Question

Skydiving If a body of mass \(m\) falling from rest under the action of gravity encounters an air resistance proportional to the square of the velocity, then the body's velocity \(v(t)\) is modeled by the initial value problem $$\begin{array}{ll}{\text { Differential equation: }} & {m \frac{d v}{d t}=m g-k v^{2}} \\ {\text { Initial condition: }} & {v(0)=0}\end{array} $$ where \(t\) represents time in seconds, \(g\) is the acceleration due to gravity, and \(k\) is a constant that depends on the body's aerodynamic properties and the density of the air. (We assume that the fall is short enough so that variation in the air's density will not affect the outcome.) (a) Show that the function $$v(t)=\sqrt{\frac{m g}{k}} \frac{e^{a t}-e^{-a t}}{e^{a t}+e^{-a t}}$$ where \(a=\sqrt{g k / m},\) is a solution of the initial value problem. (b) Find the body's limiting velocity, \(\lim _{t \rightarrow \infty} v(t)\) (c) For a 160 -lb skydiver \((m g=160),\) and with time in seconds and distance in feet, a typical value for \(k\) is \(0.005 .\) What is the diver's limiting velocity in feet per second? in miles per hour?

Short answer

The function \(v(t)=\sqrt{\frac{m g}{k}} \frac{e^{a t}-e^{-a t}}{e^{a t}+e^{-a t}}\), where \(a=\sqrt{g k / m}\), is indeed a valid solution to the differential equation with the given condition. When \(t \rightarrow \infty\), the body's limiting velocity will become \(\sqrt{\frac{m g}{k}}.\) For a skydiver with \(m g=160\) and \(k=0.005\), the limiting velocity will be approximately 356.93 feet/second or about 243.47 mph.

Step-by-step solution

Prove the function is a valid solution

Start by substituting the function, \(v(t)=\sqrt{\frac{m g}{k}} \frac{e^{a t}-e^{-a t}}{e^{a t}+e^{-a t}}\), where \(a=\sqrt{g k / m}\), into the differential equation and see if it makes the equation to be true.

Calculate the limit

To find the limiting velocity as \(t \rightarrow \infty\), calculate \(\lim _{t \rightarrow \infty} v(t)\). Because the hyperbolic functions are defined in terms of exponentials, we have to use l'Hopital's rule to calculate this limit.

Utilize real-life constants

Use given constants, \(m g=160\) and \(k=0.005\), to calculate the diver's limiting velocity in feet per second. To convert this speed to miles per hour, use the conversion factor \(1 \text{ mph} = 1.467 \text{ feet/second}\).

Key concepts

Differential Equation

A differential equation is a mathematical equation that relates a function with its derivatives. It is a powerful tool for modeling how systems change over time or in response to varying conditions. In the context of our skydiving problem, the differential equation given models the dynamic balance between the gravitational force pulling the skydiver downwards and the air resistance that opposes the motion.

The equation is represented as \( m\frac{dv}{dt} = mg - kv^2 \), where \(m\) is the mass of the skydiver, \(g\) the acceleration due to gravity, \(v(t)\) the velocity at time \(t\), and \(k\) a constant that relates to air resistance. The left side of the equation represents the change in velocity over time, also known as acceleration, and the right side represents the net force acting on the skydiver after considering air resistance. By solving this equation, we can predict the velocity of the skydiver at any given point in time.

Limiting Velocity

Limiting velocity, or terminal velocity in the context of freefall, is the maximum speed an object reaches when the force due to air resistance is equal in magnitude and opposite in direction to the gravitational force, resulting in zero net acceleration. When an object reaches its limiting velocity, it continues to fall at a constant speed.

In mathematical terms, the limiting velocity is found by setting the acceleration \( \frac{dv}{dt} \) to zero in the differential equation, leading to velocity being constant so that \( mg - kv^2 = 0 \). Solving for \(v\) gives us the formula for the limiting velocity, which depends on the mass of the object, the gravitational constant, and the proportionality constant of air resistance.

Air Resistance

Air resistance, also known as drag, is the force that opposes an object's motion through the air. In our problem, air resistance is proportional to the square of the velocity, which is a common modeling assumption for objects moving at high speeds through a fluid medium such as air. The constant \(k\) encapsulates factors such as the shape, size, and surface roughness of the falling object as well as the density of the air.

To accurately model the effects of air resistance on the skydiver's velocity, we must consider how the rapidly changing velocity affects the resistance encountered. In such a model, air resistance increases with the square of the velocity, which means that as the skydiver's speed increases, the resistance grows exponentially, eventually balancing out the gravitational force and leading to a state of constant velocity or the limiting velocity.

Hyperbolic Functions

Hyperbolic functions, such as \( \sinh(x) \) and \( \cosh(x) \), are analogs to the trigonometric functions but for a hyperbolic geometry context. They are defined using exponential functions. For example, \( \sinh(x) = \frac{e^x - e^{-x}}{2} \) and \( \cosh(x) = \frac{e^x + e^{-x}}{2} \). These functions frequently appear in solutions to differential equations that include terms involving the square of the unknown function, as is the case with our skydiving velocity model.

The solution provided for the skydiver's velocity is expressed in terms of these hyperbolic functions, indicating a strong relationship between the behavior of the skydiving problem and the properties of hyperbolic functions. Understanding these functions and their properties helps in solving and interpreting the behavior of the system being modeled.

L'Hopital's Rule

L'Hopital's rule is a technique used in calculus to evaluate limits of indeterminate forms such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). When calculating the skydiver’s velocity over time as it approaches infinity, we end up with an indeterminate form which can be resolved using l'Hopital's rule.

By applying this rule to the hyperbolic functions in our model, we can find the limit of the velocity function as time approaches infinity. This process involves differentiating the numerator and denominator of the fraction independently and then taking the limit. The resulting value is the limiting velocity, a crucial concept in understanding the maximum speed the skydiver can attain during freefall.

Velocity Modeling

Velocity modeling in physics is the process of creating mathematical formulas to describe how velocity depends on time and other physical parameters. It often involves differential equations to incorporate the forces at play, such as gravity and air resistance in this case of skydiving.

The model provided for the skydiving velocity problem is a perfect example of utilizing known physical laws and mathematical tools to predict the behavior of a falling object under certain conditions. By inputting specific values for mass, gravitational acceleration, and air resistance coefficients, you can use the model to estimate how quickly the skydiver will fall at any point during the dive, ultimately calculating the limiting velocity and ensuring the skydiver's safety.