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}
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