Calculate 𝑝̅ − 𝑟̅. Find the Unit Vector of 𝑞ത. Find the angle between the vectors 𝑞ത and 𝑟̅. Find a vector perpendicular to vectors 𝑝̅ and 𝑞ത.

These questions are designed to help you applying your learning from the nodes to the project, and
formed part of the module assessments. You are required to submit both your handwritten and MATLAB computed solutions in your individual report.

Question 1A: Hand-written
If vector 𝑝̅ = 𝚤̂ + 3𝚥̂ − 2𝑘෠ , 𝑞ത = 3𝚤̂ + 2𝚥̂ + 𝑘෠ and 𝑟̅ = 2𝚥̂ + 3𝑘෠
Determine the following:
i) Calculate 𝑝̅ − 𝑟̅.
ii) Find the Unit Vector of 𝑞ത.
iii) Find the angle between the vectors 𝑞ത and 𝑟̅.
iv) Find a vector perpendicular to vectors 𝑝̅ and 𝑞ത.
If 𝐴 = ቂ3 2
4 1ቃ , 𝐵 = ቂ5 −1 2
1 10 2ቃ , 𝐶 = ൥
2 4
1 −1
3 −2
൩ and 𝐷 = ൥
1 2 1
−1 0 4
3 −1 2
൩
v) Calculate BT – 2C + DC
vi) Calculate the determinant of AAT
vii) Find product of AA-1
viii) Find the inverse of BC

(Hint! Re-visit the Vectors and Basic Matrices node to re-cap on the Engineering Science &
Mathematics principles in solving the problem above.)

Question 1B : MATLAB Computation
Compute your calculation in Question 1A (i)-(viii) by writing a MATLAB script called Question1B.m.
(Hint! Look up the descriptions for MATLAB functions dot, cross and inv in the MATLAB Help system.)

Question 2A: Hand-written
Let the projectile trajectories of a cannon ball using equations for ideal projectile motion:
𝑦(𝑡) = 𝑦଴ − ଵ
ଶ 𝑔𝑡ଶ + ൫𝑣଴𝑠𝑖𝑛(𝜃଴)൯𝑡 Eqn. 1
𝑥(𝑡) = 𝑥଴ + ൫𝑣଴𝑐𝑜𝑠(𝜃଴)൯𝑡 Eqn. 2
where 𝑦 is the vertical distance and 𝑥 is the horizontal distance travelled by the projectile in metres,
gravitational acceleration 𝑔 = 9.8 ms-2 and 𝑡 is time in seconds. Let us also assume the initial ball
velocity 𝑣଴ = 35 ms-1, the projectile’s launching angle 𝜃଴= 5π/12 radian, initial vertical (height) and
horizontal positions are 𝑦଴ = 100m and 𝑥଴ = 0 respectively.

i) Solve for 𝑥 and 𝑦 , with 𝑡 representing the first 10 seconds. Sketch 𝑦 vs. 𝑡 and 𝑥 vs. 𝑡 in two separate graphs, and give appropriate titles to the graphs and label the axes.

ii) Find the exact time when the ball hits the ground and at what horizontal distance.

iii) To better visualise the projectile trajectory, sketch a new graph consisting both altitude on the 𝑦-axis and horizontal distance on the 𝑥-axis.

iv) Use the following adjusted angles to create two more trajectory plots on top of existing 𝑦 vs. 𝑥 sketched in (iii) and determine which launching angle results in a greater range.
𝜃଴
ଵ = ቀହగ
ଵଶ − 0.255ቁ radian Eqn. 3
𝜃଴
ଶ = ቀହగ
ଵଶ − 0.425ቁ radian Eqn. 4

v) Modify and re-write the equations (e.g. Eqn. 1 and Eqn. 2) so that the launching angles will be
insert directly as degree rather than radian.

(Hint! Re-visit the Kinematics node for the Engineering Science & Mathematics principles.)

Question 2B : MATLAB Computation
Compute your calculation in Question 2A (i)-(v) by writing a MATLAB script called Question2B.m. In
addition, compute the following as well:

vi) Adjust the launching angle to produce the greatest range (i.e. maximum horizontal distance)
but within 0° ≤ 𝜃଴ ≤ 90° in incremental of 0.1. Evaluate your results.

vii) Adding the optimal trajectory on top of existing 𝑦 vs. 𝑥 sketched in (iii).

(Hint! Look up the descriptions for MATLAB functions min, max, find and hold on/off in the MATLAB
Help system.)

Question 3A: Hand-written
In the circuit shown in Figure 1, the DC voltage sources are given as 120V, 60V and 10V, whereas the
DC current source is 36A. The value of the resistors are given as R1= 20Ω, R2= 5Ω, R3= 4Ω, R4= 6Ω, R5=
8Ω and R6= 4Ω.

Figure 1
i) Determine the total power dissipation of all the resistors e.g. RTotal?
ii) Identify the resistor with the highest power dissipation value?

(Hint! Re-visit the Basic Electrical Circuits, Resistive Circuits and Equivalent Circuits & Power Transfer node to re-cap on the Engineering Science & Mathematics principles in solving the problem above.)

(Hint! Potentially more than one approach (e.g. Nodal Analysis, Mesh Analysis, 1st and 2nd
Kirchhoff’s Law and a mixture of suggested principles) can be used to address the circuit analysis).

iii) Figure 2 is the exact circuit illustrated in Figure 2. However, we are now required to determine the current through R1= 20Ω by using Source Transformation method ONLY and finishing off with 2nd Kirchhoff’s Law.

(Hint! Recap on the Source Transformation method, simplify and analyse the circuit starting from
section that is non-restricted.)

Figure 2
IR1 = ?

Directions of currents which you choose should indicate by arrows. Both magnitude and the sign of
the currents should be provided by the answers. You should also indicate the any loops you created
for the 2nd Kirchhoff’s Law or Mesh Analysis. You need to show your working, circuits simplification,
assumptions you made and explain your reasoning.

Question 3B : MATLAB Computation
Compute your calculation in Question 3A (i)-(ii) ONLY by writing a MATLAB script called

Question3B.m.
Formulate your solution in matrix format of Ax=b and you are also required to display the
following text ‘Resistor with the highest power dissipation value is ….. ’ once you identify the hight
value.

(Hint! Look up the descriptions for MATLAB functions find and fprintf in the MATLAB Help system.)

Question 4A: Hand-written
Figure 3
i) Draw the free body diagram (FBD) of the beam ABC as illustrate in Figure 3.

ii) Calculate the reactions at the wall as a result of the loadings. (Assuming counter-clockwise as
positive value).

iii) Sketch the shear force (SF) and bending moment (BM) diagram for the beam, and showing the
values.

iv) If the beam (in Figure 1) has a cross section as show in Figure 4, what are the maximum tensile
and compressive stress due to the bending. [in SI unit]v) Are there differences between the maximum tensile and compressive stress? Why?

Figure 4
(Hint! Re-visit the Statics A and Solid Mechanics 1 node to re-cap on the Engineering Science &
Mathematics principles in solving the problem above. Also, remember to re-cap on Dimensional
Analysis for unit (e.g. SI) conversion.

100 mm
75 mm
75 mm
25 mm
x
4 kN 6 kN
6 m 2 m
AB
C

Question 4B : MATLAB Computation
Compute your calculation in Question 4A (ii)-(iv) by writing a MATLAB script called Question4B.m.
Label the axes meaningfully and give the figure a title.

(Hint! Look up the descriptions for MATLAB functions plot, xlabel, ylabel, legend and title in the
MATLAB Help system.)

Question 5A: MATLAB Computation
Estimating the costs of drilling oil wells is an important consideration for the oil industry. Table 1
illustrates the total costs and the depths of 16 offshore oil wells located in country Z. In general, the
drilling cost of drilling an oil well depends on the depth at which you are drilling e.g. drilling becomes
more expensive, per meter, as you dig deeper.
Well Platform No. Depth (m) Cost ($ million)
1 1524 2.59
2 1585 3.33
3 1829 3.18
4 1993 3.19
5 2167 4.78
6 2303 5.91
7 2440 5.77
8 2501 8.09
9 2502 4.81
10 2621 5.62
11 2751 7.74
12 2803 6.79
13 3025 7.84
14 3296 8.88
15 4206 10.49
16 4362 12.51

Table 1
Compute the following calculation by writing a MATLAB script called Question5A.m.

i) Create a scatter diagram for the data set, and label the axes and name the figure meaningfully. Also, comment on whether there is any association between the drilling cost and the well depth e.g. strong or weak correlation and positive or negative relationship.

ii) Find the regression model (via Ordinary Least Squares estimates of 𝛽መ଴ and 𝛽መଵ) with cost as the dependent variable and depth as the explanatory or independent variable. State any assumptions you made and comment on whether you believe your equation is a good fit e.g. sum of errors and proportion of variance explained.

iii) An alternate implementation would be to use the MATLAB polynomial curve-fitting function e.g. polyfit and polyval, to compute the Ordinary Least Squares estimates of 𝛽መ଴ and 𝛽መଵ. Repeat the task in (ii) again, but this time with using the built-in functions. Plot the resulting linear regression model with the data set, and label the axes and name the figure meaningfully.

iv) What cost would you predict for an oil well of 4000m?

v) What is the estimate of the error variance?

vi) What could you say about the cost (predicted) of an oil well of depth 6500m?