A spherical object (diameter d meters and mass m kg ) falling from rest experiences [FC99] a dr

step 1 In this problem I can write the expression as GT1 square by d1 is equal to Gt2 square by D2. Sol

A spherical object (diameter d meters and mass m kg )...

A spherical object (diameter $d$ meters and mass $m \mathrm{~kg}$ ) falling from rest experiences [FC99] a drag force $F_{\text {drag }}=-A v-B v^2$ newtons, where $v$ is the velocity in $\mathrm{m} / \mathrm{s}$ and $A=1.55 \times 10^{-4} d, B=0.22 d^2$. Derive the nonlinear ODE governing the velocity of the falling sphere. If the sphere is a basketball of diameter $25 \mathrm{~cm}$ and mass $0.60 \mathrm{~kg}$, analytically determine $v(t)$. Does Maple recognize the ODE as being separable? Plot $v(t)$ and show that the ball will reach a terminal velocity. Determine the terminal velocity. How long does it take the basketball to come within 1 percent of the terminal velocity?

Key Concepts

Separable Differential Equations

A separable differential equation is one in which the variables can be rearranged such that each side of the equation depends on only one variable. Recognizing that an ODE is separable is important because it allows integration and the derivation of an analytical solution, especially in initial value problems.

Analytical Solution of Initial Value Problems

An initial value problem involves solving an ODE with a given starting condition, such as an object that begins from rest. Obtaining an analytical solution provides explicit expressions for the velocity as a function of time, permitting detailed insight into the transient and steady-state behavior of the system.

Terminal Velocity

Terminal velocity is reached when the net force on the falling object becomes zero, meaning the gravitational force is exactly balanced by the drag force. It represents the constant speed at which the object descends after sufficient time has elapsed, making its analysis crucial for understanding steady-state motion.

Drag Force

Drag force is the resistive force acting opposite to the direction of motion through a fluid, and it is often modeled as a combination of terms that are linear and quadratic in velocity. This captures both viscous and inertial effects which are critical when analyzing motion in a fluid medium.

Newton's Second Law and Equation of Motion

Newton's second law, which states that the net force on an object is equal to its mass times its acceleration, provides the foundation for formulating the equation of motion. For a falling object, this means balancing gravitational force against drag to derive the nonlinear ODE governing its velocity.

Nonlinear Ordinary Differential Equations (ODEs)

When the force function includes nonlinear terms, such as the quadratic dependency on velocity from drag, the resulting ODE becomes nonlinear. Nonlinear ODEs require specialized analytical or numerical techniques for their solution and understanding their behavior is fundamental in dynamics.

Transcript

Please give Ace some feedback