Question

In this section, several models are presented and the solution of the associated differential equation is given. Later in the chapter, we present methods for solving these differential equations. One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation \(v^{\prime}(t)=g-b v,\) where \(v(t)\) is the velocity of the object for \(t \geq 0, g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity, and \(b>0\) is a constant that involves the mass of the object and the air resistance. a. Verify by substitution that a solution of the equation, subject to the initial condition \(v(0)=0,\) is \(v(t)=\frac{g}{b}\left(1-e^{-b t}\right)\) b. Graph the solution with \(b=0.1 \mathrm{s}^{-1}\) c. Using the graph in part (b), estimate the terminal velocity \(\lim _{t \rightarrow \infty} v(t)\)

Short answer

Question: Verify the given solution for the differential equation \(v'(t)=g-bv\) where \(v(t)=\frac{g}{b}(1-e^{-bt})\), graph the given solution with \(b=0.1s^{-1}\), and estimate the terminal velocity. Answer: The given solution is verified as correct. When graphing the solution with \(b=0.1s^{-1}\), the function is \(v(t) = 98(1 - e^{-0.1t})\). The estimated terminal velocity is 98 m/s.

Step-by-step solution

Verify the given solution

To verify the given solution \(v(t)=\frac{g}{b}(1-e^{-bt})\), we need to find its derivative and substitute it into the given equation \(v'(t)=g-bv\). First, find the derivative of the given solution: \(v(t) =\frac{g}{b}(1-e^{-bt})\) \(v'(t) = -\frac{g}{b}\cdot{-be^{-bt}} = ge^{-bt}\) Now, substitute \(v(t)\) and \(v'(t)\) in the given equation: \(v'(t) = g - bv(t)\) \(ge^{-bt} = g - b\left(\frac{g}{b}(1-e^{-bt})\right)\) Simplify the equation and check if it holds true: \(ge^{-bt} = g - g(1-e^{-bt})\) \(ge^{-bt} = g(1 - (1 - e^{-bt}))\) \(ge^{-bt} = ge^{-bt}\) Since both sides of the equation are equal, the given solution is verified.

Graph the solution with b=0.1s^-1

To graph the solution \(v(t) = \frac{g}{b}(1 - e^{-bt})\) with \(b = 0.1 \mathrm{s}^{-1}\), first substitute the values of \(g\) and \(b\): \(v(t) = \frac{9.8}{0.1}(1 - e^{-0.1t}) = 98(1 - e^{-0.1t})\) Now, create a graph with the horizontal axis representing time (t) and the vertical axis representing velocity (v) and plot the function \(v(t) = 98(1 - e^{-0.1t})\).

Estimate the terminal velocity

To estimate the terminal velocity, we need to find the limit of the function \(v(t)\) as t approaches infinity: \(\lim_{t \rightarrow \infty} v(t)\). Notice that the exponential term, \(e^{-0.1t}\), approaches 0 as \(t \rightarrow \infty\). Therefore: \(\lim_{t \rightarrow \infty} v(t) = \lim_{t \rightarrow \infty} \frac{g}{b}(1 - e^{-bt}) = \lim_{t \rightarrow \infty} 98(1 - e^{-0.1t}) = 98(1 - 0) = 98\) So, the estimated terminal velocity is 98 m/s. In summary, we have verified the given solution of the differential equation, graphed it with the specified value of \(b\), and estimated the terminal velocity to be 98 m/s.

Key concepts

Free Fall with Air Resistance

Understanding the dynamics of free fall with air resistance is a crucial concept in physics and engineering, particularly when studying the motion of objects through a fluid, like air. The force of air resistance opposes the gravity-driven downward motion of free fall and depends on factors including the object's shape, density, and velocity.

The differential equation used to model this scenario is given by
\[v^{\text{{prime}}}(t)=g-bv,\]where \(v(t)\) is the object's velocity at time \(t\), \(g\) is the acceleration due to gravity (9.8 m/s²), and \(b\) is a constant that takes into account both the mass of the object and the air resistance it encounters.

When an object is dropped, it starts accelerating due to gravity. However, as its speed increases, the air resistance grows until it eventually balances the gravitational force. At this point, the object ceases to accelerate and continues to fall at a constant terminal velocity. In our exercise, the process of verifying the solution to this equation involved showing that the proposed function for velocity satisfies the equation under the initial condition of starting from rest (\(v(0)=0\)).

By using calculus, particularly differentiation, one can both derive the equation modeling the fall and solve for the object's velocity as a function of time, incorporating the effects of air resistance and initial conditions.

Initial Value Problem

An essential aspect of solving differential equations, such as the equation of motion for free fall with air resistance, is to perform what is known in mathematics as an initial value problem. This technique combines a differential equation with a specific condition (an 'initial value') that the solution must satisfy at a particular instant, usually the start of the event being modeled.

In this context, we were given the initial condition
\[v(0)=0,\]which means that at time \(t=0\), the velocity of the object is zero—representing the moment just before the object begins its descent. The task of verifying the solution essentially checks whether the proposed velocity function at \(t=0\) indeed equals zero and if the function's derivative matches the rate of change of velocity specified by the differential equation. This verification assures us that we have found a legitimate solution that correctly models the object's velocity throughout its fall.

For students tackling such problems, understanding how to apply these initial conditions is critical to generate specific solutions that correspond to the physics involved in real-world situations.

Terminal Velocity

The concept of terminal velocity arises in the context of free fall and is an important physical quantity in the study of motion through resisting mediums like air or water. Terminal velocity is the constant speed attained by an object when the force of gravity is balanced by the drag force and no further acceleration occurs.

The differential equation in our exercise encapsulates this balance and allows us to estimate the terminal velocity using calculus. As seen in the solution, by finding the limit as time reaches infinity, we were able to calculate terminal velocity. In practice, this limit shows that the exponential decrease in velocity due to air resistance becomes negligible, so the velocity stabilizes at a maximum value, represented by
\[lim_{{t \rightarrow \infty}} v(t) = \frac{g}{b}\]
In our case, with \(b=0.1s^{-1}\) and \(g\) as the gravitational constant, the object's terminal velocity was estimated to be 98 m/s. Understanding terminal velocity is useful not only in theoretical physics and engineering but also in practical applications such as designing parachutes or estimating the impact speeds of falling objects.