MA2051 - Ordinary Differential Equations
Test 1 - Make-Up 1 - A96 





Instructions: Do your work on the paper provided.  Put 
your name, section number and PLA's name on that paper.  Work 
neatly.  Show your work. 


JUSTIFY YOUR ANSWERS.





1 45 pts 

  An elevator is 
rising in a tall skyscraper when its supporting cable suddenly breaks.  
As it falls, friction brakes slow it down with a force 
proportional to its speed.


    Follow the steps below to derive a model for the velocity of the 
elevator in order to help designers select a brake that slows the 
elevator sufficiently to prevent serious injury to its occupants.


    For the sake of consistency, let v(t) be the velocity of the 
elevator at time t; t = 0 is the instant the cable broke.  Use a 
coordinate system in which up is positive.


    
A
 Write an expression for the force of the brake on the 
    elevator in terms of the elevator's speed.


    
B
 Couple the previous expression with an appropriate physical 
    law to derive a model for the velocity of the elevator.


    
C
 Argue on physical grounds that there is an upper limit on the 
    speed with which the elevator can fall, no matter how tall the 
    building is.  Find that speed from your model and show that it 
    consistent with the physical reasoning you used.  What changes 
    could a designer make to reduce that maximum speed by half?


    
D
 Use your model to determine how long it takes the elevator to 
    reach the maximum speed.  How long does it take to reach  of maximum speed?  (For simplicity, assume the 
    elevator starts with zero velocity.)









2 20 pts 

  Recall that the 
logistic population equation  tex2html_wrap_inline50  predicts a steady state 
population level of a/s.  Suppose a population begins at twice that 
level, P(0) = 2a/s.  Use one step of Euler's method to estimate the 
time needed for the population to drop to the steady state value of 
a/s.  Check that your answer does indeed have units of time. Does 
your answer underestimate or overestimate the time to reach the steady state?









3 25 pts 

  Consider the 
differential equation  tex2html_wrap_inline58 .


    
 Show that it has no steady state.


    
 Find the solution that satisfies z(0) = 0.









4 10 pts 

  Maple told me 
that  tex2html_wrap_inline62  has a really ugly solution, too messy to write 
here.  Call it R(t) (for really ugly).  I did see that R(0) = 1.


    
A
 Write a solution of  tex2html_wrap_inline68  in terms of R(t). 
    Note the forcing term!!


    
B
 Write a solution of  tex2html_wrap_inline68 , z(0) = 0, in 
    terms of R(t).