G‐Force Tolerance Sample Solution Introduction When different forces are applied to an object, G-Force is a term used to describe the resulting acceleration, and is in relation to acceleration due to gravity(g). Human tolerances depend on the magnitude of the g-force, the length of time it is applied, the direction it acts, the location of application, and the posture of the body. The human body is flexible and deformable, particularly the softer tissues. A hard slap on the face may briefly impose hundreds of g locally but not produce any real damage; a constant 16 g for a minute, however, may be deadly. This portfolio will examine the tolerance humans have to both horizontal and vertical G-force. I will create a function to model the behavior of the data and discuss the apparent implications on time and G-force. Data and Graph The following table and graph 1 illustrate the tolerance of human being to horizontal G-force. Graph 1: Tolerated G-force(+Gx) vs Time +Gx (g) Time (min) +Gx (g) 0.01 35 0.03 28 30 0.1 20 25 0.3 15 20 1 11 15 3 9 10 10 6 5 30 4.5 35 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 T(time in minutes) It is obvious that T > 0 and that +Gx > 0. T, time in minutes, is the independent variable and +G(x), measured in g, is the dependent variable. There is a definite inverse relationship between time and G-Force, +Gx. The more time a human is exposed to g forces, the smaller the amount of gees they are capable of sustaining. This would imply that a rational, power, or exponential function would best fit the data. In looking at the graph, there appears to be both a horizontal and vertical asymptote, meaning that there is a certain amount of g’s that a human can tolerate indefinitely (horizontal asymptote) and an unlimited amount of g’s that a human can withstand at an infinitesimal small amount of time. Let’s look at both a rational function and a power function. Rational Function A rational function has the basic equation, y = a + c , with parameters a, b, and c. The parameter c represents the x−b horizontal asymptote. In looking at the graph it appears that a horizontal asymptote exists at y = 4, or somewhere less than 4. This would imply that humans can withstand 4 +G(x) indefinitely. There is no data on the maximum amount of indefinitely sustained g forces, but the force of gravity when you are still (for example, when you sit, stand or lie down) is considered 1 G. (reference : http://www.reference.com/browse/g+force) I will then let c = 1. The parameter b represents the vertical asymptote, which appears to be 0. This would imply that there is no upper limit to the G-force a human can tolerate given an extremely short amount of time. My equation then becomes y = The parameter a will be a positive value. Increasing a will move the graph further away from the origin. I algebraically will solve for a using data points. Example 1: using data point (.01, 35) 35 = Graph 2: The effect of changing parameter b +Gx (g) 35 f(x)=1/x+1 f(x)=.34/x+1 f(x)=4.2/x+1 f(x)=105/x+1 f(x)=10/x+1 f(x)=24/x+1 data points 30 25 a .34 + 1 , a = .34 ⇒ y = +1 .01 x 20 15 Example 2: using data point (3.9) 9= a +1. x a 24 + 1 , a = 24 ⇒ y = +1 3 x 10 5 2 Graph 2 shows the effects of using different data points to solve for a. Notice some values of a bring the graph below the data points and other values bring the graph above. 4 6 8 10 12 14 16 18 20 22 24 26 28 30 T(time in minutes) .34 + .81 + 1.9 + 4.2 + 10 + 24 + 50 + 105 = 24 .53 8 I averaged all of the a values from the 8 data points: And used 24.53 as my value of a. This gives the equation, + Gx = 24.5 + 1 to use to model the data. T Graph 3: Model created by hand +Gx (g) In looking at graph 3, the overall shape is good, but the graph does not seem to come close to the y-axis soon enough. The horizontal axis could be changed to x = -1 or x = -.5, but this does not make sense in context because T > 0. 35 f(x)=(24.53/x)+1 data points 30 25 20 15 10 5 2 I then considered a power function. 4 6 8 10 12 14 16 18 20 22 24 26 28 30 T(time in minutes) Power Function A power function has the basic equation, y = ax b , where a and b are the parameters. As stated before, a horizontal asymptote at y = 1 (indefinitely sustaining 1 +G(x)) would be beneficial and change my equation to y = ax b + 1 . Graph 4: Effects of varying the exponent in power functions The parameter b greatly effects the graph. In looking at graph 4, a power function takes on different shapes depending if b is positive, negative, integer, or decimal. A negative value for b will give an inverse relationship between the variables. y 9 f(x)=x f(x)=x^2 f(x)=x^5 f(x)=x^(-1) f(x)=x^2.3 f(x)=x^-2.3 f(x)=x^(-5.8) 8 7 6 5 The parameter a will be a positive value. Increasing a will move the graph further away from the origin. I will algebraically solve for a and b by choosing two points. 4 3 Example 1: using points (1, 11) and (.1, 20) 2 1 x 1 2 3 4 5 6 7 8 9 + Gx = aT b + 1 20 = a (.1) b + 1 11 = a(1) b + 1 10 = a substitute ⇒ 19 = 10(.1) b 1.9 = .1b ln(1.9) = b ln(.1) b= ln(1.9 ) ln(.1) = −.278 + Gx = 10T −.278 + 1 Graph 5: Power equations Example 2: using points (.01, 35) and (30, 4.5) +Gx (g) 35 30 + Gx = aT b + 1 4.5 = a (30) b + 1 35 = a(.01) b + 1 f(x)=10x^(-.278)+1 f(x)=8.89x^(-.2914)+1 data points 25 3.5 = 34 = a (.01) b 34 (.01) b 20 = a, substitute ⇒ .097 = b= 15 34 (.01) b 30b .01b (30) b .01b ln(.097 ) ln( 30 ) − ln(.01) = −.2914 + Gx = 8.89T −.2914 + 1 10 5 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 T(time in minutes) These functions can be seen in graph 5 to the left. Both fit very well and are a better match to the data than the rational equation. The inverse relationship between time and +Gx is clear and the equations hit almost every data point. I will choose + Gx = 10T −.278 + 1 to model the data for its simplicity. Technology Another equation to fit this data can be found using technology. A power model, + Gx = 11.2555 x −.24988 proved to be the best fit with a coefficient of determination, r2 = .9978. The two equations look almost identical, both proving to be an excellent model for the data. Graph 6: By hand vs Technology +Gx (g) 35 30 by hand: 10x^(-.278)+1 technology:11.255X^-.24988 data points 25 20 15 10 5 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 T(time in minutes) Vertical G‐forces The human body is considerably less adept to surviving vertical g-forces. Aircraft, in particular, exert g-force along the axis aligned with the spine. This causes significant variation in blood pressure along the length of the subject's body, which limits the maximum g-forces that can be tolerated. (http://www.reference.com/browse/g+force) Graph 7 shows vertical Gforce, +Gz and our original power model. The shape of the graph and the data are similar, but clearly changes need to be made. The fact that humans can tolerate vertical forces less can be seen by the data points falling below most of the graph. The parameter a, needs to be smaller to account for this. An equation such as, + Gz = 7.082T −.199 has a high coefficient, r2 = .9979 of determination and the parameter a has changed from10 to 7.08 Graph 7: Vertical G force and the original model +Gz (g) 35 30 original model: 10x^(-.278)+1 new model:7.08X^-.199 data points 25 20 Time (min) +Gz (g) 0.01 0.03 0.1 0.3 1 3 10 30 35 28 20 15 11 9 6 4.5 15 10 5 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 T(time in minutes) Notice how the new equation is stretched back towards the axis and the data points. The new equation still incorporates the idea of sustaining 1 G from gravity. Limitations Human tolerance of g-force in the horizontal direction can be modeled by the equation + Gz = 10T −.278 and similarly −.199 vertical g-force can be modeled by + Gz = 7.082T . Both power models can interpolate values within the domain .01 ≤ T ≤ 30 quite accurately. But how well do the models extrapolate information? Both models suggest that humans can tolerate 1g indefinitely. Is this realistic or can humans tolerate a higher amount of g? The more serious implication concerns the vertical asymptote at x = 0, suggesting humans can tolerate any extremely high amount of gees, but I don’t believe this is true. Humans can survive up to about 20 to 35 g instantaneously (for a very short period of time). Any exposure to around 100 g or more, even if momentary, is likely to be lethal, although the record is 179.8 g . (http://www.reference.com/browse/g+force) Another limitation is where the data originated – not knowing who this data is about, or how the +Gx were calculated. Problems could arise in using the model to predict if the subject or circumstances are different. To some degree, G tolerance can be trainable, and there is also considerable variation in innate ability between individuals. Overall a power model fits the data well and would be an excellent source for interpolation.
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