Module Title:

Maths and Engineering Science for Transportation

Module Leader:
Dr James Spendlove
(james.spendlove@shu.ac.uk)

Student Name:
[Enter Student Name]

Question 1: Mathematical Modelling and Analysis for a Satellite Orbiting Mars

1.1 Gravitational Potential and Field
Given
The gravitational potential around Mars is:

Where:

Let:

(a) Calculation of the Gradient of the Potential
The gradient operator is defined as:

So:

Step 1: Differentiate with respect to

Using the chain rule:

Step 2: Differentiate with respect to

Step 3: Differentiate with respect to

Final Gradient

(b) Determination of
The gravitational force is defined as:

So:

The divergence of a vector field is:

For a Newtonian gravitational field:

Result

This indicates that there are no sources or sinks of the gravitational field in empty space.

(c) Explanation
Gravitational potential is the energy linked to a mass at a specific point in a gravitational field.
The potential gradient gives the gravitational field a direction and scalar value.
The negative sign indicates that gravity is pulling.
The zero divergence indicates that gravitational fields in empty space are obeying the law of conservation.
This agrees with the classical orbital mechanics of Mars.

1.2 Orbital Parameters and Motion
Given trajectory

Substituting your values :

(a) Parameterisation of the Satellite Path
The position vector of the satellite is given by:

Using the last three digits of the student number (323):

Substituting these values gives:

So, the vector function gives you a parametric trajectory in three-dimensional space. The movement is such that it oscillates along the x and y axes while the z axis ascends linearly.
(b) Velocity
Velocity is the derivative of position:

Differentiate each component step by step:

Velocity vector

(c) Acceleration
Acceleration is the derivative of velocity:

Differentiate again:

Acceleration vector

(d) Area Swept in the Plane
Given:

Substitute into the formula:

Final Area

1.3 Orbital Dynamics and Control

Given
The total force acting on the satellite is:

Where:
is the gravitational force due to Mars
is an unknown perturbing force
The satellite has mass
From Question 1.1, the gravitational force is:

(a) Equations of Motion Using Newtons Second Law
Newtons Second Law states:

Substituting the total force expression:

Gravitational force components
From Question 1.1:

Perturbing force components
Let the perturbing force be written as:

Acceleration vector

Equations of motion (component form)
x-direction

y-direction

z-direction

Final result

Mathematics provides us with the right tools for doing physics: the mathematical form of physics.
(b) Stability and Corrective Manoeuvres for a Mars-Orbiting Satellite
During its Martian orbit, a satellite has to confront the open and the unexpected forces as well, with one of the latter being the gravitational pull. Without correction, these minor forces will steadily push the satellite beyond its intended orbital position.
Cause of perturbations
Non-uniformity in Mars mass distribution
Solar radiation pressure
Atmospheric drag (for satellites in low Mars orbit)
The attractions of Mars satellites Phobos and Deimos
Such forces are collectively referred to by the force vector pin in the motion equations

Relation to mathematical concepts
Gravity, being one of the nature's basic forces, links orbital mechanics and vector calculus. To put it differently, the gravitational force has its source in the potential's gradient.
The movement of satellites is ruled by the second-order differential equations known as the motion equations.
In the stability analysis, the deviations from the equilibrium positions are observed during a certain period.

Corrective manoeuvres
To maintain a stable orbit:
Small velocity changes (?v) are applied using thrusters
These adjustments counteract perturbing forces
Mathematically, this modifies to restore the desired trajectory
Such The control of the spacecraft's movements assures that the satellite follows a certain path, which is of utmost importance for communication, observation, and mission safety.

Conclusion
The equations derived in section (a) provide a mathematically accurate and complete structure for modeling the satellite's motion around Mars. Disruptions can be identified and managed, which allows engineers to come up with different tactics for making the orbital adjustments effectively.

Question 2: Mathematical Analysis of Aircraft Performance

2.1 Engine Thrust Projection onto Flight Path
Given
Thrust vector:

Flight direction vector (not normalised):

(a) Unit vector in the direction of flight
First, calculate the magnitude of :

Now normalise the vector:

(b) Scalar projection of thrust onto flight direction
The scalar projection is given by the dot product:

This is the effective thrust contributing to forward motion.

(c) Vector projection of thrust onto flight path
The vector projection is:

(d) Wasted thrust and percentage
First calculate total thrust magnitude:

Wasted thrust magnitude:

Percentage wasted:

Interpretation
Although the thrust vector is not quite in line with the flight path, over 99% of the total thrust generated by the propellant is for moving the vehicle ahead, which is a sign of the engines being well aligned. The minor wasted component is due to the losses in lateral and vertical thrust.

2.2 Aircraft Dynamics
Given
The aircraft trajectory is defined as:

Using your confirmed student sequence 323:

(a) Parameterised position vector
Substituting values:

This represents motion at constant speed in the -direction, with fixed lateral position and altitude.

(b) Velocity vector
Velocity is the first derivative of position:

Differentiate each component:

(c) Acceleration vector
Acceleration is the derivative of velocity:

Explanation
Since the aircraft moves with constant velocity and no change in direction, its acceleration is zero. This is consistent with straight-line, steady-speed flight.

2.3 Aircraft Communication System of Linear Equations

Given system of equations

The task is to find the point where all three equations are satisfied simultaneously.

(a) Matrix form
We first rewrite the system in matrix form.
Coefficient matrix :

Unknown vector :

Constant vector :

Matrix equation

(b) Solving the system
We solve the system using simultaneous equations:

Step 1: Eliminate variables
From equation (1):

Step 2: Substitute into equation (2)

Substitute from (4):

Expand:

Step 3: Substitute into equation (3)

Substitute from (4):

Expand:

Step 4: Solve equations (5) and (6)

Subtract (5) from (6):

Convert to common denominator:

Step 5: Find
Substitute into (5):

Step 6: Find
Substitute , into (4):

Convert to denominator 14:

Final solution

(c) Verification (VERY IMPORTANT)
Equation (1)

Equation (2)

Equation (3)

Question 3: Vehicle and UAV Dynamics and Control

3.1 First-Order ODE Car Braking with Air Resistance
Given differential equation

Where:

Initial condition:

Step 1: Rearrange the equation
Divide both sides by :

Substitute values:

(a) Separation of variables
Rearrange:

Step 2: Integrate both sides

Left-hand side:

So:

Step 3: Solve for
Divide by 20:

Exponentiate:

(b) Apply initial condition

Final velocity function

Interpretation
Velocity decays exponentially
Braking force and air resistance both contribute
Car eventually comes to rest when

3.2 Projectile Motion of a UAV Dropping a Payload

Assumptions
No air resistance
Constant gravity
Payload released with zero vertical velocity
UAV moves horizontally at constant speed

(a) Equations of motion
Horizontal motion:

Vertical motion (taking ground as ):

(b) Time when payload hits the ground
At impact:

(c) Horizontal distance travelled (range)

(d) UAV position relative to payload at impact
UAV horizontal position at :

Payload horizontal position:

Vertical position of UAV:

Vertical position of payload:

Relative position

(e) Assumptions summary
No drag forces
Flat terrain
Constant gravity
Payload does not affect UAV motion
These simplify the analysis and are standard in first-order projectile modelling.

3.3 Rocker Arm Force Analysis

(a) Free-Body Diagram
For rocker arm ABC:
Pin reactions at B: ,
Downward forces:

Spring force acting along line DF toward D
All distances and angles taken from the given figure

Sign conventions
Positive : right
Positive : upward
Positive moments: counter-clockwise

(b) Equilibrium equations

Sum of forces in

Sum of forces in

Sum of moments about point B
Moments of pin reactions are zero.

Solving:

(Where are perpendicular distances from the diagram.)

Final results
Spring force obtained from moment balance
Pin reactions obtained from force balance
All equations satisfy static equilibrium