52-56. 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.


{Use of Tech} Free fall One possible model that describes the free fall of an object in a gravitational field subject to air resistance uses the equation v'(t) = g - bv, where v(t) is the velocity of the object for t ≥ 0, g = 9.8 m/s² is the acceleration due to gravity, and b > 0 is a constant that involves the mass of the object and the air resistance.


c. Using the graph in part (b), estimate the terminal velocity lim(t→∞) v(t).

Verified step by step guidance

1. Understand that the terminal velocity is the velocity value that the object approaches as time \(t\) goes to infinity, meaning \(\lim_{t \to \infty} v(t)\).
2. Recall the given differential equation: \(v'(t) = g - b v(t)\), where \(g\) and \(b\) are constants, and \(v(t)\) is the velocity at time \(t\).
3. At terminal velocity, the velocity no longer changes, so the derivative \(v'(t)\) becomes zero. Set \(v'(t) = 0\) to find the equilibrium velocity: \$0 = g - b v_{terminal}$.
4. Solve the equation for \(v_{terminal}\): \(b v_{terminal} = g\), which gives \(v_{terminal} = \frac{g}{b}\).
5. Use the graph from part (b) to estimate the value that \(v(t)\) approaches as \(t\) becomes very large, and compare it to the theoretical terminal velocity \(\frac{g}{b}\).

Key Concepts

First-Order Linear Differential Equations

This type of differential equation has the form v'(t) + p(t)v(t) = q(t). In the given model, v'(t) = g - bv can be rewritten as v'(t) + bv = g, which is linear. Understanding how to identify and solve such equations is essential for analyzing velocity over time.

Terminal Velocity

Terminal velocity is the steady-state velocity an object reaches when the net acceleration becomes zero. Mathematically, it is the limit of v(t) as t approaches infinity, where the forces of gravity and air resistance balance out, resulting in constant velocity.

Limit of a Function as t Approaches Infinity

Evaluating lim(t→∞) v(t) involves understanding how the solution to the differential equation behaves over a long time. This concept helps determine the long-term behavior of velocity, such as reaching terminal velocity in free fall with air resistance.