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Question (1): (a) Write Mathematica NoteBook which generates a table of error function (erf) and its derivatives for
real arguments (z) between -‐3and 3 in steps of 0.25. The error function is defined by the equation below (but is
rarely evaluated by performing the integration).
(see http://mathworld.wolfram.com/Erf.html for information the error function )
The values in the table should be given with 5 decimal places. The table should have headers explaining what
the columns are. Explain how you designed the NoteBook and give an example of the output.
(b) How would you change this NoteBook if 10 significant digits were required? Mathematica NoteBook
should also be supplied
Question (2):
Write a NoteBook that reads your name in the form <first name> <middle name> <last name> and outputs the
last name first and adds a comma after the name, the first name, and initial of your middle name with a period
after the middle initial. If the names start with lower case letters, then these should be capitalized. The
NoteBook should not be specific to the lengths of your name (ie., the NoteBook should work with
anyone’s name.
As an example. An input of thomas abram herring would generate:
Herring, Thomas A.
Question (3): Write a Mathematica NoteBook that will compute the motion of a bicyclist and the energy used
cycling along an oscillating, sloped straight-‐line path. The path followed will be expressed as
H(x)  Sx  Asin(2 x / )  Bcos(2 x / )
where H(x) is the height of the path above the starting height, S is a slope in m/m, A and B are amplitudes of
sinusoidal oscillations in the path. The wavelength of the oscillations is . The forces acting on the bicycle are:
Wind Drag
Rolling Drag
Fd 1/2ArCd V2
Fr  MrgCr

where Ar is the cross-‐sectional area of the rider, Cd is the drag coefficient, r is the density of air and V
is the velocity of the bike. For the rolling drag, Mr is the mass of the rider and bike, g is gravitation
acceleration and Cr is rolling drag coefficient.
The bicyclist puts power into the bike by pedaling. The force generated by this power is given by
Rider force Fr  Pr /V
where Fr is the force produced by the rider, Pr is power used by the rider and V is velocity that the
bike is traveling (the force is assumed to act along the velocity vector of the bike). Your NoteBook
can assume that the power can be used at different rates along the path. The energy used will be the
integrated power supplied by the rider.
Assume that there is maximum value to the rider force.
Your code should allow for input of the constants above (path and force coefficients). The
NoteBook can assume a constant power scenario and constant force at low velocities.
As a test of your NoteBook use the following constants to compute:
(a)Time to travel and energy used to travel 10 km along a path specified by
S=0.01, A=5.0 m, B=0.0 m and = 2km, with constant power use of Pr =100Watts and a
maximum force available of 20N.
(b)The position and velocity of the bike tabulated at a 100-‐second interval.
(c)Add graphics to your NoteBook which plots the velocity of the bike as a function of time and
position along the path.

Assume the following values
Cd = 0.9
Cr = 0.007
Ar = 0.67 m2
r = 1.226 km/m3
g = 9.8 m/s2
Mr = 80 kg
In this case, the Mathematica NoteBook will not be of the type used for fortran and
C/C++. Look at the documentation on NDSolve for this problem.
Your answer to this question should include:
(a)The algorithms used and the design of your NoteBook
(b)The Mathematica NoteBook with your code and solution (I run your NoteBook).
(c)The results from the test case above.

This latest version of Mathematica has some issues with copying and pasting
equations and graphics from Microsoft Word. It also seems not to be able to
reproduce subscripts and superscripts from word, which previously was able to
do. It is not clear if the incompatibility is from the Microsoft Word or the
Mathematica side. This is another indication of how changes in versions of
software can have unintended consequences.
(Question 1): (a) Write Mathematica NoteBook which generates a table of error function (erf) and
its derivatives for real arguments (z) between
-3 and 3 in steps of 0.25. The error function is defined by the equation below (but is rarely
evaluated by performing the
integration).
(see http://mathworld.wolfram.com/Erf.html for information the error function
)
The values in the table should be given with 5 decimal places. The table should have headers
explaining what the columns are. Explain how you designed the NoteBook and give an example
of the output.
(b) How would you change this NoteBook if 10 significant digits were required?
Mathematica NoteBook should also be supplied
● Solution: Given that the error function is defined in Mathematica themost tricky part of the this problem is
formarting the ourput. There are many ways to approach the output problem.

Off@General::spell1D;
H* Stops message about head looking like a command*L head =
8"Argument", "Erf@zD", "dErf@zDêdz"<; derf@arg_D := 2 ê Sqrt@PiD *
Exp@-x^2D;
ents =
Table@8N@SetAccuracy@x, 10D, 3D, In[1]:=
N@SetAccuracy@Erf@xD, 10D, 6D, N@SetAccuracy@derf@xD, 6D, 6D<,
8x, -3, 3, 0.25<D;
full = Insert@ents, head, 1D; Out[6]//TableForm=
TableForm@full, TableAlignments Ø RightD
Argument Erf@zD dErf@zDêdz
-3.00 -0.999978 0.00014
-2.75 -0.999899 0.00059
-2.50 -0.999593 0.00218
-2.25 -0.998537 0.00714
-2.00 -0.995322 0.02067
-1.75 -0.986672 0.05277
-1.50 -0.966105 0.118930
-1.25 -0.922900 0.23652
-1.00 -0.842701 0.41511
-0.750 -0.711156 0.64293
-0.500 -0.520500 0.87878
-0.250 -0.276326 1.06001
0.µ 10-10 0.µ 10-10 1.12838
0.250 0.276326 1.06001
0.500 0.520500 0.87878
0.750 0.711156 0.64293
1.00 0.842701 0.41511
1.25 0.922900 0.23652
1.50 0.966105 0.118930
1.75 0.986672 0.05277
2.00 0.995322 0.02067
2.25 0.998537 0.00714
2.50 0.999593 0.00218
2.75 0.999899 0.00059
3.00 0.999978 0.00014
(b) Output of table to 10 deciminal places.

Off@
General::spell1D
; H
* Stops message about head looking like a command*L head =8"Argument",
"Erf@
zD
", "dErf@
zD
êdz"<;
In[22]:=
derf@
arg_D := 2 êSqrt@
PiD* Exp@
-x^2D
;
ents =Table@
8
N@
SetAccuracy@
x, 10D
, 3D
, N@
SetAccuracy@
Erf@
xD
, 10D
, 10D
, Out[26]//TableForm=
N@SetAccuracy@derf@xD, 10D, 10D<, 8x, -3, 3, 0.25<D;
full =Insert@
ents, head, 1D
; TableForm@full, TableAlignments Ø RightD
Argument Erf@zD dErf@zDêdz
-3.00 -0.999977910 0.000139253
-2.75 -0.999899378 0.0005862772
-2.50 -0.999593048 0.002178284
-2.25 -0.998537283 0.0071423190
-2.00 -0.995322265 0.0206669854
-1.75 -0.986671671 0.0527749959
-1.50 -0.9661051465 0.1189302892
-1.25 -0.9229001283 0.2365211224
-1.00 -0.8427007929 0.415107497
-0.750 -0.7111556337 0.642931069
-0.500 -0.5204998778 0.878782579
-0.250 -0.2763263902 1.060014129
0.µ 10-10 0.µ 10-10 1.128379167
0.250 0.2763263902 1.060014129
0.500 0.5204998778 0.878782579
0.750 0.7111556337 0.642931069
1.00 0.8427007929 0.415107497
1.25 0.9229001283 0.2365211224
1.50 0.9661051465 0.1189302892
1.75 0.986671671 0.0527749959
2.00 0.995322265 0.0206669854
2.25 0.998537283 0.0071423190
2.50 0.999593048 0.002178284
2.75 0.999899378 0.0005862772
3.00 0.999977910 0.000139253

period after the middle initial. If the names start with lower case letters, then these should be
capitalized. The NoteBook should not be specific to the lengths of your name (ie., the NoteBook
should work with anyone’s name.
As an example. An input of thomas abram herring
would generate:
Herring, Thomas A.
● This problem is not too bad to solve. This solution works in >5.0 Mathematica and does
explicitly some things such as splitting a string apart that are now Mathemat- ica commands
(StringSplit).

In[27]:=
H
* Define a function that will convert chararacter of a string to upper case *L
confirst@a_D := StringReplacePart@a, ToUpperCase@StringTake@a, 81<DD,81, 1<D;
H
* Get the name from the user*L
inname =InputString@"Enter Name Hfirst, middle, lastL "D;
H
* Convert whole string to lower case*L fullname =
ToLowerCase@innameD;
H
* Now get the list of blanks in the string*L posblanks =
StringPosition@fullname, " "D;
H
* Get the position of first blank*L
posfirst = Extract@
Extract@
posblanks, 1D
, 1D
; firstname =
StringTake@
fullname, posfirst- 1D
;
firstname = confirst@
firstnameD
; H
* Use our confirst routine *L H
* Get Middle Name
*L
posmid = Extract@
Extract@
posblanks, 2D
, 1D
;
midinit = StringTake@fullname, 8posfirst+1, posfirst +1<D;midinit =
confirst@midinitD;
H
* Get Last Name *L
lastname = StringTake@fullname, 8posmid+1, StringLength@fullnameD<D; lastname =
confirst@lastnameD;
H
* Output the string *L
finalname = lastname <> ", " <> firstname <> " " <> midinit <> "."; outline = "Input
name " <> inname <> " converted to: " <> finalname; H
* Output the results*L
Print@outlineD;
Input name thomas abram herring converted to: Herring, Thomas A.
Write a Mathematica NoteBook that will compute the motion of a bicyclist and the energy used cycling
along an oscillating, sloped straight-‐line path. The path followed will be expressed as
H(x)  Sx  Asin(2x /)  B cos(2x / )

where H(x) is the height of the path above the starting height, S is a slope in m/m, A and B are
amplitudes of sinusoidal oscillations in the path. The wavelength of the oscillations is . The forces
acting on the bicycle are:
Wind Drag Rolling
Drag
Fd 1/2ArCdV 2
Fr  MrgCr
where Ar is the cross-‐sectional area of the rider, Cd is the drag coefficient, r is the density of air and V is the velocity
of the bike. For the rolling drag, Mr is the mass of the rider and bike, g is gravitation acceleration and Cr is rolling
drag coefficient.
The bicyclist puts power into the bike by pedaling. The force generated by this power is given by
Rider force Fr  Pr /V
where Fr is the force produced by the rider, Pr is power used by the rider and V is velocity that the bike is traveling
(the force is assumed to act along the velocity vector of the bike). Your NoteBook can assume that the power can
be used at different rates along the path. The energy used will be the integrated power supplied by the rider.
Assume that there is maximum value to the rider force.
Your code should allow for input of the constants above (path and force coefficients). The NoteBook can assume a
constant power scenario and constant force at low velocities.
As a test of your NoteBook use the following constants to compute:
(a) Time to travel and energy used to travel 10 km along a path specified by S=0.001, A=5.0 m, B=0.0 m and =
2km, with constant power use of Pr =100Watts

and a maximum force available of 20N.
(b)
(c)
The position and velocity of the bike tabulated at a 100-‐second interval. Add
graphics to your NoteBook which plots the velocity of the bike as a
function of time and position along the path.
Assume the following values
Cd = 0.9
Cr = 0.007
Ar = 0.67 m2
 = 1.226 kg/m3
g = 9.8 m/s2
Mr = 80 kg
In this case, the Mathematica NoteBook will not be of the type used for fortran and C/C++. Look at the
documentation on NDSolve for this problem.
Your answer to this question should include:
(a) The algorithms used and the design of your NoteBook
(b)The Mathematica NoteBook with your code and solution (I run your NoteBook).
(c) The results from the test case above.
● There are several ways of approaching this problem and two solution are presented below. The problem itself is
dvided into a number of cells that allow parts of problem to be re-execuated. There two basic steps to the
solution:
(1)Use NDSolve to solve the second order differential equation that describes the problem. The solution
effectively becomes an equation that returns values at any specified time.

(2) Use FindRoot to determine when the bike has reached the end of track.
cr = 0.007;
area
mass
● Set up the defaults first. This cell should be evaluated
Clear@t, x, xd, hx, vx, xp, xd, th, daccD; cd =
0.90 ;
H*Print@"Parameter cd ",cdD*L
= 0.67; H* m^2 *L
= 80; H* kg *L
prider = 100; H* Power Watts*L
fmax = 20;
slope = 0.001;
as = 5.0;
bs = 0.0;
lambda = 2000;
tracklen = 10 000;
outint = 100.0;
H* Define the acceleration functions we will need*L grav =
9.8 ;
rhoair = 1.226; Print@"Default Values
set"D
Default Values set
● Now allow the user to enter values of there own. This cell does not need to be evaluated if
the default values are desired.

In[82]:=
Input@"Drag Coefficient ", cdD; Input@"Rolling friction coefficient ",
crD;
tracklen = Input@"Length of track HkmL ", tracklen ê 1000D *
1000.0;
slope = Input@"Track slope ", slopeD; as = Input@"Sin
Amplitude HmL ", asD; bs = Input@"Cos Ampliude Hm L", bsD;
lambda = Input@"Periodic wavelength HkmL ", lambda ê 1000D *
1000.0;
mass = Input@"Rider+Bike Mass HkgL ", massD; area =
Input@"Rider Area Hm^2L ", areaD;
cd =
cr =
outint = Input@"Output interval HsL", outintD;
o Now set up the force model equations
In[92]:=
theta@xp_D :=
ArcTan@slope + as * Cos@2 * Pi * xp ê lambdaD * 2 * Pi ê lambda -
bs * Sin@2 * Pi * xp ê lambdaD * 2 * Pi ê lambdaD htrack@xp_D := slope * xp +
as * Sin@2 * Pi * xp ê lambdaD +
bs * Cos@2 * Pi * xp ê lambdaD;
H* Second derivative of surface for computing cenripical
acceleration *L
dy2dx2@xp_D := - as * Sin@2 * Pi * xp ê lambdaD * H2 * Pi ê lambdaL^2 -
bs * Cos@2 * Pi * xp ê lambdaD * H2 * Pi ê lambdaL^2; vmag@xd_, zd_D :=
Sqrt@xd^2 + zd^2D;

H* Acceleration due to drag *L
- rhoair * vmag@xd, zdD * area * cd *
- rhoair * vmag@xd, zdD * area * cd *
dragx@xd_, zd_D :=
xd ê H2 * massL;
dragz@xd_, zd_D :=
zd ê H2 * massL;
H* Rolling
rollx@xp_D
rollz@xp_D
H* Gravity
gravx@xp_D
gravz@xp_D
force. This act along surface *L
:= - grav * cr * Cos@theta@xpDD;
:= - grav * cr * Sin@theta@xpDD;
force acting perpendicular to surface *L
:= -grav * Cos@theta@xpDD * Sin@theta@xpDD;
:= +grav * Cos@theta@xpDD^2;
H* Centripetal acceleration *L
H* Could divide by Sqrt@H1+Tan@theta@xpDD^2L^3DD
factor to curvature term. This change does not seem
to make solution closer to track shape*L

centx@xp_,xd_,zd_D :=
Hxd^2+ zd^2L* Sin@theta@xpDD* dy2dx2@xpD; centz@xp_,xd_,zd_D :=
Hxd^2+ zd^2L* Cos@theta@xpDD* dy2dx2@xpD;
H* This commented code has expression for radius of curvature
centx@xp_,xd_,zd_D :=
Hxd^2+zd^2L*Sin@theta@xpDD*
dy2dx2@xpDêSqrt@H1+Tan@theta@xpDD^2L^3D; centz@xp_,xd_,zd_D :=
Hxd^2+zd^2L*Cos@theta@xpDD*
dy2dx2@xpDêSqrt@H1+Tan@theta@xpDD^2L^3D;
*L
H* To see the effects of the centripetal force, remove the comments below
centx@xp_,xd_,zd_D :=0;
centz@xp_,xd_,zd_D :=0;
*L
H* Now the rider force *L
ridrx@xp_,xd_,zd_D :=
HMin@If@vmag@xd,zdD> 0, priderêvmag@xd,zdD,fmaxD, fmaxDêmassL*
Cos@theta@xpDD;
ridrz@xp_,xd_,zd_D :=
HMin@If@vmag@xd,zdD> 0, priderêvmag@xd,zdD,fmaxD, fmaxDêmassL*
Sin@theta@xpDD;
Print@AccelerationfunctionsSetD
AccelerationfunctionsSet
● The following cell can be used to test that the acceleration functions above generate numeric
results when called with distance and velocity. This test is useful if NDSolve does not return
answer.

In[121]:=
dx = 1500.; txv = 2.0; tzv = 0.01; Print@"Slope ",
theta@dxD, " 2nd derivative ",
dy2dx2@dxD, " Height ", htrack@dxDD
Print@"Drag X ", dragx@txv, tzvD, " Z ", dragz@txv, tzvDD
Print@"Roll X ", rollx@dxD, " Z ", rollz@dxDD Print@"Gravity
X ", gravx@dxD, " Z ", gravz@dxDD Print@"Centripetal X ",
centx@dx, txv, tzvD, " Z ",
centz@dx, txv, tzvDD
Print@"Rider X ", ridrx@dx, txv, tzvD, " Z ",
ridrz@dx, txv, tzvDD
Slope 0.001 2nd derivative 0.000049348 Height -3.5
Drag X -0.0184822 Z -0.0000924109 Roll X -0.0686 Z -
0.0000686 Gravity X -0.00979999 Z 9.79999
Centripetal X 1.97397 µ 10- 7 Z 0.000197397
Rider X 0.25 Z 0.00025
● Now set up the solution for NDSolve. There are two equations in the x and z accelerations
and initial conditions for x and z and x' and z' at time zero. The solution is set to solve for a
maximum of 10000 seconds. This is OK for the standard case but may need to be modified
for other longer running cases.
The evaluaton of the solution is saved for x and z positions and z and z velocities.
(NDSolve contains examples of setting up these solutions.
We can then find the length of the time need by solving the equation xp[t]-tracklen
== 0. This is not with FindRoot.

In[177]:=
solution =
NDSolve@
8x''@tDã ridrx@x@tD,x'@tD,z'@tDD+ dragx@x'@tD,z'@tDD+
rollx@x@tDD+ gravx@x@tDD+ centx@x@tD,x'@tD,z'@tDD,
z''@tDã ridrz@x@tD,x'@tD,z'@tDD+ dragz@x'@tD,z'@tDD+
rollz@x@tDD+ gravz@x@tDD- grav+
centz@x@tD,x'@tD,z'@tDD, x@0D== 0,z@0Dã 0, x'@0Dã 0,z'@0Dã 0<,8x,z<,8t,0,10000<D;
xp@t_D := First@Evaluate@x@tD ê. solutionDD; zp@t_D:= First@Evaluate@z@tDê.solutionDD;
xv@t_D:= First@Evaluate@x'@tDê.solutionDD; zv@t_D:= First@Evaluate@z'@tDê.
solutionDD;
H* Compute the error in the distance *L
endt = t ê. FindRoot@xp@tD- tracklenã 0,8t,1,3000.0<D;
H* Now integrate to get work done *L
work = NIntegrate@mass* Sqrt@xv@tD^2+ zv@tD^2D*
Sqrt@ridrx@xp@tD,xv@tD,zv@tDD^2+
ridrz@xp@tD,xv@tD,zv@tDD^2D,8t,0,endt<, AccuracyGoalØ 4D;
Print@"Time to reach end of track ",endt," sec, Speed ", xv@endtD," mês"D;
Print@"Work done ",work," Joules, ",workê4.1868ê10^3, " kcal"D;
NIntegrate::ncvb :
NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in t near 8t<= 8678.279<. NIntegrate obtained
174398.48726962117`
and 0.39321119853291564` for the integral and error estimates. à
Time to reach end of track 2255.17 sec, Speed 2.65009 mês Work done 174398. Joules, 41.6544 kcal
● The case above gives an example of how tricky Mathematica can be in telling you what it is actually doing.
In the integration above for work a numerical rounding error message is printed. The message suggests
there is some problem around 678 seconds. The cell below divides the calculation into 2 parts normally split
at around 678 seconds and in this case no numerical error warning is printed. It’s not at all clear why the
program is behaving this way.

In[166]:=
tt = 678.279;
worksplit =
NIntegrate@mass * Sqrt@xv@tD^2 + zv@tD^2D *
Sqrt@ridrx@xp@tD, xv@tD, zv@tDD^2 +
ridrz@xp@tD, xv@tD, zv@tDD^2D, 8t, 0, tt<D +
NIntegrate@mass * Sqrt@xv@tD^2 + zv@tD^2D *
Sqrt@ridrx@xp@tD, xv@tD, zv@tDD^2 +
ridrz@xp@tD, xv@tD, zv@tDD^2D, 8t, tt, endt<D; Print@"Work
computed in one step ", work,
" and split at time ", tt, " " , worksplit,
". Difference in results ", work - worksplitD;
Work computed in one step 174 399. and split at time
678.279 174 399.. Difference in results 0.
●Final output for version A (full solutions)
In[185]:=
Print@"12.010 HW 4: Mathematica Bike Problem"D;
Print@"Solution Parameters"D;
Print@"Track Length ", tracklen ê 1000.0, " km"D;
Print@"Track Slope ", slope, " Sin Cos ", as, " ",
bs, " m, Lambda ", lambda ê 1000., " km"D; Print@"Rider
Power ", prider, " Watts, Max Force ",
fmax, " N"D;
Print@"Time to reach end of track ", endt, " sec, Speed ",
xv@endtD, " mês"D
Print@"Work done by rider ", work, " Joules, ", work ê
4.1868 ê 10^3, " kcal" D

12.010 HW 4: Mathematica Bike Problem
Solution Parameters Track Length 10. km
Track Slope 0.001 Sin Cos 5. 0. m, Lambda 2. km Rider Power 100
Watts, Max Force 20 N
Time to reach end of track 2255.17 sec, Speed 2.65009 mês Work done by
rider 174 398. Joules, 41.6544 kcal

● Add some graphics to the results: Velocity versus time, Z position versus X position, and
difference from track shape. The latter plot shows how well our integration matched the actual
shape of the track.
In[241]:=
Plot@ xv@tD, 8t, 0, endt<,
AxesLabel Ø 8"Time HsecL", "Velocity HmêsL"<D
ParametricPlot@ 8xp@tD ê 1000 , zp@tD<, 8t, 0, endt<, AxesLabel Ø 8"X
position HkmL", "Z positionHmL"<, AspectRatio Ø 1 ê 1D
ParametricPlot@ 8xp@tD ê 1000 , Hzp@tD - htrack@xp@tDDL * 1000.<,
8t, 0, endt<,
AxesLabel Ø 8"X position HkmL", "Error in Z HmmL"<, AspectRatio Ø 1 ê
1D
Out[241]=
500 1000 1500 2000
Time
3
2
1
4
Velocity HmêsL
7
6
5

Out[242]=
2 4 6 8 10 X position
5
10
Z positionHmL

2 4 6 8 10
X position
- 0.4
- 0.3
- 0.1
Out[243]=
- 0.2
0.1
Error in Z HmmL
0.2
-0.5
● We now use table and table form to have to put the positions and velocities as
a function of the interval selected by the user. (The default interval is 100
seconds). To generate the table with 100s output, and to add the final value, we
generate the table in two parts and save the table in a list called outlst. We use the
append function to join the two tables together.

In[274]:=
outlst = Table@8t, xp@tD, zp@tD, xv@tD, zv@tD<,
8t, 0, endt, outint<D;
H* Now add the final entry to the tabe; *L
outlst =
Append@outlst,
Transpose@Table@8t, xp@tD, zp@tD, xv@tD, zv@tD<,
8t, endt, endt, outint<DDD;
TableForm@outlst,
TableHeadings Ø
8None, 8"Time HsL", "X Pos HmL", "Z pos HmL",
"X Vel HmêsL", "Z Vel HmêsL"<<D
Out[276]//TableForm=
Time HsL
0
X Pos HmL
0.
Z pos HmL
0.
X Vel HmêsL
0.
Z Vel HmêsL
-5.29396µ 10-23
100 79.9901 1.32329 1.48869 0.0241386
200 306.763 4.41341 3.36696 0.0335387
300 826.797 3.41531 6.79761 -0.0845564
400 1513.81 -3.48135 5.87698 0.00988077
500 1951.68 1.19559 2.95947 0.0489105
600 2202.42 5.17194 2.65422 0.0361969
700 2608.52 7.32075 5.8297 -0.0247871
800 3292.9 -0.685554 6.88819 -0.0586485
900 3834.83 1.35511 3.88251 0.0568391
1000 4117.37 5.91931 2.35353 0.0368374
1100 4428.12 9.30101 4.53331 0.0204786
1200 5046.56 4.31778 7.24973 -0.105412
1300 5681.7 1.47426 4.97372 0.0471862
1400 6035.25 6.58758 2.48085 0.0412101
1500 6294.17 10.2845 3.32052 0.0347469
1600 6803.7 9.69517 6.71443 -0.0793262
1700 7493.54 2.49435 5.98646 0.00407604
1800 7941.44 7.0264 3.03341 0.0498764
1900 8193.43 11.0479 2.60451 0.0361917
2000 8588.77 13.3952 5.72489 -0.0190286
2100 9269.22 5.52648 6.96317 -0.0655712
2200 9821.37 7.16001 3.99206 0.0570797
2255.17 10000. 9.99951 2.65009 0.0442763

tance along the sirface. These could be derived given that we have the eqautions. A
simpler solution to solve for the horizontal motion and compute the z-motion consistent
wth staying on the track. The z-motion is needed because drag and rider force depend on
the total velocity not just the horizontal velocity.
We use the same constants above but we re-define the accerations and NDSolve here.
In[284]:=
theta@xp_D :=
ArcTan@slope + as * Cos@2 * Pi * xp ê lambdaD * 2 * Pi ê lambda -
bs * Sin@2 * Pi * xp ê lambdaD * 2 * Pi ê lambdaD
H* Get total velocity given running on track *L
vtot@xd_, xp_D := Sqrt@xd^2 + Hxd * Tan@theta@xpDDL^2D;
dacc@xd_, xp_D := -cd * rhoair * vtot@xd, xpD^2 *
area ê H2 * massL;
racc@th_D := - grav * cr * Cos@thD; facc@xd_, xp_D
:=
Min@If@ vtot@xd, xpD > 0, prider ê vtot@xd, xpD, fmaxD, fmaxD ê
mass
gacc@th_D := - grav * Sin@thD; Print@"1-D
accelerations set"D
1-D accelerations set
● Alternative solution, Here we solve the one dimensional problem which is basi-
cally the roller coaster solution that keeps the bike on the ground. In the Fortran and
C-versions we computed the forces along the sloped surface and integrated
horizontal motions from the motion along the slope. If a similar approach is fol-
lowed here, then we need expressions for height and slope as functions of the dis-

H* Set up differential equations to be be solved, x@tD is
horizontal position,
There is still a problem here with along track versus horizontal
distance
Here we use y for the dependent variable*L solnb =
NDSolve@ 8y''@tD ã
Hdacc@y'@tD, y@tDD + gacc@theta@y@tDDD +
racc@theta@y@tDDD + facc@y'@tD, y@tDDL *
Cos@theta@y@tDDD,
y@0D ã 0, y'@0D ã 0.0<, y, 8t, 0.0, 10 000.0<,
AccuracyGoal Ø 10D;
py@t_D := Evaluate@y@tD ê. solnbD; vy@t_D :=
Evaluate@y'@tD ê. solnbD;
H* Compute the error in the distance *L
endb = t ê. FindRoot@py@tD - tracklen ã 0 , 8t, 1, 3000.0<D; work =
NIntegrate@mass * facc@First@vy@tDD, First@py@tDDD *
vtot@First@vy@tDD, First@py@tDDD, 8t, 0, endt<D; Print@"Time to
reach end of track ", endb, " sec, Speed ",
First@vy@endtDD, " mês"D;
Print@"Work done ", work, " Joules, ", work ê 4.1868 ê 10^3, " kcal" D;
Time to reach end of track 2255.05 sec, Speed 2.64848 mês Work
done 174 405. Joules, 41.6558 kcal
Plot@8vy@tD, Sqrt@xv@tD^2 + zv@tD^2D<, 8t, 0, endb<, PlotRange
Ø All,
AxesLabel Ø 8"Time HsecL", "Velocity HmêsL"<D
Plot@Hvy@tD - Sqrt@xv@tD^2 + zv@tD^2DL * 1000, 8t, 0, endb<, PlotRange
Ø All,
AxesLabel Ø 8"Time HsecL", "Diff Vel HmmêsL"<D
In[311]:=

Out[311]=
500 1000 1500 2000
3
2
1
4
Velocity HmêsL
7
6
5
Out[312]=
500 1000 1500 2000
Time
- 4
- 2
2
4
Diff Vel HmmêsL
6
● Now output table of values

In[308]:=
outl1d = Table@8t, py@tD, vy@tD<, 8t, 0, endb, outint<D;
H* Now add the final entry to the tabe; *L
outl1d =
Append@outl1d,
Transpose@Table@8t, py@tD, vy@tD<,
8t, endb, endb, outint<DDD;
TableForm@outl1d,
TableHeadings Ø
8None, 8"Time HsL", "X Pos HmL", "X Vel HmêsL"<<D
Out[310]//TableForm=
Time HsL X Pos HmL X Vel HmêsL
0 0. 0.
100 80.0296 1.48938
200 306.907 3.36857
300 827.102 6.79856
400 1514.08 5.87554
500 1951.82 2.95863
600 2202.58 2.65548
700 2608.89 5.83165
800 3293.34 6.88692
900 3835.08 3.8804
1000 4117.54 2.35409
1100 4428.5 4.537
1200 5047.16 7.25006
1300 5682.12 4.97145
1400 6035.47 2.48028
1500 6294.5 3.32362
1600 6804.39 6.7169
1700 7494.16 5.98318
1800 7941.75 3.03121
1900 8193.74 2.6065
2000 8589.47 5.72868
2100 9270.08 6.96082
2200 9821.86 3.98801
2255.05 10000. 2.65031

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