Hi,
I'm doing an assignment and the problem I have outlined for myself goes like this. There's a lander craft who's acceleration is described by a function:
a(t) = T(t) - g, where T(t) is an exponential function that describes the crafts thrust (mass = 1kg thus simplified as acceleration).
T(t) = b*t*ec*t , where b and c are constants.
The crafts thrust is generated by burning fuel, which for the purpose of this assignment will equal the rate of unit thrust generated.
Now there's a 'Cost Function' which returns the total amount of fuel burnt. This is given by:
F(t) = Definite Integral of T(t) , beginning at 0 and ending at period P.
Period P is defined as the time it takes for the craft to land; which is when the first derivative of acceleration equals 0 (the craft starts at rest, so solution at x= 0 is ignored)
a'(t) = v(t) => v(P) = 0
Here's what I want to do:
I want to graph the Cost function on a y-axis (e.g. how much fuel is burnt), and on the x and z axis there will be different values for the constants of b and c.
Basically I will then use derivates of the cost function to find what strategies (strategy determined by constants) are the best (minimizes fuel burnt).
However, naturally this depends on time. And I know how to get time (period P). But I've never graphed multivariable functions before and I don't know how to express this.
E.g.: How may I graph the cost function using only two variables, constant b and c?
I'm doing an assignment and the problem I have outlined for myself goes like this. There's a lander craft who's acceleration is described by a function:
a(t) = T(t) - g, where T(t) is an exponential function that describes the crafts thrust (mass = 1kg thus simplified as acceleration).
T(t) = b*t*ec*t , where b and c are constants.
The crafts thrust is generated by burning fuel, which for the purpose of this assignment will equal the rate of unit thrust generated.
Now there's a 'Cost Function' which returns the total amount of fuel burnt. This is given by:
F(t) = Definite Integral of T(t) , beginning at 0 and ending at period P.
Period P is defined as the time it takes for the craft to land; which is when the first derivative of acceleration equals 0 (the craft starts at rest, so solution at x= 0 is ignored)
a'(t) = v(t) => v(P) = 0
Here's what I want to do:
I want to graph the Cost function on a y-axis (e.g. how much fuel is burnt), and on the x and z axis there will be different values for the constants of b and c.
Basically I will then use derivates of the cost function to find what strategies (strategy determined by constants) are the best (minimizes fuel burnt).
However, naturally this depends on time. And I know how to get time (period P). But I've never graphed multivariable functions before and I don't know how to express this.
E.g.: How may I graph the cost function using only two variables, constant b and c?