An object of mass m is moving horizontally through a medium...

An object of mass $m$ is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is, $$m \frac{d^{2} s}{d t^{2}}=m \frac{d v}{d t}=f(v)$$ where $v=v(t)$ and $s=s(t)$ represent the velocity and position of the object at time $t,$ respectively. For example, think of a boat moving through the water. (a) Suppose that the resisting force is proportional to the velocity, that is, $f(v)=-k v, k$ a positive constant. (This model is appropriate for small values of v.) Let $v(0)=v_{0}$ and $s(0)=s_{0}$ be the initial values of $v$ and $s$ Determine $v$ and $s$ at any time $t .$ What is the total distance that the object travels from time $t=0 ?$ (b) For larger values of $v$ a better model is obtained by supposing that the resisting force is proportional to the square of the velocity, that is, $f(v)=-k v^{2}, k>0 .$ (This model was first proposed by Newton.) Let $v_{0}$ and $s_{0}$ be the initial values of $v$ and $s .$ Determine $v$ and $s$ at any time $t .$ What is the total distance that the object travels in this case?

An object of mass $m$ is moving horizontally through a medium which resists the motion with a force that is a function of the velocity; that is,
$$m \frac{d^{2} s}{d t^{2}}=m \frac{d v}{d t}=f(v)$$
where $v=v(t)$ and $s=s(t)$ represent the velocity and position of the object at time $t,$ respectively. For example, think of a boat moving through the water.
(a) Suppose that the resisting force is proportional to the velocity, that is, $f(v)=-k v, k$ a positive constant. (This model is appropriate for small values of v.) Let $v(0)=v_{0}$ and $s(0)=s_{0}$ be the initial values of $v$ and $s$ Determine $v$ and $s$ at any time $t .$ What is the total distance that the object travels from time $t=0 ?$
(b) For larger values of $v$ a better model is obtained by supposing that the resisting force is proportional to the square of the velocity, that is, $f(v)=-k v^{2}, k>0 .$ (This model was first proposed by Newton.) Let $v_{0}$ and $s_{0}$ be the initial values of $v$ and $s .$ Determine $v$ and $s$ at any time $t .$ What is the total distance that the object travels in this case?

Key Concepts

Asymptotic Analysis and Total Distance Calculation

In dynamics, determining the total distance traveled by an object often involves integrating the velocity function over time and examining its behavior as time approaches infinity. Asymptotic analysis is used to assess whether the integral converges to a finite value (indicating a finite stopping distance) or diverges, which is critical in understanding the long-term behavior of systems subject to resistive forces.

Separation of Variables and Integration

Many motion problems involving resistive forces lead to separable differential equations. The method of separation of variables allows one to rewrite the differential equation so that all terms involving the dependent variable (e.g., velocity) are on one side and the independent variable (e.g., time) on the other. This facilitates integration to obtain explicit solutions for velocity and, subsequently, position, given initial conditions.

Exponential Decay in Linear Drag Systems

When the resistive force is proportional to the velocity, solving the differential equation typically results in exponential decay of the velocity over time. This exponential behavior is indicative of a system where the rate of change decreases rapidly at first and then levels off, leading to a gradual approach towards rest, and is a common motif in various physical processes governed by first-order linear differential equations.

Newton's Second Law and Differential Equations

This principle states that the net force acting on an object equals its mass times the acceleration, which naturally leads to formulating the motion of the object in terms of differential equations. In problems involving acceleration and resistive forces, the law provides the starting point for setting up second-order (or reduced first-order) differential equations that describe the evolution of velocity and position as functions of time.

Resistive (Drag) Force Models

Resistive force models are used to represent the forces that oppose the motion of an object through a medium. Two common models are those in which the resistive force is proportional to the velocity (linear drag) and to the square of the velocity (quadratic drag). These models are crucial for accurately describing dynamics in different regimes, such as low-speed versus high-speed conditions, and are applied to real-life scenarios like motion through air or water.

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