Reference no: EM132284644

Question 1. Newton's law of cooling of a uniform body with temperature T in an environment with constant temperature T_e is given by the following:
dT/dt = -k(T - T_e)

where t is time and k is a positive constant.

a) Explain the significance of the minus sign in front of k (i.e. why it is not positive).

b) The solution is T(t) = T_e + A_e^-kt for A ∈ R an arbitrary constant. Determine the value of k for a body that has T(0) = 4T, and has cooled to half that temperature in 10 minutes.

Question 2. The size of a bacteria population P(t) in units of millions of individuals, where t is time in days, grows and decreases according to the following model.

dP/dt = [(2Π/365) cos(2Πt/365)]P

a) Explain (in words) how the relative rate of change predicted by the model could account for a seasonal influence. (What is the period of the cosine function?)

b) Solve this differential equation to give P as a function of t, given that P(0) = 2 (million individuals).

c) Use a Runge-Kutta 4 numerical solution with an increment size of 1 day to calculate the population size after 3 years. Firstly write the necessary equations for predicting P_j+1 and t_j+1 from the values of P_j and t_j, then implement these in a spreadsheet program. (Or repeat
the calculations 1095 times.) Be sure to write the RK4 numerical scheme explicitly on your assignment paper.

d) Plot the numerical solution together with a graph of the analytical solution.

Question 3. A paper glider will often fly in a wavy path if it is launched at a slow speed and can loop the loop if launched at a fast speed (provided it has flaps to create lift). The following non-linear system of differential equations model the flight of a glider.

dV/dt = - sin L - dV²

dL/dt = (V² - cos L)/ V

V is the velocity and L is the angle of the nose to the horizon, so that if L = 0 the glider is flying horizontally. The constant d is the coefficient of drag (or air friction). Drag is proportional to V₂ as is lift. The terms sin L and cos L represent the component of gravity in a direction parallel to and perpendicular to the direction of the glider's noe. The second equation is divided by V to account for angular rather than linear movement.

Using a chain rule with V(t) and L(t) both functions of t we can show that, if d = 0 then E = V³ - 3V cos L is a constant quantity.

a) Write the formulation of Euler's method for calculating V_j+1 and L_j+1 from values for V_j and L_j for a time increment of h = Δt = 0.01.

b) Implement this system using a spreadsheet that will allow you to change initial values of V and L as well as d. Also use the spreadsheet to calculate the value of E. After having made the calculation for part c) include a printout of page 1 of the spreadsheet.

c) For d = 0 and L = 0 vary your initial values of 0 < V ≤ 2 to suggest a condition on E such that the glider will either loop the loop or not. Make calculations for 0 ≤ t ≤ 5. Plot L versus V for a wavy flight and a loop the loop flight on the one graph.

d) Plot V and L versus time for the two cases below and discuss the difference shown.
i. L = 0,V = 1 and d = 0
ii. L = 0,V = 1 and d = 0.5