91Ó°ÊÓ

In the context of a falling body, velocity is a measure of how fast the object is moving through a resisting medium such as air or water. It indicates both the speed and direction of motion. The given differential equation describes how the velocity of a body changes over time under specific conditions.

• The differential equation \( \frac{d v}{d t} = a - b v \) shows the relationship between the rate of change of velocity \( v \) and constants \( a \) and \( b \).
• Constant \( a \) represents the initial driving force acting on the body, pushing it to accelerate.
• Constant \( b \) is a damping factor that slows the velocity due to resistance from the medium.

Understanding how these elements work together helps us determine how velocity evolves from rest and reaches its terminal velocity over time. This forms the basis for solving the exercise.

Integration is a mathematical process used to find the antiderivative or the original function that describes a rate of change. In this exercise, we use integration to solve the differential equation related to the velocity of a falling body.

• We rearrange the differential equation \( \frac{d v}{d t} = a - b v \) into an integrable form: \( \frac{d v}{a-b v} = d t \).
• The purpose is to separate variables, making it possible to apply the integration process and find a function for velocity \( v \).

Integration Steps

Performing the integration gives us:

• The left-hand side integral becomes \(-\frac{1}{b} \ln|a-b v| + C_1\), where \( C_1 \) is a constant of integration.
• The right-hand side integral resolves to \( t + C_2 \), another constant \( C_2 \).

Combining these results allows us to solve for the velocity function over time.

Initial conditions are specific values provided in mathematical problems, which are used to solve differential equations uniquely. They help determine the constants of integration that appear after integrating the differential equation.

• In this exercise, the initial condition is given by noting that the velocity \( v = 0 \) when \( t = 0 \).
• This condition is crucial for finding the constant \( C \) in our integrated equation: \(|a-bv| = Ce^{-bt} \).

Using the Initial Condition

Applying \( v = 0 \) at \( t = 0 \) gives us:

• \(0 = \frac{a}{b} - \frac{C}{b}\)
• This simplifies to \( C = a \)

Substituting \( C = a \) back into the equation, we have the final velocity function that satisfies both the differential equation and the initial condition.

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are used to model growth or decay processes, essential for describing systems like the velocity of a falling body over time.

• In our solution, \( e^{-bt} \) represents the exponential decay of velocity as time progresses.
• The term demonstrates how the initial velocity's effect diminishes, tending towards zero as \( t \) increases.

Role of Exponential Function

The exponential function is significant in forming the final velocity expression:

• Final velocity: \( v = \frac{a}{b}(1-e^{-bt}) \)
• This indicates that initially, the exponential term influences the velocity heavily, but decreases over time towards stabilization (terminal velocity).

The exponential decay helps in understanding the transient behavior within a resistance medium, showcasing how velocity eventually approaches a constant value.