Mathematical model for cooling process

Question

General Information

i) All questions in the tasks must be completed precisely with adequate detail to gain the pass criteria.

ii) All submissions to be electronic in the MS Word format. All answers should be clearly identified as to that task and question they refer to.

Task 1

Analyse engineering problems and formulate mathematical model using first order differential equations

Question1. A heated object is allowed to cool in a room temperature which has a constant temperature of T₀:

a. Evaluate the cooling process

b. Formulate mathematical model for cooling process.

Question2. At time   t= 0 water begins to leak from a tank of constant cross-sectional area A.  The rate of outflow is proportional to the h, the depth of water into the tank at time t.  Write the constant of proportion kA where k is constant.

a. Evaluate the tank leaking process.

b. Formulate mathematical model for leaking process.

Write conclusions based on your formulated mathematical model for the leaking process.

Task 2

Solve first order differential equations by using numerical and analytical methods.

Question3. Find out the solution of following equations:

a). di/dt = t(2-5i), i(0) = 10

b). t² dQ/dt + tQ =2

Question4. Use the Euler method with the step size shown to advance four steps from given initial condition with given differential equation
dV/dt = 2t + v, V(0) = 1; h = 0.1

Task 3

Analyse engineering problems and formulate mathematical model by using second order differential equations.

Question5. For simple model of a shock observer shown in figure below:

a. Analyse the model, vertical motion of the mass.

b. Formulate mathematical model of the model

Task 4

Resolve second order homogeneous and non- homogenous differential equations.

Question6. Find out the general solutions of the following equations:

i. d²y/dx² + dy/dx – 2y = 0

ii. d²y/dx² + 9y = 0

Question7. Find out the general solutions of the following equation that satisfy the given initial conditions.

d²q/dt² + dq/dt + q = t² - 1
q(0)=0
dq/dt = 0, t=0

Task 5

Apply first and second order differential equations to solution of engineering situations

Question8. The velocity v of a rocket attempting to escape from the earth’s gravitational field is given by:
v  dv/dr = g R²/r²
Where:
r    is its distance from the centre of the earth and
R    is the mean radius of the earth

Find out a formula for V(r) and verify the minimum launch velocity V₀ in order that rocket escapes.