x1v1

D1 (2)

where ”l” is the stride length before slowing down.v0, v1 and D0, D1 are the speeds before and after slowing downand their related duty factors.

• ”l” is measured as participants run with a regular pace in the court.

• v0 is measured as the highest speed in the court.

• v1?


Study design

Momentum - Impulse Theorem

fs∆t = m∆v (3)

vi+1 = vi −fs,im

∆t, Limitation : measurement of fs,i . (4)

vi+1 = vi −µsN

m∆t, Estimation : fs,i = fmax ,s,i = µsN. (5)

• Normal force N is measured in high jump by 2-axis force plate.

• Approximate static friction coefficient between basketball shoesand wooden floor is 1.0 ∼ 1.2 [3, 4].

[3] John McLester, Peter St. Pierre, Applied Biomechanics: Concepts andConnections.[4] Benno Maurus Nigg, Brian R. MacIntosh, Joachim Mester, Biomechanics andBiology of Movement.


Measurement

• 11 basketball players participated in the investigation in fall 2016.

Among the 11 participants, one of them played college basketball,seven of them played varsity high school basketball for an averageof 3.3 years, and four of them played basketball only recreationally.

• Data collection

Prior to the game: athlete’s height, weight, stride length, groundreaction force in high jump, reaction time.

Throughout the game: athlete’s speed, number of occasionsparticipants went out of bounds, and the distance they travelled.


Force plate

[1]

[1] J. M. Donelan, R. Kram, The Journal of Experimental Biology 203, 2405-2415(2000)


Results and conclusions

Data from direct measurement


Results and conclusionsData from modelingBuffer zone space xtotal were calculated.


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bu er zones in basketball - university of new...

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