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Chapter 3: Problem 14

An object of mass m is released from rest and falls under the influence of gravity. If the magnitude of the force due to air resistance is $$b v^{n}$$. where b and n are positive constants, find the limiting velocity of the object (assuming this limit exists). [Hint: Argue that the existence of a (finite) limiting velocity implies that $$d v / d t \rightarrow 0 \text { as } t \rightarrow+\infty . ]$$

Short Answer

Expert verified
The limiting velocity of the object as time approaches infinity is \[v = \left(\frac{mg}{b}\right)^{\frac{1}{n}}\]

Step by step solution

01

Analyze the Forces Acting on the Object

First, determine the forces acting on the object. The force due to gravity is \(mg\) where \(m\) is the mass and \(g\) is the acceleration due to gravity. The force due to air resistance is given by \(-bv^{n}\) where \(b\) and \(n\) are positive constants, \(v\) is the velocity of the object and the negative sign represents the opposing direction of this force.
02

Set up Differential Equation

Apply Newton's second law, \(F = ma\), where \(F\) is the total force, \(m\) is the mass and \(a\) is the acceleration. The total force acting on the object is the sum of the gravitational force and the force due to air resistance, i.e., \(F = mg - bv^{n}\). Acceleration a, is the change in velocity with respect to time, i.e., \(a = dv/dt\). Hence, the equation becomes \(m dv/dt = mg - bv^{n}\).
03

Solve the Differential Equation

To solve the differential equation, \[m \frac{dv}{dt} = mg - bv^{n}\] it can be rearranged as \[\frac{dv}{dt} = g - \frac{b}{m} v^{n}\]. We are interested in the limiting velocity as time t approaches infinity. So we assume that the existence of a finite limiting velocity implies that \(\frac{dv}{dt}\) approaches 0 as \(t\) approaches infinity. So when \(\frac{dv}{dt} = 0\) which gives \[0 = g - \frac{b}{m} v^{n}\] which can further be rearranged to find the limiting velocity \(v\).
04

Find the Limiting Velocity

The limiting velocity, \(v\), can now be found by rearranging the previous equation to solve for \(v\). This gives us \[v = \left(\frac{mg}{b}\right)^{\frac{1}{n}}\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations
Differential equations are mathematical equations that involve functions and their derivatives. They describe how a particular quantity changes with respect to another. In the context of this problem, the differential equation is used to model how the velocity of the falling object changes over time under the influence of gravitational force and air resistance.
  • The equation provided is \( m \frac{dv}{dt} = mg - bv^{n} \), which describes the motion of the object.
  • Here, \( \frac{dv}{dt} \) is a derivative, representing the rate of change of velocity with respect to time.
  • By solving this equation, we can understand how the velocity of the object behaves over time.
Differential equations are essential in predicting the object's behavior and are used extensively in physics and engineering to model dynamic systems.
Understanding how to manipulate these equations allows us to calculate essential properties like the limiting velocity, which is the focus of this exercise.
Newton's Second Law
Newton's second law is one of the fundamental principles of classical mechanics. It states that the force acting on an object is equal to the mass of the object multiplied by its acceleration: \( F = ma \).
  • In this problem, Newton's second law is applied to relate the forces acting on the object, both due to gravity and air resistance, to its acceleration.
  • The total force acting on the object is the sum of the gravitational force (\( mg \)) and the force of air resistance (\( -bv^{n} \)).
  • This relationship allows us to set up the differential equation, \( m \frac{dv}{dt} = mg - bv^{n} \), to solve for the velocity as time progresses.
By applying this law, we know that these forces result in a change in velocity over time, helping us ultimately determine the limiting velocity of the object.
Air Resistance
Air resistance, also known as drag, is a force that opposes the motion of an object through the air. It is proportional to some power of the velocity, \( v^{n} \), in this exercise.
  • This resistance depends on factors like the shape and speed of the object, as well as the density of the air.
  • In our case, the force due to air resistance is given by \( -bv^{n} \).
  • The negative sign indicates that this force acts in the opposite direction of the object's motion.
Air resistance is crucial in determining how quickly an object will reach its limiting velocity, as it counters the force of gravity. Understanding this concept helps us in setting up the differential equation needed to find the solution for the motion of the object.
Gravitational Force
Gravitational force is the attractive force that pulls objects towards the center of the Earth. It is calculated by multiplying the mass of an object by the gravitational acceleration, \( g \).
  • In this scenario, the gravitational force acting on the object is \( mg \).
  • This force causes the object to accelerate towards the Earth when released from rest.
  • Gravitational force impacts the object continuously as it falls, and must be countered by air resistance to reach a limiting velocity.
Understanding gravitational force is key to solving for the motion of the object, providing a starting point for the differential equation. It ensures that the forces are balanced when calculating the limiting velocity.

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Most popular questions from this chapter

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