Vector Field Problem
This is a word problem that was given to my group as a project and the problem itself is rather lengthy so I will give the portions that are relevant to the problem.
The set up to the problem is that your group was studying for finals and suddenly you pass out and wake up on planet Xanthar. You meet a man named Victor Field at a Tavern and he tells you the only way to get back home is by using a catapult to launch the holy hand grenade of antioch into the castle of the evil wizard Div Curl.
1. The problem statement, all variables and given/known data
I am paraphrasing from the problem:
"[Victor Fields tells you that]...the perimeter of [Div Curl's] castle compound is impenetrable on the ground, surrounded by a 30 foot tall magic wall. He also says the forests surrounding the compound is surrounded by dragons, that patrol an area 1000 feet from the compound wall. (The compound is a circular compound with a radius of 1 mile). he hands you a copy of blueprints for a great catapult (from the ACME company), which can be used to launch the Holy Hand Grenade of Antioch. He tells you that the catapult fires its object with an elevation angle of 60 degrees and an initial speed of 2000 ft./sec. He also tells you that the catapult will not function unless it is on perfectly level ground and cannot be placed either above or below basic ground level of the magic forest. He hands you a book of matches and a copy of a book written in an obscure ancient XantHurian dialect. finally he hands you a bird in a cage which you immediately recognize to be an African Swallow. He also tells you that the atmospheric pressure on Xanthar behaves as it does on earth, with identical density but that the gravity near the castle operates radially from the center with the equation:
g(t)= (-2 + sint)i + (-32 + cost)j ft/sec^2 where t is in seconds.
He also tells you that the drag acceleration due to air resistance in the Xanthurian atmosphere is numerically equal to the velocity and you know this to be a vector equation."
2. Relevant equations
gravity = g(t)= (-2 + sint)i + (-32 + cost)j
3. The attempt at a solution
My group used g(t) as our vector field.
Our goal is to find what distance away from the compound do you need to be able to fire the holy hand grenade of antioch into the compound.
That is how far we have gotten so far. We have worked on this problem for days and we still have not gotten it.
Our professor said that there are missing elements that we need to complete the problem and that we need to find what these elements are ourselves.
The issue is that we don't know what we are missing. If anyone can point us in the right direction then it will be a great help. We do not need the answer, we just need to know what we are missing. Thank you so much for any help!
I have a number of clarifying questions I would ask:
1. Are the castle and the compound the same thing?
2. In what direction are i and j relative to your point of view?
3. As a hunch, you're probably going to get a second-order differential equation that you might be able to integrate once exactly. Do you need an exact answer?
1. Yes the compound and the castle are the same thing.
2. The problem does not specify where we are with respect to the compound.
3. Our class haven't studied second-order differential equations yet, so that might be an issue if we do end up with a second-order differential equation. Since we need the distance away from the compound, I think we do need an exact answer.
I hope I answered your questions fully. If I am missing something just tell me and I will do my best to answer. I will look into differential equations and see if something might help me. Thank you for your help!
** oh and according to our professor, the punishment for not solving this problem is having to spend an eternity solving third order non-linear differential equations with inverse hyperbolic coefficients. xD
You're going to have to analyze the forces on the object and use Newton's laws to get the equations of motion.
Force 1: gravity, we'll call it Fg. You already have the acceleration due to gravity defined for you. Let's let the mass of the grenade be m.
Force 2: air resistance, we'll call it Fr. This is exactly equal to the velocity, according to the problem statement. However, it's going to be in the opposite direction of the velocity, like air resistance normally is. That is: (Fr/m) = -v, a vector equation. Hence, Fr = - m v = -m (vx i + vy j), to put it in component form.
Now, Newton's law states that sum F = m a, where dv/dt = a. We sum up the forces thus:
m ((-2 + sin(t))i + (-32 + cos(t))j) - m (vx i + vy j) = m ((dvx/dt) i + (dvy/dt) j).
Obviously, the masses cancel. In addition, since this is a vector equation, you can set each component equal individually:
-2 + sin(t) - vx = dvx/dt
-32 + cos(t) - vy = dvy/dt.
You have a system of first-order ordinary differential equations (ODE, or DE for short) here. One great simplification is that they are not coupled. You can solve each by itself.
So how would you solve the DE's?