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1.8 Summary

•veriﬁcation:providingevidencethattheprogramworkscorrectly

•debugging:locatingandcorrectingerrorsinaprogram

1.8.2Example:Trajectoryofaball

Problem. Whatisthetrajectoryofaballthatisthrownorkickedwith

aninitialvelocityv

makinganangleθwiththehorizontal?Thisproblem

canbesolvedbybasichighschoolphysicsasyouareencouragedtodo

inExercise1.13.Theballwillfollowatrajectoryy=f(x)throughthe

airwhere

f(x)=xtanθ−

2v2

gx2

cos2θ

(1.6)

Inthisexpression,xisahorizontalcoordinate,gistheaccelerationof

gravity,v

isthesizeoftheinitialvelocitythatmakesanangleθwith

thexaxis,and(0,y

)istheinitialpositionoftheball.Ourprogramming

goalistomakeaprogramforevaluating(1.6).Theprogramshouldwrite

outthevalueofalltheinvolvedvariablesandwhattheirunitsare.

Weremarkthattheformula(1.6)neglectsairresistance.Exercise1.11

exploreshowimportantairresistanceis.Forasoftkick(v

=10km/h)

ofafootball,thegravityforce isabout 120timeslargerthanthe air

resistance.Forahardkick,airresistancemaybeasimportantasgravity.

Solution. WeusetheSIsystemandassumethatv

isgiveninkm/h;

g=9.81m/s

;x,y,andy

aremeasuredinmeters;andθindegrees.The

programhasnaturallyfourparts:initializationofinputdata,importof

functionsandπfrommath,conversionofv

andθtom/sandradians,

respectively,andevaluationoftheright-handsideexpressionin(1.6).We

choosetowriteoutallnumericalvalueswithonedecimal.Thecomplete

programisfoundintheﬁletrajectory.py:

g = = 9.81

# m/s**2

v0 = = 15

# km/h

theta = = 60 0 # # degrees

x = = 0.5

# m

y0 = = 1

# m

print """\

= %.1f f km/h

theta = = %d degrees

= %.1f f m

= %.1f f m\

""" % % (v0, , theta, y0, x)

from math import pi, , tan, cos

# Convert v0 to m/s and theta to o radians

v0 = = v0/3.6

theta = = theta*pi/180

y = = x*tan(theta) ) - 1/(2*v0**2)*g*x**2/((cos(theta))**2) + + y0

print ’y

= %.1f f m’ % % y

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1 Computingwithformulas

Thebackslashinthetriple-quotedmulti-linestring makesthe string

continueonthenextlinewithoutanewline.Thismeansthatremoving

thebackslashresultsinablanklineabovethev0lineandablankline

betweenthexandylinesintheoutputonthescreen.Anotherpointto

mentionistheexpression1/(2*v0**2),whichmightseemasacandidate

forunintendedintegerdivision.However,theconversionof v0tom/s

involvesadivisionby3.6,whichresultsinv0beingfloat,andtherefore

2*v0**2beingfloat.Therestoftheprogramshouldbeself-explanatory

atthisstageinthebook.

We can execute the program in IPythonor an ordinary y terminal

windowandwatchtheoutput:

Terminal

= 15.0 km/h

theta = = 60 0 degrees

= 1.0 m

= 0.5 m

= 1.6 m

1.8.3Abouttypesettingconventionsinthisbook

Thisversionofthebookappliesdiﬀerentdesignelementsfordiﬀerent

typesof“computertext”.Completeprogramsandpartsofprograms

(snippets)aretypesetwithalightbluebackground.Asnippetlookslike

this:

a = = sqrt(4*p p + + c)

print ’a =’, a

Acompleteprogramhasanadditionalverticallinetotheleft:

C = = 21

F = = (9.0/5)*C + + 32

print F

Asareaderofthisbook,youmaywonderifacodeshownisacomplete

programyoucantryoutorifitisjustapartofaprogram(asnippet)so

thatyouneedtoaddsurroundingstatements(e.g.,importstatements)

totrythecodeoutyourself.Theappearanceofaverticallinetotheleft

ornotwillthenquicklytellyouwhattypeofcodeyousee.

AninteractivePythonsessionistypesetas

>>> from m math h import *

>>> p p = = 1; c c = = -1.5

>>> a a = = sqrt(4*p p + + c)

Runningaprogram,sayball_yc.py,intheterminalwindow,followed

bysomepossibleoutputistypesetas

Terminal

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1.9 Exercises

ball_yc.py

At t=0.0417064 s and 0.977662 2 s, , the height is 0.2 m.

RecallfromSection1.5.3thatwejustwritetheprogramname.Areal

executiondemandspreﬁxingtheprogramnamebypythoninaterminal

window,orbyrunifyouruntheprogramfromaninteractiveIPython

session.WerefertoAppendixH.2for morecompleteinformationon

runningPythonprogramsindiﬀerentways.

Sometimesjusttheoutputfromaprogramisshown,andthisoutput

appearsasplaincomputertext:

h = = 0.2

order=0, error=0.221403

order=1, error=0.0214028

order=2, error=0.00140276

order=3, error=6.94248e-05

order=4, error=2.75816e-06

Filescontainingdataareshowninasimilarwayinthisbook:

date

Oslo

London

Berlin

Paris

Rome

Helsinki

01.05 18

21.2

20.2

13.7

15.8

01.06 21

13.2

14.9

01.07 13

26.2

14.5

StyleguideforPythoncode. ThisbookpresentsPythoncodethatis

(mostly)inaccordancewiththeoﬃcialStyleGuideforPythonCode

knowninthePythoncommunityasPEP8.Someexceptionstotherules

aremadetomakecodesnippetsshorter:multipleimportsononeline

andlessblanklines.

1.9Exercises

Whatdoesitmeantosolveanexercise? Thesolutiontomostofthe

exercisesinthisbookisaPythonprogram.Toproducethesolution,you

ﬁrst need understand the problem and what the program is supposed

to do, and then you need to understand how to translate the problem

description into a series of Python statements. Equally important is

the veriﬁcation (testing) of the program. A complete solution to a pro-

gramming exercises therefore consists of two parts: 1) the program text

and 2) a demonstration that the program works correctly. Some simple

programs, like the ones in the ﬁrst two exercises below, have so obviously

correct output that the veriﬁcation can just be to run the program and

record the output.

In cases where the correctness of the output is not obvious, it is

necessary to prove or bring evidence that the result is correct. This can

be done through comparisons with calculations done separately on a

calculator, or one can apply the program to a special simple test case

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1 Computing with formulas

with known results. The requirement is to provide evidence to the claim

that the program is without programming errors.

The sample run of the program to check its correctness can be inserted

at the end of the program as a triple-quoted string. Alternatively, the

output lines can be inserted as comments, but using a multi-line string

requires less typing. (Technically, a string object is created, but not

assigned to any name or used for anything in the program beyond

providing useful information for the reader of the code.) One can do

Terminal

Terminal> python myprogram.py > result

and use a text editor to insert the ﬁleresult inside the triple-quoted

multi-line string. Here is an example on a run of a Fahrenheit to Celsius

conversion program inserted at the end as a triple-quoted string:

F = 69.8

# Fahrenheit degrees

C = (5.0/9)*(F - 32)

# Corresponding Celsius degrees

print C

’’’

Sample run (correct result is 21):

python f2c.py

21.0

’’’

Exercise 1.1: Compute 1+1

The ﬁrst exercise concerns some very basic mathematics and program-

ming: assign the result of 1+1 to a variable and print the value of that

variable. Filename: 1plus1.py.

Exercise 1.2: Write a Hello World program

Almost all books about programming languages start with a very simple

program that prints the textHello, World! to the screen. Make such a

program in Python. Filename: hello_world.py.

Exercise 1.3: Derive and compute a formula

Can a newborn baby in Norway expect to live for one billion (109)

seconds? Write a Python program for doing arithmetics to answer the

question. Filename: seconds2years.py.

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1.9 Exercises

Exercise 1.4: Convert from meters to British length units

Make aprogram where you set a length given in meters and then compute

and write out the corresponding length measured in inches, in feet, in

yards, and in miles. Use that one inch is 2.54 cm, one foot is 12 inches,

one yard is 3 feet, and one British mile is 1760 yards. For veriﬁcation, a

length of 640 meters corresponds to 25196.85 inches, 2099.74 feet, 699.91

yards, or 0.3977 miles. Filename: length_conversion.py.

Exercise 1.5: Compute the mass of various substances

The density of a substance is deﬁned as� =m/V, wherem is the mass

of a volumeV . Compute and print out the mass of one liter of each of

the following substances whose densities ing/cm

are found in the ﬁle

src/files/densities.dat6:iron,air,gasoline,ice,the humanbody,

silver, and platinum. Filename: 1liter.py.

Exercise 1.6: Compute the growth of money in a bank

Letp be a bank’s interest rate in percent per year. An initial amountA

has then grown to

�

100

�

aftern years. Makea program for computing how much money 1000euros

have grown to after three years with 5 percent interest rate. Filename:

interest_rate.py.

Exercise 1.7: Find error(s) in a program

Suppose somebody has written a simple one-line program for computing

sin(1):

x=1; print ’sin(%g)=%g’ % (x, sin(x))

Create this program and try to run it. What is the problem?

Exercise 1.8: Type in program text

Type the following program in your editor and execute it. If your program

does not work, check that you have copied the code correctly.

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1 Computing with formulas

from math import pi

h = 5.0

# height

b = 2.0

# base

r = 1.5

# radius

area_parallelogram = h*b

print ’The area of the parallelogram is %.3f’ % area_parallelogram

area_square = b**2

print ’The area of the square is %g’ % area_square

area_circle = pi*r**2

print ’The area of the circle is %.3f’ % area_circle

volume_cone = 1.0/3*pi*r**2*h

print ’The volume of the cone is %.3f’ % volume_cone

Filename: formulas_shapes.py.

Exercise 1.9: Type in programs and debug them

Type these short programs in your editor and execute them. When they

do not work, identify and correct the erroneous statements.

a) Does sin2(x) +cos2(x) = 1?

from math import sin, cos

x = pi/4

1_val = math.sin^2(x) + math.cos^2(x)

print 1_VAL

b) Computesinmeterswhen s= v

t+0,5at

,withv

=3 m/s,t = 1

s, a = 2 m/s

v0 = 3 m/s

t = 1 s

a = 2 m/s**2

s = v0.t + 0,5.a.t**2

print s

c) Verify these equations:

(a +b)

+2ab +b

(a −b)

−2ab +b

a = 3,3

b = 5,3

a2 = a**2

b2 = b**2

eq1_sum = a2 + 2ab + b2

eq2_sum = a2 - 2ab + b2

eq1_pow = (a + b)**2

eq2_pow = (a - b)**2

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1.9 Exercises

print ’First equation: %g = %g’, % (eq1_sum, eq1_pow)

print ’Second equation: %h = %h’, % (eq2_pow, eq2_pow)

Filename: sin2_plus_cos2.py.

Exercise 1.10: Evaluate a Gaussian function

The bell-shaped Gaussian function,

f(x) =

√

2πs

exp

�

−

�

x− m

�

(1.7)

is one of the most widely used functions in science and technology. The

parametersm ands> 0 are prescribed real numbers. Make a program

for evaluating this function whenm = 0,s = 2, andx = 1. Verify the

program’s result by comparing with hand calculations on a calculator.

Filename: gaussian1.py.

Remarks. Thefunction(1.7)isnamedafterCarlFriedrichGauss

,1777-

1855, who was a German mathematician and scientist, now considered as

one of the greatest scientists of all time. He contributed to many ﬁelds,

including number theory, statistics, mathematical analysis, diﬀerential ge-

ometry, geodesy, electrostatics, astronomy, and optics. Gauss introduced

the function (1.7) when he analyzed probabilities related to astronomical

data.

Exercise 1.11: Compute the air resistance on a football

The drag force, due to air resistance, on an object can be expressed as

�AV

(1.8)

where� is the density of the air,V is the velocity of the object,A is

the cross-sectional area (normal to the velocity direction), andC

is the

drag coeﬃcient, which depends heavily on the shape of the object and

the roughness of the surface.

The gravity force on an object with mass m is F

= mg, where

g= 9.81ms

−2

We can use the formulas forF

andF

to study the importance of

air resistance versus gravity when kicking a football. The density of air

is� = 1.2kg m−3. We haveA =πa2 for any ball with radiusa. For a

football, a = 11 cm, the mass is 0.43 kg, and C

can be taken as 0.2.

Make a program that computes the drag force and the gravity force

on a football. Write out the forces with one decimal in units of Newton

7http://en.wikipedia.org/wiki/Carl_Gauss

1 Computing with formulas

(N =kgm/s2). Also print the ratio of the drag force and the gravity force.

DeﬁneC

,�,A,V,m,g,F

,andF

as variables, and put a comment

with the corresponding unit. Use the program to calculate the forces on

the ball for a hard kick,V = 120km/h and for a soft kick,V = 10km/h

(it is easy to mix inconsistent units, so make sure you compute withV

expressed in m/s). Filename: kick.py.

Exercise 1.12: How to cook the perfect egg

As an egg cooks, the proteins ﬁrst denature and then coagulate. When

the temperature exceeds a critical point, reactions begin and proceed

faster as the temperature increases. In the egg white, the proteins start

to coagulate for temperatures above 63 C, while in the yolk the proteins

start to coagulate for temperatures above 70 C. For a soft boiled egg,

the white needs to have been heated long enough to coagulate at a

temperature above 63 C, but the yolk should not be heated above 70 C.

For a hard boiled egg, the center of the yolk should be allowed to reach

70 C.

The following formula expresses the timet it takes (in seconds) for

the center of the yolk to reach the temperature T

(in Celsius degrees):

M2/3cρ1/3

Kπ2(4π/3)2/3

�

0.76

−T

�

(1.9)

Here,M,ρ,c, andK are properties of the egg:M is the mass,ρ is the

density,c is the speciﬁc heat capacity, andK is thermal conductivity.

Relevant values areM = 47 gfor a smallegg andM =67 gfor alarge egg,

ρ=1.038 gcm−3, c=3.7 Jg−1K

−1

,andK = 5.4· 10−3 Wcm−1K

−1

Furthermore,T

is the temperature (in C degrees) of the boiling water,

andT

is the original temperature (in C degrees) of the egg before being

put in the water. Implement the formula in a program, setT

=100

CandT

=70 C, and computet for a large egg taken from the fridge

=4 C) and from room temperature (T

=20 C). Filename: egg.py.

Exercise 1.13: Derive the trajectory of a ball

The purpose of this exercise is to explain how Equation (1.6) for the

trajectory of a ball arises from basic physics. There is no programming

in this exercise, just physics and mathematics.

The motion of the ball is governed by Newton’s second law:

=ma

(1.10)

=ma

(1.11)

1.9 Exercises

whereF

andF

are the sum of forces in thex andy directions, respec-

tively,a

anda

are the accelerations of the ball in thex andy directions,

andm is the mass of the ball. Let (x(t),y (t)) be the position of the ball,

i.e., the horizontal and vertical coordinate of the ball at timet. There

are well-known relations between acceleration, velocity, and position: the

acceleration is the time derivative of the velocity, and the velocity is the

time derivative of the position. Therefore we have that

d2x

dt2

(1.12)

d2y

dt2

(1.13)

If we assume that gravity is the only important force on the ball,F

and F

=−mg.

Integrate the two components of Newton’s second law twice. Use the

initial conditions on velocity and position,

x(0) = v

cos θ,

(1.14)

y(0) = v

sinθ,

(1.15)

x(0) = 0,

(1.16)

y(0) = y

(1.17)

to determine thefour integration constants. Write up the ﬁnalexpressions

forx(t) andy(t). Show that ifθ =π/2, i.e., the motion is purely vertical,

we get the formula (1.1) for they position. Also show that if we eliminate

t,weendupwiththerelation(1.6)betweenthe xand ycoordinates

of the ball. You may read more about this type of motion in a physics

book, e.g., [16]. Filename: trajectory.*.

Exercise 1.14: Find errors in the coding of formulas

Some versions of our program for calculating the formula (1.3) are listed

below. Find the versions that will not work correctly and explain why in

each case.

C = 21;

F = 9/5*C + 32;

print F

C = 21.0; F = (9/5)*C + 32;

print F

C = 21.0; F = 9*C/5 + 32;

print F

C = 21.0; F = 9.*(C/5.0) + 32;

print F

C = 21.0; F = 9.0*C/5.0 + 32;

print F

C = 21;

F = 9*C/5 + 32;

print F

C = 21.0; F = (1/5)*9*C + 32;

print F

C = 21;

F = (1./5)*9*C + 32;

print F

1 Computing with formulas

Exercise 1.15: Explain why a program does not work

Find out why the following program does not work:

C = A + B

A = 3

B = 2

print C

Exercise 1.16: Find errors in Python statements

Try the following statements in an interactive Python shell. Explain why

some statements fail and correct the errors.

1a = 2

a1 = b

x = 2

y = X + 4 # is it 6?

from Math import tan

print tan(pi)

pi = "3.14159’

print tan(pi)

c = 4**3**2**3

_ = ((c-78564)/c + 32))

discount = 12%

AMOUNT = 120.-

amount = 120$

address = hpl@simula.no

and = duck

class = ’INF1100, gr 2"

continue_ = x > 0

rev = fox = True

Norwegian = [’a human language’]

true = fox is rev in Norwegian

Hint. Itiswisetotestthevaluesoftheexpressionsontheright-hand

side, and the validity of the variable names, separately before you put the

left- and right-hand sides together in statements. The last two statements

work, but explaining why goes beyond what is treated in this chapter.

Exercise 1.17: Find errors in the coding of a formula

Given a quadratic equation,

+bx+ c = 0,

the two roots are

−b +

√

b2 −4ac

, x

−b−

√

b2 − 4ac

(1.18)

What are the problems with the following program?

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