91Ó°ÊÓ

A linear differential equation is a type of differential equation that can be written in the form: \( \frac{dy}{dt} + p(t)y = q(t) \). In the context of this exercise, we deal with such an equation: \( m \frac{dv}{dt} = mg - cv \). This equation describes how the velocity \( v \) of an object changes over time \( t \). Here, \( m \) is the mass of the object, \( g \) is the acceleration due to gravity, and \( c \) is a constant representing the proportionality to the velocity, often due to air resistance.
By converting the equation to: \( \frac{dv}{dt} + \frac{c}{m}v = g \), it sets up a first-order linear differential equation suitable for solving with the method of integrating factors. The homogeneous structure and linear terms make it possible to solve for velocity using simple integration methods and techniques.

Newton's Second Law is a foundational principle in physics that explains how the velocity of an object changes when it is subjected to external forces. It is commonly expressed as \( F = ma \), where \( F \) is the net force acting on an object, \( m \) is the object's mass, and \( a \) is its acceleration.
In this exercise, the law is applied to an object in free fall with air resistance. The equation \( m \frac{dv}{dt} = mg - cv \) expresses the forces acting on the object:

• \( mg \) is the force due to gravity, pulling it downwards.
• \( cv \) is the force of air resistance opposing the motion.

The result is a dynamic balance between these two forces, dictating how the object's velocity changes as it accelerates towards its limiting velocity.

An integrating factor is a function used to simplify the solving process of linear differential equations. It is typically represented as \( \mu(t) = e^{\int p(t)dt} \), which helps in converting a non-exact differential equation into an exact one, enabling straightforward integration.
In our exercise, the integrating factor is derived as \( e^{\frac{c}{m} t} \). By multiplying the entire differential equation by this factor, we prepare it for direct integration. This transforms the left-hand side into a single derivative term, making it integrable:

• \( e^{\frac{c}{m} t} \frac{dv}{dt} + \frac{c}{m}e^{\frac{c}{m} t}v = ge^{\frac{c}{m} t} \).

Integrating both sides gives us the expression that leads to the solution for velocity, particularly beneficial in solving physics problems like motion with resistance.

Limiting velocity, often referred to as "terminal velocity," is the constant speed that a freely falling object eventually reaches when the force due to gravity is balanced by the drag force due to air resistance.
In this scenario, the limiting velocity is found by observing the expression for velocity \( v = \frac{mg}{c}(1-e^{-\frac{c}{m}t}) \) as \( t \rightarrow \infty \). As time progresses, the exponential term \( e^{-\frac{c}{m}t} \rightarrow 0 \), simplifying the equation to:

• \( v = \frac{mg}{c} \)

This signifies that the object continues to fall at this constant speed without further acceleration or deceleration, a very important concept in understanding dynamic motion in physics.