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).

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Understand that the terminal velocity is the velocity value that the object approaches as time \(t\) goes to infinity, meaning \(\lim_{t \to \infty} v(t)\).

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 \(t\).

At terminal velocity, the velocity no longer changes, so the derivative \(v'(t)\) becomes zero. Set \(v'(t) = 0\) to find the equilibrium velocity: \(0 = g - b v_{terminal}\).

Solve the equation for \(v_{terminal}\): \(b v_{terminal} = g\), which gives \(v_{terminal} = \frac{g}{b}\).

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 \(\frac{g}{b}\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.