MA2051 - Ordinary Differential Equations
Test 1 Old Makeup #1 - A96 



  


Instructions: Do your work on the paper provided.  Put your
name and section number on that paper.  Work neatly.  Show your work. 
JUSTIFY YOUR ANSWERS.







1 50 pts 

  Shotgun pellets and 
other small spherical metal particles are formed by dropping a bit of 
molten metal down a long hollow tube, called a shot tower.  As the mass falls, 
air resistance retards its motion with a force proportional to the SQUARE of its velocity.


    A) Let m denote the mass of the particle and v its velocity.  Let r 
be the proportionality constant for the force of air resistance.  Take 
UP as the POSITIVE direction.


    
 B) Write an expression for the force of gravity on the pellet.


    
 C) Write an expression for the force of air resistance on the 
pellet.  BE CAREFUL WITH SIGNS!.


    
 D) Use Newton's Law ( tex2html_wrap_inline48 ) to derive a model for 
the velocity of the pellet.


    
 E) Will the pellet fall faster and faster or will it attain a 
constant terminal velocity?  If the latter occurs, find the value of the 
terminal velocity.  BE CAREFUL WITH SIGNS!.


    
 F) Is this terminal velocity stable?









2 30 pts 

  Solve the following. 
If no initial condition is given, find a general solution.  If an initial 
condition is given, solve the initial value problem.


    
 u' - 12tu = 0


    
 u' - 12tu = -24t


    
 u' - 12tu = -24t, u(0) = 0









20 pts 

  The initial value 
problem c' = -kc,  tex2html_wrap_inline60 , with k > 0 models the decay of a 
mass c of a radioactive isotope.  Use one step of Heun's method to 
find an approximate expression for the time when the isotope mass is 
one half its initial amount.


 





  

    

© 1996 by Will Brother.
All rights Reserved. File last modified on September 19, 1996.