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A Boxer’s Punch

A Senior Project presented to the Faculty of the Physics Department
California Polytechnic State University, San Luis Obispo

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In Partial Fulfillment of the Requirements for the Degree
Bachelor of Science, Physics

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by

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Jacob A. Ekegren
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April 2014
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Advised by Dr. Matthew Moelter
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© 2014 Jacob A. Ekegren 


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Table of Contents

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Introduction and Setup

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Theory Applied

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Experimental Procedure

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Results and Analysis

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Conclusion!

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References

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Introduction and Setup
For over a year now, I have been interested in the sport of boxing. This fascination led me to explore what occurs to a human head upon impact from a boxer’s punch. It is known that a knockout occurs when blood circulation to the brain is compressed. This compression results from the sudden acceleration and deceleration of the head[1]. Therefore, the primary focus of this experiment explores the relative effort necessary to cause significant movement to a head about a neck. Figure 1 - Picture of modeled head and spine secured to a table

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To achieve this, a simplistic mechanical model of a human head, a socket, and a spine was built. A volleyball was used to simulate a head. A garage door spring with a diameter of
4.0 ± .05cm was used as a spine. Lastly, a small wooden block acted as a socket between the head and the spine. A small support board held the spring in place for the experiment and allowed it to be moved up or down to simulate different neck lengths. Wearing a standard 14oz boxing glove, I punched the model and recorded the subsequent motion utilizing a Fujiyama high speed camera at 240 frames per second (fps). This function of the camera allowed for a detailed analysis of the motion undergone by the model and the forces involved. Striking the head applied a torque on the spring about the socket. Different neck lengths were used to compare the various strengths of these forces applied and the motion resulting from them.

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Theory Applied

First, it is important to know Hooke’s Law[2]. Where τ is the torque applied, Κ is the spring constant, and θ is the angle of deflection of the spring.
Eq. 1 τ = -Κθ
Κ was determined by utilizing a standard scale. The above apparatus, without the head, was placed on the side of a table parallel to the ground. Standing on the scale, subtracting out my weight, and then pushing up on the spring, the force, in pounds, that the spring pushed back with was measured. This relationship comes from Newton’s third law[2], shown in Eq. 2.
Eq. 2 Fa = -Fb
Also, Newton’s second law
Eq. 3 F = ma and, Eq. 4 τ = F x r = Frsinφ = Fr⟂ where r, the moment arm, is perpendicular to the direction of the restoring force of the spring.
Thus,
Eq. 5 Fr⟂ = -Κθ
After measuring the restoring force in pounds, converting to newtons, and then calculating the angle in radians that the spring was bent, a simple plot was created with a fit linear curve, shown in figure 3.

Figure 2 - Linear fit to solve for K

Since we expect the relationship in Eq. 5 to be linear, in the form y = mx + b, m = K/r⟂.
Solving for K shows the value to be 14.01 ± .005 kg·m2/s2. This value for K is with respect to

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the spring at 11.1 ± 0.05 cm. It is reasonable to assume a inversely proportional relationship of K with respect to the length of the spring, l, such that,
Eq. 6 K(l) = K11.1·l0/l where l0 = .111 m and K11.1 is the calculated value at a 11.1 cm. The following table shows the calculated value for K ± .05 kg·m2/s2 for each respective neck length used for the remainder of the experiment as in accordance with Eq. 6.

Table 1 - Solve for K(l)

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Experimental Procedure

Returning to the original configuration, shown in fig. 1, the experiment began by recording the head being punched. I was the subject wearing the boxing glove. The camera was placed on a tripod at an acceptable distance away as to show complete movement of the head and maintain the references in the frame. As in Table 1, various neck lengths were used and three punches at each were recorded.
Because of the cameras high frame rate, 240 fps, each video was between 60 and 105 s in length when in actuality the punch and subsequent movement lasted only about a second. To compensate for this, each video was edited using QuickTime[3]. The nonessential portions of the video before the impact and after the motion ended were cut out. To keep from damaging the equipment, all punches are a 20% - 30% estimate of my overall strength.
Finally, one more punch at approximately 50% - 70% of my overall strength was recorded. For uniformity, any video where the head rotated too drastically on the spring was discarded.
This is due to the fact that the reference dot on the ball would not begin and end at close to the same starting point.

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Results and Analysis
Pro[4],

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Using Logger an X-Y coordinate system and distance reference was setup within the frame of the camera. A virtual reference point, in the form of a blue dot, was placed on the ball’s reference point to allow tracking it throughout the video. Going frame by frame, as the ball’s dot moved, the virtual dot was manually moved and placed over it. Figure 3 shows a still frame of this process with all of the virtual dots after the ball was struck and had returned to rest.

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Figure 3 - 6.0 cm neck tracked after having been struck and returned to rest. The blue dots represent the motion of the red reference dot in figure 1 from the beginning of the video until the motion of the head stopped.

Using the distance reference and the recorded length of time, Logger Pro is able to calculate X(t) and Y(t). Taking these calculated numbers, an X-Y plot was created. If the formulas and following calculations were correct, then the plot should resemble figure 3.

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Plot 1 - An X-Y plot supporting that Logger Pro correctly tracks the movement of the ball after impact.

By utilizing the internal derivative function of Logger Pro, Vx(t), Vy(t), Ax(t), and Ay(t) we subsequently calculated. Taking the largest tangential acceleration, Ax(t), and using Newton’s second law, Eq. 3, the force experienced by the ball at the moment was calculated. Because of
Newton’s third law, Eq. 2, this is also the force from the punch. The mass of the ball and socket was measured to be 1.04 ± .005 kg. The average force calculated for all of the recorded punches except for the last punch at 50% - 70% power on the ball was 334.305 ± .005 N. To put this into
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a more useful unit, this is 75.738 ± .001 lb. The punch at the estimated 50% - 70% power was calculated to be 94.172 ± .001 lb.
Given that the neck length is constant for the duration of a single impact, it makes more sense to plot θ(t) instead of X(t) and Y(t). Using trigonometry, θ, denoted as increasing from 0 clockwise from the Y-axis, was calculated using Eq. 7.
Eq. 7 θ = tan-1(X/Y)
Plotting θ(t) shows the oscillatory motion of the spring. Then returning to Eq. 1, the Torque applied on the spring can be calculated and plotted as well.

Plot 2 - Theta (θ) in radians vs Time (s) in seconds - 6.0cm neck

Plot 3 - Torque (τ) in N*m vs Time (s) in seconds - 6.0 cm neck

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As expected, the relative motion witnessed and forces applied by the spring were oscillatory. Plot 2 shows the deflection angle (θ) with respect to the y-axis as seen in figure 3. Because the spring applies a restoring force in the form of a torque (τ), damped oscillatory motion was expected and is consistent with Plot 2. Utilizing Eq. 1, the torque (τ) applied by the spring to return it to equilibrium is shown in Plot 3. The period of oscillation was between 0.16 and 0.20
± 0.005 s. Table 2, below, shows the recorded periods for each punch. The damped oscillations returned to rest approximately one second after the initial impact. Though the rest of the motion goes as expected, it is not reasonable to assume that a person’s head will oscillate like a spring.
This is not because the spine is not acting like an organic spring, but rather due to the conscious human response. A person may willfully evoke muscular responses after an impact. As such, from experience, a punch on a person is likely to resemble the first oscillation but will deviate afterwards. The Moment of Inertia (I) is the measurement of resistance of an object around a moment arm. This can be calculated in two ways. Either directly, or through the measurement of the period of oscillation. For a torsion pendulum the period is[2],
Eq. 8 T = 2π√(I/K)
Solving Eq. 6 for I gives,
Eq. 9 I = (T/2π)2K
Table 2 also shows the calculated Moments of Inertia (I) using Eq. 9. Though averaging the
Moments of Inertia for each neck length might seem reasonable, because each punch was not exactly the same and the range of power is only an estimate of my overall strength, each calculation was kept separate.

Table 2 - Moments of Inertia (I) calculated from the period of oscillation θ(t) vs time.
The uncertainty for I is ± 0.0003 kg·m2.

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From the above assertion, the human response will alter the oscillatory motion of the head. Therefore, the most important and real world applicable piece of information would be the impulse applied at the moment of impact. Impulse is defined by a change in momentum[2],
Eq. 10 J = Δp = ∫F dt = Faverage · Δt
The relationship from Eq.10 remains the same for angular momentum, shown in Eq. 11.
Eq. 11 Jangular = ΔL = r x Δp = ∫r x F dt = ∫τ dt = τaverage · Δt
Plot 4 shows the relative change in angular momentum from the moment of impact until just as the glove lost contact with the modeled head on the 6.0 cm neck.

Plot 4 - Calculated change in angular momentum for 6.0 cm neck. From the moment of impact until the glove lost contact with the modeled head.

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Table 3 shows the calculated ΔL for each video captured at their respective neck lengths.

Table 3 - Calculated ΔL

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Again, because each punch was not exactly the same, rather than average the change, they were left separate. The variations observed when the neck length was the same are most likely attributed to the variations in strength utilized at the moment of impact. Though the numbers seem small, to put them into a better perspective, it is important to realize that they are over an extremely small period of time and that the actual movement of the head is only a few centimeters.

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Conclusion

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Since I was punching at an estimated 20% - 30% of my overall estimated strength, it is clear from the previous table that little effort is needed to cause significant instantaneous changes in momentum to a human head. Clearly, further study can be done on this subject.
A knockout typically results from the brain oscillating inside of the skull after impact from a punch. Because all momentum is conserved, the relative motion observed from this simplistic model must translate to similar motion in a model of a head with a brain. The brain would oscillate inside of the head merely damped by the fluids inside. Further analysis can reveal the impulses being experience by the brain during such motion.

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References

1. Enamait, Ross. “The Chin, The Beard, The Knockout.” published, 2003 rossboxing.com
© 2006.

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2. Nave, Carl R. “http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html” partment of Physics and Astronomy, Georgia State University. © C.R. Nave, 2012.

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3.
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De-

QuickTime Player, a popular video playing software. Version 10.3 © 2009-2013, Apple Inc.

4. Logger Pro, a premier data-collection software. © 1996-2014 Vernier Software & Technology.

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