**Question 1: Rocket Flight (35 Marks) [Updated 30/08/2019]**

The flight of a test rocket weighing 25 metric tonnes can be modelled as follows. At time t = 0, the rockets engines generate exactly the right amount of upward thrust to maintain the rocket against gravity. From t = 0 to 10s, the thrust increases linearly to a maximum of 50000 kgf (kilogram-force). The engines then shut down and the rocket moves only under the force of gravity. After it reaches the apex the rocket starts to fall back down. When the magnitude of the downwards velocity reaches v = 30 m/s, a parachute opens and the rocket continues to move down at a constant speed of vchute = 20 m/s until it hits the ground (assume parachute deployment and change in velocity are instant).

Write a MATLAB program that calculates and plots the speed and altitude of the rocket as a function of time during the flight.

Tips for analysis of the problem

• The force of gravity, g = 9.81 m/s2

• The rocket may be assumed to be a particle that moves along a straight line in the vertical plane.

• Make sure all your values are converted to consistent SI units

A. Stage 1: The first 10s when the rocket engine is on. During this period, the rocket moves up with an acceleration determined by,

where, F is the instantaneous thrust of the engines. The velocity and height as a function of time are ð‘£ð‘£(ð‘¡ð‘¡) = 0.4905ð‘¡ð‘¡2 and â„Ž(ð‘¡ð‘¡) = 0.1635ð‘¡ð‘¡3.

The initial velocity and initial position are both zero. The time, velocity, and height at the end of this stage are t1, v1, and h1

B. Stage 2: The motion from when the engine stops until the parachute opens. In this stage, the rocket moves with a constant deceleration, g. The speed and height of the rocket as a function of time are given by:

The time and height at the end of this stage are t2, and h2

C. Stage 3: The motion from when the parachute opens until the rocket hits the ground. In this stage, the rocket moves with constant velocity (zero acceleration). The height as a function of time is given by:

where the chute is the constant velocity after the parachute opens.

D. Does the rocket reach a height of 1km measured from its starting position? If yes, what is the overshoot? If not, then what maximum thrust is required to achieve this height?

Note: You have to re-derive the expressions for ð‘£ð‘£(ð‘¡ð‘¡) and â„Ž(ð‘¡ð‘¡) in Stage 1 if you change maximum thrust.

**Question 2: Solution of equation (30 Marks)**

Mechanical engineers, as well as most other engineers, use thermodynamics extensively in their work. The following polynomial can be used to relate the zero-pressure specific heat of dry air, ð‘ð‘ð‘ð‘ kJ/(kg K), to temperature (K):

Tips for analysis of the problem

• Look at the lecture notes from Week 2 for methods to solve for roots of an equation

• Choose a method based on its merits as detailed in the lecture. Note there is no one correct answer. You have to justify your choice.

• Take help from the MATLAB code provided with the lecture and the tutorial to implement your chosen method

A. In MATLAB, plot the graph of ð‘ð‘ð‘ð‘ vs. T to get a rough idea of the behavior of this function.

B. Develop an M-file, without the use of MATLAB built-in function “fzero”, to determine the temperature that corresponds to specific heat of 1.16 kJ/(kg K). Justify your choice of method used to solve the above problem.

C. Use the MATLAB built-in function fzero to determine the temperature that corresponds to specific heat of 1.16 kJ/(kg K). Is there any difference between this solution and the one found in the previous step B? If yes, then give reasons as to why this may be the case, and how you may reduce or eliminate these differences.

**Question 3: Cable tension analysis (35 Marks)**

An object of mass M kg is hung from the end of a rigid 2.5m long horizontal pole of negligible weight. The pole is attached to the wall by a pivot and is supported by a 2.2m cable that is attached to the wall at a higher point as shown in the figure below. The tension in the cable can be calculated as,

where, ð‘‡ð‘‡ is the tension in the cable (N), M is the mass of the object (kg), ð‘™ð‘™ð‘ð‘ is the length of the cable (m), ð‘™ð‘™ð‘ð‘ is the length of the pole (m) and ð‘‘ð‘‘ is the distance between the wall and the attachment point of the cable to the pole (m). Set M as the sum of the ages of all team members in your team.

Tips for analysis of the problem

• Look at the lecture notes from Week 4 for optimization methods

• Golden-section search is similar to bracketing methods. Use the code provided for bracketing methods as inspiration for your own solution.

• Look up and understand the MATLAB find function (type “help find” in MATLAB) and how it works.

A. In MATLAB, plot a graph of cable tension as a function of ð‘‘ð‘‘, with ð‘‘ð‘‘ going from 0.5 to 2.0m.

B. Develop a MATLAB file that implements the Golden-Section search method to find the value of ð‘‘ð‘‘ that produces the minimum tension in the cable.

C. Use the MATLAB built-in function find to solve the same problem as in B. (i.e. find ð‘‘ð‘‘ that produces minimum tension). Is there any difference between this solution and the one found in the previous step B? If yes, then give reasons as to why this may be the case, and how you may reduce or eliminate these differences.