Vector Calculus for Engineers
CME100, Fall 2004
Problem Set #8
(Line Integrals, Green’s Theorem)
Date: 11/17/2004 Due: 11/24/2004
Reading: Thomas 13.2-13.4
Exercises:
Section 13.2: p. 1068 Exercises 10, 14, 18, 20, 26, 40
Section 13.3: p. 1078 Exercises 2, 10, 14, 30, 38
Section 13.4: p. 1090 Exercises 4, 8, 16, 18, 22, 31, 34
MATLAB Workbook (optional):
Exercises 22, 23
Problem 1 An aerospace company is developing a mission to launch a series of communication satellites into low Earth orbit to an altitude of 125 km. As a rocket scientist, you have been given a task to perform a trade study to select between three different launch trajectories. Your objective is to maximize the amount of payload into orbit, i.e. minimize the amount of fuel spent to achieve orbital velocity. The three trajectories being considered are specified by the following parametric equations:
where g is a parameter that ranges between 0 to 1, t is time, x is the horizontal position, and y is the vertical position from the launch site, both measured in meters. The values of C and n for the three cases are given in the table below.
Case No. / C / n1 / 3000 / 3.0
2 / 2775 / 2.8
3 / 2545 / 2.6
The energy delivered by the engines is used to increase the potential and kinetic energy of the vehicle as well as to fight atmospheric drag. Your objective is to choose out of the three trajectories the one for which the energy loss due to drag is the lowest.
The drag force acting on the vehicle can be computed from the following relation:
where is the air density, A is the vehicle frontal area, is the velocity vector, and is the drag coefficient. For the purposes of this problem assume that , m2, and that the density of air is a function of altitude y: kg/m3.
a)Plot each of the three trajectories on the same set of axes, i.e. horizontal vs. vertical displacement
b)In each case compute and plot the speed of the vehicle as a function of time [Hint: note that and . Then use the gradient function to differentiate x, y and t with respect to g to find the components of the velocity vector]
c)Plot the drag force in each of the three cases as a function of time on the same set of axes. Comment on the shape of the curves.
d)Assuming the drag force acts in the direction opposite to the direction of the velocity vector, , the work done by the drag force can be found from the following expression:
Use the trapz function (trapezoidal integration) to compute the work done by the drag force in each of the three cases. Based on your results, which of the three trajectories would you recommend and why ?