brochesspro said:
This is another version of the work I did after starting fresh. How do you think it fares? I think there is something fishy about it but I just can not seem to point my finger at it. Please evaluate it,
The result, ##\frac{2dw}{w^2-v^2}## has some of the right behavior. But yes, it is fishy. Somehow the variables seem to have been swapped.
One assumes that ##w## is the wind speed and that ##v## is the airplane's velocity relative to the air.
If ##w=v##, ##w^2 = v^2##, the denominator becomes zero and an infinite round trip time is computed. This makes sense since the upwind leg of the journey will indeed take forever.
If ##v=0##, we get a result of ##\frac{2d}{w}##. That would be the correct number for a round trip at velocity ##w##. But ##w## is the wind speed.
Let us back up and see if we can spot the error...
You start with some stuff that I find hard to follow because you are not documenting your variable names. That is a pet peeve of mine. Slowing down to decipher that, I get:
##\vec{V_\text{pw}}## is the velocity of the plane with respect to the wind.
##\vec{V_\text{pm}}## is the velocity of the plane with respect to the mass of the Earth (I guess).
##\vec{V_\text{wm}}## is the velocity of the wind with respect to the mass of the earth.
You assert that the velocity of the plane with respect to the wind is the vector difference between the plane's velocity and the wind's velocity, both relative to the fixed ground frame: ##\vec{V_\text{pw}} = \vec{V_\text{pm}} - \vec{V_\text{wm}}##. That is a valid assertion.
You then carefully look at your sign conventions and use the positive speed ##v## to replace ##\vec{V_\text{pw}}## and the positive wind speed ##w## for ##\vec{V_\text{wm}}##.
The conclusion is that the plane's ground velocity (##\vec{V_\text{pm}}##) for the downwind leg is ##v+w##. Indeed, that seems obviously correct.
You proceed to re-use the same variable names for the upwind leg. Again, you are careful with the sign conventions. This time the plane's velocity relative to the ground is the negation of its ground speed.
The conclusion is that the plane's ground velocity (##\vec{V_\text{pm}}##) for the upwind leg is ##w-v##. This is the negation of ##v-w## and since the plane is leftward moving this time, that seems obviously correct again.
Ahhh, there it is. I think I see the error.
You assert that the total time is the sum of the downwind flight time plus the upwind flight time. But before writing that assertion down, you immediately rewrite the downwind flight time as ##\frac{d}{v+w}## and the upwind flight time as ##\frac{d}{w-v}##.
But that second term is not correct. The displacement to be traversed is not ##d##. It is ##-d##. A negative displacement covered at what will (hopefully) be a negative velocity.
[My hat is off to teachers who have to reverse engineer this kind of stuff every time they grade papers. It boggles the mind that you do not all go bonkers]
