
Calculus Calculus Math Forum 
 LinkBack  Thread Tools  Display Modes 
May 31st, 2015, 06:19 PM  #1 
Newbie Joined: May 2015 From: Waterloo, Canada Posts: 2 Thanks: 0 Math Focus: Calculus  A math problem that involves first order ODE
Hi all, I came across this problem today. This is my attempted solution. As you can see, I stumbled upon an equation that is in implicit form. I found C by substituting in the initial values given, however unable to find v when y = 0 because it is in implicit form. Am I doing this wrong? 
June 2nd, 2015, 02:09 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,816 Thanks: 1460 
let the downward direction be positive with $y_0 = 0$, $v_0=0$, and $y_f=30$ $\displaystyle m \frac{dv}{dt} = mg  kv$ $\displaystyle \frac{dv}{dt} = g  \frac{k}{m}v$ $\displaystyle \frac{dv}{dt} = g  0.8v$ $\displaystyle \frac{dv}{g0.8v} = dt$ $\displaystyle \lng0.8v = 0.8t + C$ $\displaystyle g0.8v = Ae^{0.8t}$ $\displaystyle v(0) = 0 \implies A = g$ ... $\displaystyle 0.8v = g(1  e^{0.8t})$ $\displaystyle v = \frac{5g}{4}(1  e^{0.8t})$ ... note terminal velocity, $\displaystyle v_T = \frac{5g}{4} \, m/s$ $\displaystyle \frac{dy}{dt} = \frac{5g}{4}(1  e^{0.8t})$ $\displaystyle y = \frac{5g}{4}(t + 1.25e^{0.8t} + C)$ $\displaystyle y(0) = 0$ $\displaystyle 0 = \frac{5g}{4}(1.25 + C) \implies C = 1.25$ $\displaystyle y = \frac{5g}{4}[t + 1.25(e^{0.8t}  1)]$ $\displaystyle 30 = \frac{5g}{4}[t + 1.25(e^{0.8t}  1)]$ using $\displaystyle g = 9.8 \, m/s^2$ and solving with a calculator ... $\displaystyle t \approx 3.63 \, sec$ $\displaystyle v(3.63) \approx 11.82 \, m/s$ 

Tags 
calculus, differential equation, first order, involves, math, ode, order, problem 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Multiplication or Division in Daily Life Problem Which Involves Negative Number  Monox D. IFly  Elementary Math  19  May 24th, 2016 02:35 AM 
In what order should I learn math?(Math is daunting)  E01  Academic Guidance  7  February 9th, 2014 09:25 AM 
topological space involves open sets  fairyrak  Real Analysis  5  December 30th, 2011 06:39 PM 
problem involves limit of sequences  jackie  Real Analysis  1  February 25th, 2008 11:30 AM 