Math Is Fun Forum

Hello. I have been working various problems and I think I am doing alright. I am not sure what to do with this one, however.

"A sailboat is running along a straight course and under a light wind at 1 m/sec. The wind wuddenly picks up, applying a constant force of 600 N onto the boat. The only other force that is acting on the boat is water resistance where k=100 N-sec/m that is proportional to the velocity of the boat. The mass of the boat is 50 kilograms. First find an equation for the motion of the sailboat, then determine its limiting velocity under these wind conditions."

Here is what I have so far: Ft is the total force on the boat, Fw is the force of the wind, Fr is the resistance on the boat by the water, k is the proportionality of the water resistance, m is the mass of the boat, v(t) is the velocity of the boat.

Looking it up, I find that acceleration is the force divided by the mass (right?) so...

How does that look so far? Am I heading in the right direction? What does N-sec/m mean, anyway?

Hello.

I am practicing a variety of general problems and I was hoping I could get my solutions verified so I can see whether I am understanding the concepts or not. I am putting them all into this one post so not to flood the forum with individual posts. My solutions will be in the posts following.

Thanks in advance.

1. Solve the following:

b.

c.

2. Plot the phase portrait of the system and answer a - c.

a. What are the equilibrium points of the system?
b. What is the long term behavior of the system with initial conditions x(0) = 3 and y(0) = 1?
c. Plot the x(t) and y(t) planes.

3. Find the equilibrium points:

4. Plot the phase portrait and do a - c.

a. Determine the long term behavior of the system.
b. Plot y(t).
c. Solve the differential equation.

5. Consider the partially decoupled system:

a. Derive the general solution.
b. Find the solution that satisfies the initial condition (x0,y0) = (-1,3)

6. Solve the differential equation:

7. Use Euler's Method for Systems to find y(1) given x(0) = 0, y(0) = 2, with a step size of .1 for the system

I'll be working these out throughout today and tomorrow. Thanks again!

"Newton's Second Law, F=m*a, may be viewed as W = m*g where W is the weight in pounds and g is gravity. A sled weighing 100 lbs is being pushed in a straight line against the wind by a force of 10 lbs. Friction is negligible but there is an air resistance whose magnitude in pounds is equal to twice the velocity of the sled in feet per second.

"If the sled starts from rest, find the velocity and the distance traveled at the end of one second."

These physics-type questions throw me off. I have never had physics. We know the sled weighs 100 pounds and it's being pushed by 10 pounds of force on the horizontal... and we know that the wind has a magnitude of twice the velocity of the sled.

How is a problem such as this started? Does one find a position function and then differentiate? What is the difference between mass and weight? How should this solution be started?

Thanks for the help.

I am trying to understand how to 'read" a set of equations and it isn't really making sense. Here is an example of the sort of problem I am trying to learn about:

"In one of these systems, the prey are animals very large in size and the predators are very small animals. Thus it takes many predators to eat one prey, but each prey eaten is a tremendous benefit for the predator population. The other system has very large predators and very small prey. Determine which system is which and provide a justification for your answer."

Systems:

and

Apparently we are not only concerned with the evolution of the predators and the prey within some habitat but somehow it is supposed to depend on the size of the two? The book gives elephants and mosquitoes as an example of large and small predators and prey... but the idea of elephants eating mosquitoes is silly. That, however, is the objective.

How is this determined?