Question

Flight formula for Indian spotted owlets The following table shows the average body mass \(m(t)\) (in \(g\) ) and average wing chord length \(w(t)(\text {in } m m),\) along with the derivatives \(m^{\prime}(t)\) and \(w^{\prime}(t),\) of \(t-\)week-old Indian spotted owlets. The flight formula function \(f(t)=w(t) / m(t)\) which is the ratio of wing chord length to mass, is used to predict when these fledglings are old enough to fly. The values of \(f\) are less than \(1,\) but approach 1 as t increases. When \(f\) is close to \(1,\) the fledglings are capable of flying, which is important for determining when rescued fledglings can be released back into the wild. (Source: ZooKeys, 132,2011 ) $$\begin{array}{lrrrr} t & m(t) & m^{\prime}(t) & w(t) & w^{\prime}(t) \\ \hline 1 & 23.32 & 41.45 & 10.14 & 14.5 \\ 1.5 & 50.59 & 64.94 & 20.13 & 26.17 \\ 2 & 82.83 & 57.95 & 36.7 & 39.86 \\ 2.5 & 105.13 & 31.08 & 58.92 & 47.11 \\ 3 & 115.48 & 12.48 & 81.55 & 41.38 \\ 3.5 & 119.4 & 4.44 & 98.99 & 27.94 \\ 4 & 120.76 & 1.51 & 109.75 & 15.74 \\ 4.5 & 121.22 & 0.51 & 115.5 & 7.99 \\ 5 & 121.37 & 0.17 & 118.34 & 3.85 \\ 5.5 & 121.42 & 0.06 & 119.69 & 1.8 \\ 6 & 121.44 & 0.02 & 120.32 & 0.84 \\ 6.5 & 121.45 & 0.01 & 120.61 & 0.38 \end{array}$$ State the units associated with the following derivatives and state the physical meaning of each derivative. a. \(m^{\prime}(t)\) b. \(w^{\prime}(t)\) c. \(f^{\prime}(t)\)

Short answer

Answer: The units and physical meanings are as follows: a. \(m^{\prime}(t)\): The units are grams per week \((\frac{g}{week})\), and the physical meaning is the growth rate of the owlets' body mass with respect to their age. b. \(w^{\prime}(t)\): The units are millimeters per week \((\frac{mm}{week})\), and the physical meaning is the growth rate of the owlets' wing chord length with respect to their age. c. \(f^{\prime}(t)\): The units are \(\frac{mm}{week^2}\), and the physical meaning is the rate of change of the flight formula function, indicating how fast the owlets are approaching their flying readiness.

Step-by-step solution

a. Units and Physical Meaning of \(m^{\prime}(t)\)

The given mass function, \(m(t)\), has the unit of grams (g). The derivative \(m^{\prime}(t)\) represents the rate of change of mass with respect to the age of the owlets in weeks. Therefore, the units for \(m^{\prime}(t)\) will be grams per week \((\frac{g}{week})\). The physical meaning of \(m^{\prime}(t)\) is the growth rate of the Indian spotted owlets' mass with respect to their age.

b. Units and Physical Meaning of \(w^{\prime}(t)\)

The given wing chord length function, \(w(t)\), has the unit of millimeters (mm). The derivative \(w^{\prime}(t)\) represents the rate of change of wing chord length with respect to the age of the owlets in weeks. Thus, the units for \(w^{\prime}(t)\) will be millimeters per week \((\frac{mm}{week})\). The physical meaning of \(w^{\prime}(t)\) is the growth rate of the Indian spotted owlets' wing chord length with respect to their age.

c. Units and Physical Meaning of \(f^{\prime}(t)\)

The flight formula function \(f(t) = \frac{w(t)}{m(t)}\) is unitless because it's the ratio of wing chord length (in millimeters) to mass (in grams). To find the units of \(f^{\prime}(t)\), we differentiate \(f(t)\) with respect to \(t\) and simplify as follows: $$ f'(t) = \frac{d}{dt} \left( \frac{w(t)}{m(t)} \right) = \frac{w^{\prime}(t)m(t) - w(t)m^{\prime}(t)}{(m(t))^2} $$ Since the numerator contains a product of wing length and mass derivatives, and both of these have different units, we cannot simplify the units of \(f^{\prime}(t)\). The units of \(f^{\prime}(t)\) are \(\frac{mm \cdot g}{g \cdot week^2}\) or \(\frac{mm}{week^2}\). The physical meaning of \(f^{\prime}(t)\) is the rate of change of the flight formula function, which helps determine how fast the owlets are approaching their flying readiness with respect to their age.

Key concepts

Physical Meaning of Derivatives

When we discuss the physical meaning of derivatives in the context of biology or any science, we are often referring to the rate at which some quantity changes over time. A derivative provides us with the instantaneous rate of change of one variable with respect to another. In the example of the Indian spotted owlets, the derivatives represent growth rates.

For instance, the derivative of the body mass function, denoted by \( m'(t) \), tells us how quickly the mass of the owlets is increasing per week. Likewise, the derivative of the wing chord length function, \( w'(t) \), indicates how rapidly the wing length grows over time. Understanding these rates of change is crucial as they can be indicative of health, development stages, and readiness for milestones such as the fledglings' ability to fly. The derivative is a fundamental concept in calculus that reveals dynamic processes in a diverse array of scientific fields.

Rate of Change

The rate of change is a concept that quantifies how a variable alters over a period. In biology, this can relate to growth rates, such as the change in mass or size of an organism as it develops. The rate of change is particularly vital in the study of living organisms, where growth patterns can signal normal or abnormal development.

In the exercise with the Indian spotted owlets, the rate of change of both the body mass \( m'(t) \) and wing chord length \( w'(t) \) is tracked over weeks. The former is measured in grams per week, while the latter is measured in millimeters per week. These rates provide biologists with information to predict milestones like the ability to fly, based on the changes in the birds' physical characteristics over time. As fledglings grow, monitoring the rates of change allows informed decisions regarding animal care and the release of rehabilitated birds back into the wild.

Flight Formula Function

The flight formula function, \( f(t) = \frac{w(t)}{m(t)} \), is a significant part of the study, showcasing the ratio of wing chord length to mass. This ratio is a critical determinant for assessing the flight readiness of birds. In the case of the Indian spotted owlets, the flight formula function increases over time and approaches the value of 1 as the fledglings get older.

When the function's value is close to 1, it implies a physiological state where the birds' wings are likely strong and long enough relative to their body mass to support flight. The derivative of this function, \( f'(t) \), sheds light on how quickly the ratio of wing length to mass is changing. As the flight formula's derivative tends towards zero, it indicates the birds are nearing flight capability. This information is instrumental for wildlife conservationists to time the release of birds that were in care back into their natural habitat.