Chapter 3: Methods
This chapter describes methods that make problem solving easier, that allow clear thinking, and that lead to correct answers. These methods are essential when you need to solve problems that you do not know how to solve at first glance. They apply to all parts of the solution of a problem.
The next sections are an example of a problem and its solution. The solution has been done using the standard problemsolving methods that we will discuss in this chapter. When we introduce these methods, we will use parts of the solution as illustrations.
Example 1: Sears Tower Elevator
The Skydeck in the Sears Tower in Chicago is on the 103^{rd} floor. The Skydeck elevator leaves the lobby and accelerates upward with a constant acceleration of 0.80 m/s^{2} until it reaches its maximum velocity of 18.2 miles per hour. Then it travels at its maximum velocity until it passes the 94th floor, 1263 feet above the lobby.
a. How long does the elevator take to accelerate to its maximum velocity?
b. How high above the lobby is the elevator just as it gets to its maximum velocity?
c. How many seconds after it leaves the lobby does the elevator pass the 94th floor?
Solution for Example 1
Sears Tower Elevator
D. Scarl
12 July, 2003
Drawing
Definitions
At the lobby  
time 
= t_{0}  = 0 s 
height above street 
= y_{0}  = 0 m 
velocity 
= v_{0}  = 0 m/s 
From lobby until elevator reaches maximum velocity  
acceleration 
= a_{0}  = 0.80 m/s^{2} 
At time that elevator reaches maximum velocity  
time 
= t_{1}  
height above street 
= y_{1}  
While elevator is traveling at maximum velocity  
acceleration 
= a_{1}  = 0 m/s^{2} 
velocity 
= v_{1}  = 18.2 mi/hr 
When elevator passes 94th floor  
time 
= t_{2}  
height above street 
= y_{2}  =1263 ft 
Convert units
v_{1} = 18.2 mi/hr (1609 m/1 mi) (1 hr/3600 s) =8.13 m/s
y_{2} = 1263 ft (0.305 m/ft) = 385 m
a) Find time to reach maximum velocity
General equations:
For motion with constant acceleration
y  = y_{0} + v_{0}t +(1/2)at^{2} 
v  = v_{0} + at 
When elevator reaches max velocity
v_{1}  = v_{0} + a_{1}t_{1} 
= 0 + a_{1}t_{1}  
t_{1}  = v_{1}/a_{1} 
=(8.13 m/s)/(0.80 m/s^{2})  
t_{1}  = 10.16 s 
The time for the elevator to reach its maximum velocity
is t_{1} =
10.16 s

b) Find height of elevator when it reaches its maximum velocity.
y =y_{0} + v_{0}t+ (1/2) at^{2}
y_{1} = 0 + 0 t_{1} + (1/2) a_{1}t_{1}^{2}
= (1/2) (0.80 m/s^{2} ) (10.16 s)^{2} = 41.29 m
When it reaches its maximum velocity, the elevator is 41.29 m above the lobby.
c) Calculate time to pass 94th floor.
While the elevator is moving with its maximum velocity
y  = y_{0} + v_{0}t +(1/2)at^{2} 
y_{2}  = y_{1} + v_{1}(t_{2}  t_{1}) + 0 
(The elapsed time while the elevator is moving with its maximum velocity is t_{2}  t_{1} .)
t_{2}  t_{1}  = (y_{2}  y_{1})/v_{1} 
t_{2}  = t_{1} + (y_{2}  y_{1})/v_{1} 
= 10.16 s + (385 m  41.29 m)/(8.13 m/s)  
= 10.16 s + 42.30 s  
t_{2}  = 52.46 s 
Divide into parts
Dividing the solution into parts is the first step in solving any problem. It is useful for problems at all levels and is essential for large problems. (Professional programmers know that one of the hardest tasks in writing a large program is dividing the program into parts that can be worked on separately and recombined smoothly.) When starting a problem, divide the description of the problem and its solution into the smallest possible parts and work on each part separately.
The parts into which most introductory science and engineering problem solutions can be divided are

Heading
Labeled drawings
Symbol definitions
Data
Preliminary equations
Science equations
Calculation
Results.
The first four of these parts are a restatement of the problem. They will be described in Chapter 4. The next three, the calculation of the solution, will be described in Chapter 5. The last part, presentation of the results, will be described in Chapter 6.
After you have divided the problem into parts, divide each of the parts into parts. One way you can do this is by dividing each part in space and time. If there are several different masses, describe the conditions for each mass separately. If there is a circuit with many amplifiers, first draw and analyze each amplifier separately. If the events in the problem take place at two or more different times, describe separately what is happening at the first time, the second time, and so forth, then write the equations that connect the events at the different times
Do the parts separately
First divide a problem into parts then force yourself to think about the parts separately. Do not try to solve the whole problem at once. As you start to work on a problem, it is tempting to try to figure out the answer or to worry about what equations apply and how to solve them. These are not the parts of the problem to think about at the beginning. First do the simple and automatic steps that begin the solution of every problem: write a heading, draw a labeled diagram, define your symbols. None of these steps requires knowledge of how you will eventually solve the problem.
Professional problem solvers learn not to worry about the parts of the solution that they cannot do, while they work on the parts they can do. If a problem is difficult, describing it clearly and dividing it into parts is a useful contribution, even if some of the parts cannot be done by you or by anyone else. Describing what needs to be done is a first step toward getting it done.
Dividing a solution into parts is useful on tests. If you sit and worry about the part of the problem you can't do, you will get no credit for writing down the parts you can do. After writing all of the automatic steps, such as the definition of symbols and the geometry equations, you may understand the problem well enough to be able to solve it.
Divide the solution into parts and do the parts you can do.
Label each part
Begin each part of the solution with a short title saying what you are starting to do:

Drawing
...
Definitions
...
Unit conversion
...
Geometry
...
General equations
...
Writing a title for each part of the solution helps in thinking about that part of the solution and helps even more when you come back to the problem later or show the problem to someone else. On tests, writing titles helps your instructor understand what you are doing and leads to a higher grade. These titles are part of the documentation of the problem; they make clear what you are doing on each line.
A special case of labeling each part of the solution is writing a one line description of the problem at the beginning of your work:
Sears Tower Elevator
This title helps those absentminded professors who do not remember exactly what problem they assigned and helps you remember exactly what the problem was when, before an exam, you look back at a problem that you solved at the very beginning of the course.
Algebra is a very condensed way of expressing ideas. Break up the algebra with a few words describing what you are about to do:
...
Find time to reach maximum velocity
...
Finally, some labels are just English sentences saying what your result was:
This sentence makes it easier to understand your result, helps you to think about whether it makes sense, and gives your result its proper importance.