Suggestions for Calculus I honors projects?

[QuantumCurt]

Hey everyone, I was hoping I could get some input on this. I'm taking an honors section of Calculus I this semester, and part of the course involves a cumulative honors project. It has to be a minimum 8 page paper, at least a third of which has to be actual calculations, about a topic that goes above and beyond the actual scope of the class.

I'm a physics major, so I'm trying to find a physics related topic that would be within the scope of this project. I was considering doing something involving the Calculus of rainbows, but the more I've looked into it, it seems like it would be an extremely complicated topic. A couple of my friends are trying to convince me to do it on Lagrangian Mechanics...but that's seeming like it would be an very complex project. I haven't even started calculus based physics yet, so I feel like that may be too much to take on.

I'm having trouble coming up with ideas. I'm doing a project on the formation, life cycle and properties of black holes for the cumulative honors project in my physics class, so something that kind of ties into that concept would be cool. But it by no means has to be related to that project at all.

Does anyone have any ideas? Any suggestions would be much appreciated!

 

[jfizzix]

In calculus 1, you have only just covered derivatives of single variable functions, yes?

in 1D kinematics, let's say you throw a ball up vertically in the air.

If we neglect air drag position function will be approximately quadratic :y(t) = y(0) +v(0)t -1/2 gt^2.
What is the velocity as a function of time v(t)?
What is the acceleration a(t)?

Since the velocity is the derivative of the position, and the acceleration is the derivative of the velocity, you can figure out what v(t) and a(t) are, no matter what y(t) is.

What about if we don't neglect air drag?
I know that if you assume the air drag is proportional to the velocity, you can solve the differential equation to get the position y(t). In your case, I would look up the position function, and take derivatives to see what the velocity and acceleration do?

 

[Mandelbroth]

What about if we don't neglect air drag?
I know that if you assume the air drag is proportional to the velocity, you can solve the differential equation to get the position y(t). In your case, I would look up the position function, and take derivatives to see what the velocity and acceleration do?

I wish to elaborate on this.

Let's take a really simple example. Suppose we have an object of mass $m$, with initial speed zero, in free fall. Thus, if we were to draw a free body diagram, we'd have two forces: weight pulling down with magnitude $mg$ and a resistive force pushing up with magnitude proportional to the speed of the object. Thus, we have the net force, assuming mass doesn't change, as $\sum \vec{F}=m\vec{a}=mg-kv$, where I've chosen down as the positive direction. We know that acceleration is the derivative of velocity, so, dividing by $m$ gives us $v'=g-\frac{k}{m}v$, which is separable. With a little knowledge of integration, you can solve for velocity.

 

[QuantumCurt]

So far we've covered a chapter on limits, and we're just about done with the chapter on differentiation. We've mostly covered derivatives in one variable, but the last section we did was on implicit differentiation of equations in two variables. We're just about done with this chapter. We only cover 4 chapters in this class. The next chapter is on applications of the derivative and the last chapter will be on integration. I'm likely going to have to jump ahead and learn integration sooner though, to complete just about any project I could do for the class.

I'm strongly considering something related to classical mechanics. I was talking to my physics professor earlier today after I turned in the proposal for my honors projects in my physics class, and he recommended rocket propulsion, which seemed like a good idea to me. It would involve Newton's 2nd Law very heavily, obviously. The force is constantly changing as the rocket is accelerating, due to the change in mass resulting from spent fuel. It would involve some integration and some basic differential equations, but he said that it should be well within the realm of what I could reasonably accomplish in this project.

That would tie in fairly closely to what you guys are recommending, because air resistance would have to be factored in.

Does that seem like a good idea?

 

[Mandelbroth]

So far we've covered a chapter on limits, and we're just about done with the chapter on differentiation. We've mostly covered derivatives in one variable, but the last section we did was on implicit differentiation of equations in two variables. We're just about done with this chapter. We only cover 4 chapters in this class. The next chapter is on applications of the derivative and the last chapter will be on integration. I'm likely going to have to jump ahead and learn integration sooner though, to complete just about any project I could do for the class.

I'm strongly considering something related to classical mechanics. I was talking to my physics professor earlier today after I turned in the proposal for my honors projects in my physics class, and he recommended rocket propulsion, which seemed like a good idea to me. It would involve Newton's 2nd Law very heavily, obviously. The force is constantly changing as the rocket is accelerating, due to the change in mass resulting from spent fuel. It would involve some integration and some basic differential equations, but he said that it should be well within the realm of what I could reasonably accomplish in this project.

That would tie in fairly closely to what you guys are recommending, because air resistance would have to be factored in.

Does that seem like a good idea?

That sounds like an excellent idea. Just remember that, when mass changes, Newton's second law becomes $\sum \vec{F}=\dot{p}$, where $\dot{p}$ is the derivative of momentum with respect to time.

 

[QuantumCurt]

That sounds like an excellent idea. Just remember that, when mass changes, Newton's second law becomes $\sum \vec{F}=\dot{p}$, where $\dot{p}$ is the derivative of momentum with respect to time.

Thanks for the tip. I've got a lot of research to get done. My calculus teacher has tentatively approved the topic though. He wants me to specify a little bit more the exact type of problem I'd be doing.

I think I'm going to model it after the Saturn V rocket, since I imagine there's a plethora of data available on it.

Does anyone have any links to some good articles/papers about rocket physics that would be relevant to this? Any recommendations on how to further narrow the topic down?