(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t))
(endMath)
/*????????????????????????????
>>>>>>>>>>>> Confirmation of ∂[∂(t): Rpcs(POIo(t),t)]
12Oct2019 is this wrong? /%
1369:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t))
/* The work below was done independently of my previous work, to see if I :
- obtaine the same answers
- notice [missing issues, factors]
The original checks can be found at :
- "?documenht?"
- ?section?
/* see "$d_PROJECTS""Lucas - Universal Force/Images/Howell - Chapter 4 - dp[dt _ Rpcs(POIo,t)] - cropped.png"
"Figure "∂[∂(t): Rpcs(POIo(t),t) ]""
In the figure, I have
/%
delta(Rpcs)
= Rpcs(POIo(t),t2) - Rpcs(POIo(t),t1)
= Vonv(PART)*delta(t)*cos(pi - Aθpc(POIo(t),t2))
/* Reminders : /%
cos(pi - Aθpc(POIo(t),t)) = -cos(Aθpc(POIo(t),t))
sin(pi - Aθpc(POIo(t),t)) = -sin(Aθpc(POIo(t),t))
cos(pi/2 - Aθpc(POIo(t),t)) = ?[+,-]?*sin(Aθpc(POIo(t),t))
sin(pi/2 - Aθpc(POIo(t),t)) = ?[+,-]?*cos(Aθpc(POIo(t),t))
/* +-----+
Case 1 : the particle is "ahead" of the Point of Interest in the Observer frame (POIo)
ie the particle is flying away from the POIo on a parallel track
and Rpcs is GROWING
/%
Aθpc(POIo(t),t) >= pi/2, so
cos(Aθpc(POIo(t),t2)) <= 0
∂[∂(t): Rpcs(POIo(t),t) ] >= 0
delta(Rpcs)
= Rpcs(POIo(t2),t2) - Rpcs(POIo(t1),t1)
/* but there is NO change in the AθPI2pc(POIo(t),t) perpendicular direction /%
Rpcs(POIo(t2),t2)*sin(Aθpc(POIo(t2),t2)) = Rpcs(POIo(t1),t1)*sin(Aθpc(POIo(t1),t1))
/* and /%
Vons(PART)*(t2 - t1)
= Rpcs(POIo(t2),t2)*cos(Aθpc(POIo(t2),t2)) - Rpcs(POIo(t1),t1)*cos(Aθpc(POIo(t1),t1))
/* This is the delta form of the Chain Rule for differentiation
Imagine a box that expands at the advancing edges
/%
= Vons(PART)
*{ Rpcs(POIo(t1),t1)*{cos(Aθpc(POIo(t2),t2)) - *cos(Aθpc(POIo(t1),t1))}
+ (t2 - t1)*cos(Aθpc(POIo(t1),t1))
- 1/2*(t2 - t1)*{cos(Aθpc(POIo(t2),t2)) - *cos(Aθpc(POIo(t1),t1))}
}
/* But as deltas -> 0, ignore the third term, so /%
∂[Rpcs(POIo(t),t)]
= limit as delta(t) -> 0 { Vons(PART) *d[t2*cos(Aθpc(POIo(t2),t2)) - t1*cos(Aθpc(POIo(t1),t1))] }
= Vons(PART)*{ ∂[t]*cos(Aθpc(POIo(t),t)) + t*∂[cos(Aθpc(POIo(t),t))]
∂[∂(t): Rpcs(POIo(t),t)]
= ∂[∂(t): Vons(PART)*{ ∂[t]*cos(Aθpc(POIo(t),t)) + t*∂[cos(Aθpc(POIo(t),t))] }
= Vons(PART)*{ cos(Aθpc(POIo(t),t)) + t*∂[∂(t): cos(Aθpc(POIo(t),t))] }
/* with
|cos(Aθpc(POIo(t),t))|
28Oct2019 YIKES!!! I didn't put in the || signs.
/* WRONG!! so this result is the same as /%
1369:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t))
I need to change this!!!! -> huge, cascading] corrections needed in derivations!
/* Note that for Case 1, /%
Aθpc(POIo(t) >= pi/2,
/* so /%
cos(Aθpc(POIo(t),t)) <= 0
∂[∂(t): Rpcs(POIo(t),t)] >= 0
/* +-----+
Case 2 : POIo is ahead of particle
ie the particle is flying toward the POIo on a parallel track
and Rpcs is SHRINKING!
/%
Aθpc(POIo(t),t) <= pi/2, so
cos(Aθpc(POIo(t),t2)) >= 0
∂[∂(t): Rpcs(POIo(t),t) ] <= 0
delta(Rpcs) is <= 0
= (-1)* cos(Aθpc(POIo(t),t1)) *(Vons(PART)*(t2 - t1))
= (-1)*|cos(Aθpc(POIo(t),t1))|*Vons(PART)*delta(t)
∂[∂(t): Rpcs(POIo(t),t)]
= limit as delta(t) -> 0 :
= cos(Aθpc(POIo(t),t)) *Vons(PART)
= (-1)*|cos(Aθpc(POIo(t),t))|*Vons(PART)
/* this result is the same as Case 1 and : /%
1369:(mathH) ∂[∂(t): Rpcs(POIo(t),t)] = -Vons(PART)*cos(Aθpc(POIo(t),t))
/* Note that for Case 2, /%
Aθpc(POIo(t) <= pi/2,
/* so /%
cos(Aθpc(POIo(t),t)) >= 0
∂[∂(t): Rpcs(POIo(t),t)] <= 0
So the original results are confirmed.
28Oct2019 CHANGE!!!
Using :"vector" notation for [+,-] angles :
In order to have a unified expression for both conditions, for clarity I SHOULD use :
∂[∂(t): Rpcs(POIo(t),t)] = (-1)*Vons(PART)*|cos(Aθpc(POIo(t),t))|
This looks the "same" as what I've been using, but check how that affects derivatives!