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SMJM 1013
Engineering Mathematics 1
ASSIGNMENT 2
APPLICATIONS OF MATHEMATICAL CONCEPTS
IN ENGINEERING
Title:
PLANE NAVIGATION
NAME MATRIC SIGNATURE
NO.
1. LIM JIA ZHENG A22MJ0076
2. LUM JIA MING A22MJ0082
3. MOHAMAD FAQIH AQHARI BIN AHMAD A22MJ0086
4. MOHAMAD NURALIMI BIN MD ZUKRI A22MJ5041
5.
NAME OF LECTURER SECTION GROUP
Dr. Amir Syafiq Syamin Shah Amir Hamzah 1 2
A.P. Dr. Nur’azah Abdul Manaf & Dr. Hafizah Farhah Saipan @ Saipol 3 4 3
Dr. Hafizah Farhah Saipan @ Saipol 5 6
As an example, you may choose the topic of complex number and the case study that you have
selected might be related to how the concept is being used to calculate power dissipation of an
element in a circuit involving alternating currents. If possible, include some sample
calculations as to how the concept are used in your case study, and also some computer
simulations to validate your calculations. Include reference in the report.
Prepare a report with a minimum 10 pages and maximum of 15 pages using the following
format:
1. Introduction
2. Literature Review
3. Methodology
4. Analysis of results and Discussions
5. Conclusions
6. References
7. Appendix (if any)
Type using MS Word, font Times New Roman with size 12, Line spacing 1.5.
• Submit only one (1) report for each group in PDF format with cover page and marking
criteria table.
• The report submission is via this link: https://forms.gle/AQdwehKr4wdjTR678 latest on
20th January 2022 (Friday, 12PM).
SMJM1013 ASSIGNMENT 2 1-2022/2023
Marks for the assignment will be given based on the criteria explained in Table 1:
Table 1: Marking Criteria
Evaluation Scale
Weightage Student
Criteria Marks Between Acceptable to
(%) Marks Below expectation Outstanding
Outstanding
Displayed a poor grasp of Displayed some grasp of Displayed an excellent
the material. the material. grasp of the material.
Topic Demonstrated a Demonstrated adequate Demonstrated good mastery
Knowledge superficial handling of mastery of content, of content, application and
3 30
PO1 CO1 content, application and application and implications. Good research
WP1 WK1 implications. Little depth implications. Research depth.
of research. not very deep.
Mark (16-20) Mark (21-25) Mark (26-30)
Method & Merely restating given Adequate method & Well written method &
Application method & poor ability to ability to apply theories. excellent ability to apply
1.5 15
PO2C02 apply theories. theories.
WP1 Mark (7-9) Mark (10-12) Mark (13-15)
Simply restating gathered Adequate analysis and Complete analysis with
Analysis and data and information. discussion with some good supporting evidence.
Interpretation supporting evidence. Describe relationship, trends
1.5 15
PO2C02 and etc. Discuss the finding
WP2 WK1 in relation to theory.
Mark (7-9) Mark (10-12) Mark (13-15)
Demonstrates little or no Demonstrates substantial Demonstrates extensive
Importance of understanding of the understanding of the understanding of the
lifelong learning 1 10 importance of lifelong importance of lifelong importance of lifelong
PO11CO4 learning. learning. learning.
Mark (2-4) Mark (5-7) Mark (8-10)
Demonstrates little or no Demonstrates appropriate Demonstrates in-depth level
Decision
level of decision-making decision-making skills. of engagement and decision-
making 1 10
skills. making skills.
PO11CO4
Mark (2-4) Mark (5-7) Mark (8-10)
Report Total 8 80
Peer
Assessment 2 20 Marks are given according to rubric for Peer Assessment.
PO10CO3
Report must be typed and submitted latest on 20th January 2022 (Friday, 12PM).
TOTAL 10% 100 Use the cover page given.
• Military Usage - Any piece of artillery that fires a projectile by employing gun power or
any other type of typically explosive-based propellant is considered to be a cannon.
• Projectile - The baseball vector is utilized automatically by players in sports like basketball
and baseball.
• Vector in Gaming - Vectors are utilized in the storage of locations, directions, and
velocities in video games.
• Roller Coaster - The majority of the motion that occurs during a roller coaster ride is a
reaction to the gravitational pull that the earth exerts.
• Crosswind - A wind that blows in a direction that is perpendicular to the path that one is
travelling is referred to as a crosswind.
LITERATURE REVIEW
A vector in a plane is represented by a directed line segment (an arrow). The endpoints of
the segment are called the initial point and the terminal point of the vector. An arrow from the
initial point to the terminal point indicates the direction of the vector. The length of the line
segment represents its magnitude. We use the notation ∥v∥ to denote the magnitude of the vector
v. A vector with an initial point and terminal point that are the same is called the zero vector,
denoted 0. The zero vector is the only vector without a direction, and by convention can be
considered to have any direction convenient to the problem at hand. Vectors with the same
magnitude and direction are called equivalent vectors. We treat equivalent vectors as equal, even
if they have different initial points. Thus, if v and w are equivalent, we write v = w.
The use of boldface, lowercase letters to name vectors is a common representation in print,
but there are alternative notations. When writing the name of a vector by hand, for example, it is
easier to sketch an arrow over the variable than to show it is a vector: v⃗. When a vector has initial
point P and terminal point Q, the notation PQ is useful because it indicates the direction and
location of the vector.
PROBLEM STATEMENT
A plane flies through wind that blows with a speed of 45 miles per hour in the direction 43° North
to West. In still air: the plane has a speed of 550 miles per hour; the plane is headed in the direction
60° North to East.
In this assignment, the method of navigation by calculating the true speed and direction of a plane
using vector is studied. A situation is made up to simulate the real situation of a plane flying at a
certain velocity and direction. The main mathematical concept utilized in this assignment are
addition of vectors.
First of all, the speed and direction of the plane in still air and those of wind are identified and
tabulated. The directions of the plane and wind are in the form of bearings, and are to be converted
into the form of standard direction. In order to convert the bearings into standard direction, the
bearing in the second quadrant is added with 90 degrees, while the bearing in the first quadrant is
subtracted from 90 degrees. The standard directions are then to be converted into their analytic
forms by using the formula v =< |𝑣 |𝑐𝑜𝑠𝜃 , |𝑣| 𝑠𝑖𝑛𝜃 >, where v denotes the speed and 𝜃 denotes
the standard direction.
A rough sketch of the vectors are prepared to obtain the true velocity of the plane in vector form
by using the concept of addition of vectors. The true speed of the plane is then calculated by using
Pythagoras Theorem, while the true direction of the plane is calculated using the formula:
PART 1: Identify speed and direction for: plane in still air; wind.
⃗⃗⃗⃗ )
Wind (𝑊 45 mph N 43° W
θw = 133° θp = 30°
PART 3: Convert from standard direction and magnitude to analytic form.
⃗⃗⃗⃗ =< |𝑊
𝑊 ⃗⃗⃗⃗ | cos 𝜃𝑤 , |𝑊
⃗⃗⃗⃗ | sin 𝜃𝑝 > 𝑃⃗⃗ =< |𝑃⃗⃗|𝑐𝑜𝑠𝜃𝑝 , |𝑃⃗⃗|𝑠𝑖𝑛𝜃 >
=< 45 cos 133 , 45 sin 133 > =< 600 cos 30 , 600 sin 30 >
=< 45 cos 133 , 45 sin 133 > + < 600 cos 30 , 600 sin 30 >
= < −30.6899,32.9109 > + < 519.6152 , 300 >
= < 488.9253 ,332.9109 >
332.9109
tan 𝜃|𝑃+𝑊| =
488.9253
332.9109
𝜃|𝑃+𝑊| = 𝑡𝑎𝑛−1
488.9253
= 34.2511°
PART 8: Drawing component of vector
⃗⃗⃗⃗ + 𝑃⃗⃗| =
|𝑊 ⃗⃗⃗⃗ |
|𝑊
591.5045mph = 45𝑚𝑝ℎ
34.2511°
|𝑃⃗⃗ | = 600𝑚𝑝ℎ
DISCUSSION
We found that by adding up the analytical form of the speed of the plane in still air and that
of the wind, the true velocity of the plane can be calculated. This was done by applying Pythagoras
Theorem by using the pair of values we obtain from the addition of analytical values of the two
vectors. Besides, the true direction, θ of the plane was obtained from the value of tan 𝜃|𝑃+𝑊| .
The utilization of vector addition in this problem is in accordance with the properties of
vector which we have learnt in lectures, and is important in investigating its roles in vector
navigation of airplane.
CONCLUSION
Vectors are very important in our daily life because they help us in understanding the
quantities like speed and direction. Applications in everyday life would be velocity of a car, or
throwing a ball. Vector quantities can be applied in other practical applications like sports,
travelling, daily transport, space travel, and lots more. In sports such as baseball, it's the vector that
helps the player and even in football, it's the knowledge of directions and magnitude that allow
you to make a perfect pass and a classy goal or even a fantastic save and that knowledge is nothing
without knowing and understanding vectors. It’s also a known fact that pilots and sailors need
vectors in their line of work so they can find out things such as headwind and tailwind, wind speed,
ground speed, etc.
So as engineering students, vectors also play a big role in the calculations we have to carry
out. This project is a perfect example of how vectors are applied. Calculations involving vectors
are used to calculate the true speed and direction of an object.
Overall, vectors are probably the most important tool to learn in all of physics and
engineering. Today, mathematics is an inevitable part of science and it is used in every field be it
engineering, economics, health and science and art. Applied mathematics has always been
leading to important discoveries and giving birth to new disciplines.
REFERENCES
• Morelej17, U. (2019, March 25). Vector addition in aircraft navigation. Linear Algebra
Applications S19. Retrieved January 17, 2023, from Vector Addition in Aircraft
Navigation – Linear Algebra Applications S19 (wordpress.com)
• NASA (n.d.). Exploring Space Through MATH: Applications in Precalculus.
www.nasa.gov. Retrieved January 3, 2023, from
https://www.nasa.gov/pdf/740534main_Precal-ED_Space%20Vectors.pdf
• Roberts, D. R. and F. (2013). Applications with Navigation and Vectors. Navigation and
vectors - mathbitsnotebook(geo - CCSS math). Retrieved January 14, 2023, from
https://mathbitsnotebook.com/Geometry/TrigApps/TANavigationVectors.html
• Zhu, F. (n.d.). 3.5 Orbital Mechanics. A Guide to CubeSat Mission and Bus Design.
Retrieved January 4, 2023, from https://pressbooks-
dev.oer.hawaii.edu/epet302/chapter/3-5-orbital-mechanics/