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Re: [Phys-l] ballistic arc length



Hello,

Sorry about the confusion...

We wanted the arc length independent of time; i.e., find y(x) and then find the total distance traveled along the arc -- hence the 'messy' result.

Yes, we were not trying to be overly difficult, the projectile started and stopped at the same height.

It started out as an attempt to 'cross-over' that was well- intentioned. I thought the problem would be fairly straightforward and a good way to do this. Now, through sheer 'cussedness', I don't want to give up on it.

Thanks,
Peter

On Dec 29, 2009, at 12:15 AM, jhowell@earlham.edu wrote:

Two questions:

First: You ask the students to:
1. Find the equation for the path of the trajectory (arc or the
flight path).

Do you mean arc length as a function of time? x and y as a function of
arc length? I'm not sure what you'd be asking.

Second: Does the projectile start and end at the same height, or is it
shot off the edge of a cliff (or up a hill of constant slope)?

For what it's worth, an alternative problem (from Fowles "Analytical
Mechanics") has a projectile shot upward at angle theta from the base of a
hill of constant slope of angle alpha. The student is to find the value
of theta that will make the projectile go furthest up the hill.

Good luck - John

Hello,

For the next semester one of our calculus instructors and I have
decided to try and formulate a "team problem" to tie the physics and
math content together -- and maybe enhance the transfer of skills
between our subjects a bit more.

With just a bit of thought I proposed the ballistics problem, with a
twist:
A projectile is launched with an initial speed v_0 at an angle
_theta_. Gravity is chosen to be g = 9.8 m/s^2 and is directed
vertically downward. Using the equations of motion, the 'normal'
solution is to find things about the projectiles x and y motion.
Instead we want to find something about the actual path of flight:

1. Find the equation for the path of the trajectory (arc or the
flight path).

2. Find the angle that would minimize/maximize this path.

Now, using basic rules of calculus I can find the first answer of an
arc length (which is a very complex experession). Differentiating
this to find 2 is proving daunting.

Some advice:

1. Is there an easier way to approach this? (Hamiltonian with
appropriate coordinate system?)

2. If not, would this be considered too much for them to 'tackle'?
(Our students are in at least Calculus III -- of a 3 semester sequence
-- and in Physics II [E&M, etc.])

Thanks,
Peter
_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l



_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l