In many of the following problems, it will be

Chapter 3, Problem 20E

(choose chapter or problem)

In many of the following problems, it will be essential to have a calculator or computer available. You may use a software package† or write a program for solving initial value problems using the improved Euler’s method algorithms on pages 127 and 128. (Remember, all trigonometric calculations are done in radians.)

Falling Body. In Example 1 of Section 3.4, we modeled the velocity of a falling body by the initial value problem

\(m \frac{d v}{d t}=m g-b\),                       \(v(0)=v_{0}\).

under the assumption that the force due to air resistance is \(-bv). However, in certain cases the force due to air resistance behaves more like \(-b v^{r}\), where

\(r(>1)\) is some constant. This leads to the model

(14) \(m \frac{d v}{d t}=m g-b v^{r}\),                       \(v(0)=v_{0}\).

To study the effect of changing the parameter 𝑟 in (14), take \(m=1,\ g=9.81,\ b=2\), and \(v_{0}=0\). Then use the improved Euler’s method subroutine with \(h=0.2\) to approximate the solution to (14) on the interval \(0 \leq t \leq 5\) for \(r=1.0\), 1.5, and 2.0. What is the relationship between these solutions and the constant solution \(v(t) \equiv(9.81 / 2)^{1 / r}\)?

Equation Transcription:

Text Transcription:

m {dv}over{dt}=m g-b

v(0)=v_{0}

-bv

-bv^{r}

r(>1)

m{dv}{dt}=mg-bv^{r}

v(0)=v_{0}

m=1, g=9.81, b=2

v_{0}=0

h=0.2

0</=t</=5

r=1.0

v(t) equiv(9.81/2)^{1/r}