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.])