Topic 2 Summary :
Things CHANGE
They were NOT allowed to change in the old Topic 1 scenario ... (except location)
the NEW approach , in Topic 2 , which allows them to change ,
is to distinguish the "Object" that is being investigated : "inside the system"
from the "Environment" that object is immersed within : "outside the system" .
Topic 1 statements are all about the object system itself (presuming no environment).
. . . it is traditional in Physics to treat the object as "passive recipient" , and the environment as the "active subject".
The mass of water in your cup can change ... if some of it leaves the cup ... or if some enters the cup.
. . . The "object system" is your cup ; there is a "system boundary" that separates "in the cup" from "outside it".
If there is less water inside than before, it means that water flowed across this boundary in an outward direction.
Δminside / Δt = −ρA·v . . . where A is the outward-pointing Area that had water flowing through it , with velocity v .
Important Statement #2 about How the Universe Works :
Force being applied to an object for a time duration CAUSES the object's momentum to change.
... the Force is applied by the subject - distinct from object - as the object's environment .
. . . While the Force is being applied , a process is occurring (Impulse) : on objectFby subject Δt = Δp .
Re-arranging the two changes into a ratio => on objectFby subject = Δp/ Δt
. . . Force depends on location ... IF F(r) has no dis-continuity , the limit as Δt→0 will exist ... (even if F(r) is not smooth)
=> F = dp/dt , as a derivative .
Since the object's mass is constant ,
=> F = m· Δv / Δt = m·a . . .
This acceleration a has direction ... same direction is positive as for x , and v , and p .
. . . (by now you habitually indicate the "+'ve" direction on your sketch ...)
this a is the average value that exists for the duration Δt of the applied Force . . .
. . . connects the earlier condition (v1) with the later condition (v2) .
Since momentum , as a function of time has to be smooth through time (with no "corners") , the limit as Δt→0 exists , and
=> a(t) = dv(t)/dt , as a derivative .
(Reminder, definition of average velocity) Important Kinematic Equation #1 : vavg ≡ Δr / Δt
=> place story line : rstart + vavg·Δt = rend (acceleration is not explicit)
(definition of average acceleration) Important Kinematic Equation #2 : aavg ≡ Δv / Δt
=> motion story line: vstart + aavg·Δt = vend (displacement is not explicit)
IF acceleration is constant during the interval Δt : vavg = vstart + ½ Δv = vend − ½ Δv = ½ ( vstart + vend ) .
=> place story w/ accel : Kinematic Equation set #3 : has 3 different "wordings" mixing the info of #1 and #2 )
. . . (a) . . . rstart + vstart·Δt + ½ aavg (Δt)2 = rend (ending velocity is not explicit)
. . . (b) . . . rstart + vend·Δt − ½ aavg (Δt)2 = rend (starting velocity is not explicit)
. . . (_) . . . rstart + ½ (vend2 − vstart2) / a = rend (how to divide by a vector? Re-Write!)
. . . (c) . . . v2start + 2 aavg·Δr = v2end (time duration is not explicit)
CAUTION : avoid form "c" until you understand multiplication of vectors ... the directions of a and Δr are important !
caution : avoid form "c" until you find out how to identify v² on a diagram ... it discards direction of both velocities
- - - NOTICE - - - which quantity - - - does NOT explicitly show - - - in each equation above - - -
Solvable Kinematic scenarios can only have 2 __TWO___ of these quantities as unknown
. . . make sure the other 3 are __ON_YOUR_DIAGRAM__ . . .(4, if you have xi and xf , instead of Δx )
Your diagram should tell you whether you need to "sub-contract" to obtain a value from a preliminary scenario ...
Solve part (a) for a's unknown , by using whichever equation is _missing_ part b's unknown .
Read the equation as words ... appropriate to the scenario ...
... then , do symbolic algebra to isolate a's unknown . . . RE-READ YOUR RESULT as words
... when satisfied that the relationship is reasonable, plug numbers [ WITH UNITS ! ] and compute your result
. . . THINK ABOUT whether it's ABOUT RIGHT , size-and-unit-wise !
... finally , you can use any of the OTHER equations to calculate b's unknown
. . . even those which need to use a's _now_known_ .
Dynamic scenarios _imply_ a by telling F and m . . . or Δv , by telling Δp and m . . . or combinations with Δt
- - - - the KEY is - - - to TRANSLATE the STORY-LINE - - - to a SYMBOLIC STATEMENT - - -
Quantities which have direction are called vectors. Vectors are drawn as arrows (with labels, of course).
... we will pattern our treatment of all object vectors on Displacement .
We will always use coordinates with axes that are perpendicular to each other ; then,
motion in any direction is independant of motion in all other directions
. . . an event condition relates these directions by its time
. . . algebraically, the key is to keep each component (x) separate from the others (y and z) !
then , a "big & scary" 2-dimensional scenario becomes 2 "small" 1-dimensional scenarios ... connected via time .
Vector representations : first choose an origin and coordinate system (showing each positive direction).
drawing : a location vector is drawn as an arrow with its tail at the origin, its tip at that location.
this represents a location as if it is a radially-pointing range at some angle from the coordinate axes ... hence the abbreviation r .
. . . vectors of other quantities can have their tail at the object being described,
but must point in the direction of the quantity (e.g, v in velocity's direction).
bold italic letters represent vector quantities !
component : The ordered set of the location coordinates ( x , y , z ) . . . = (right , forward , up) or (East , North , Zenith)
is the algebraic way to write this location vector (Cartesian component form)
. . . a velocity vector is written as the set of velocity components ( vx , vy , vz ) in the same order ... .
In almost every case, it is best to :
0) diagram the "Before" scenario ... include arrows (and labels!) for the vector condition quantities.
½) notice which vector quantities might be different in the "Meanwhile" and the "After" diagram
1) draw and label the important process quantities , from Before to After ,along the path taken.
2) choose a coordinate system ; often parallel (and perp.to) the path v or p ... except in free-fall .
3) split important vector quantities into (parallel , perpendicular) or ( horizontal , vertical ) components.
4) write how the each direction's vector quantities relate to one another ... or to TIME.
Addition : drawing : usually done on a small coordinate system , separate from the main diagram (so it doesn't get cluttered).
the first arrow tail is at the origin of the addition coordinate axes (x=0,y=0,z=0).
its tip points the same direction from its tail as it does in your sytory-book diagram ... to ( x1 , y1 , z1 ).
. . . the vector being added to the previous has its tail placed at the previous tip
its tip points the same direction from its tail as it does in your story-book diagram ... to (x1 + x2 , y1 + y2 , z1 + z2 ).
. . . vector arrows are appended tail-to-previous-tip in sequence, keeping each aligned with its arrow on the first diagram.
The Sum ("Result for the addition") of the vectors is the arrow from first tail (origin) to last tip .
algebraically : the Result of the addition is the list of components : (x1+x2+x3... , y1+y2+y3... , z1+z2+z3... ) .
Adding a vector to itself results in a vector twice as long , in the same direction ; i.e, v + v = 2 v ...
. . . the negative of a vector points in the opposite direction
... a vector added to its opposite ... Δx + ( − Δx ) ... makes the "zero length vector" ... = 0 = (0, 0, 0) .
Process quantities ... displacement (Δr) , boost (Δv) , Impulse (FΔt) , Action (p·Δx) ... add sequentially
make sure the durations being added DO NOT OVERLAP . . . change = change so , there is always one Δ in each term !
Condition vectors ... location , velocity , acceleration , momentum , Force ... are added at the instant of that condition
. . . it is the Sum of simultaneous Forces that cause the effected mass to accelerate (at that instant) : Σ F = m a .
(to 
