USE UNITS all thru your answer ... don't just append them onto the final number !

DRAW a DIAGRAM for each scenario . . . label the diagram with SYMBOLS , as you read
. . . condition points , words like   "start" , "end" , "turn-around" . . . with vector arrows
. . . process spans as brackets or lines , words like   "average" , "moved" , "change" , "Δ" , "t=0→2" ...

Write statements as symbols first ; manipulate symbols before plugging numbers.
. . . each statement should have a SUBJECT . . .
. . . keep track of adjectives such as   "initial" , "average" , "final" , "stopped" . . . "change"

subscripts for "total" or "system" , "part A" ...

keep track of the SOURCE for the external Forces applied to the (passive) object

Recognize whether you are predicting based on theory . . . reasoning from causes to their effects ... (what should it do?)
      or whether you are deducing from observation . . . (what does this imply about it?)

Get un-stuck (often) by wondering "why isn't it the same as it used to be?"

Topic 3 Summary :

Quantities which have direction are called vectors. Vectors are drawn as arrows (with labels, of course).
... we will pattern our treatment for 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 ( v_x , v_y , v_z ) in the same order ... .

Use Trig (right) triangle formulas to obtain components ... and/or   extract a diagonal from a component
. . . s^o_h c^a_h t^o_a   ... for right triangles
. . . Pythagoras : diagonal   d² = a² + b² + c²   ... for orthogonal components

if NOT a right triangle, a·b = a² + b² + 2ab cos(θ_a,b)

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) .

spring Force : depends on the stiffness konstant k of the spring , which is a scalar property (almost) intrinsic to a particular spring
      . . . (although a real spring's stiffness does change with age or abuse)
. . . and also depends on the distance and direction which that end of the spring has been stretched : s .
      the spring's end pulls in the opposite direction from its stretch vector (or pushes opposite its compression vector)
      . . . this is parallel to the length of the spring !     neglect the mass of ideal (textbook) springs
=> F_spring = − k s .

Two springs hooked end-to-end in series (anchor...spring1...spring2...Force , as : |-v^v^v-·-^v^v^v→ )
. . . will stretch farther than either of them ...the stretch distances add , so s_total = s₁ + s₂ ,   but their Tensions are the same
=> 1 / k_series = 1 / k₁ + 1 / k₂ .
this implies that longer springs have smaller stiffness k than otherwise similar short springs .

If two springs are both connected to the anchor and to the external Force, in parallel, ( |‾‾‾→ ) , their Forces add
. . . their stretch distance is the same, but Forces add :
=> k_parallel = k₁ + k₂ .
this implies that thicker wire springs have larger stiffness k than otherwise similar thin-wire springs .

elastic modulus : even straight steel wire stretches a little bit if pulled with enough Tension ...
. . . the cause of the stretch is stress : Tension spread across the cross-sectional wire Area , in [N/m²] or [N/atom²]
      the effect of stress is strain : fractional length increase , ΔL /L , due to increased average atom-to-atom distance parallel to T
. . . the ratio : stress/strain is Young's modulus Y , which is a property of the material (not just the object)
      Y describes the Pressure that would cause its length to double (if it didn't break first)
=> P = Y ΔL/L , for Pressure in solids .

for fluids, which have no intrisic shape, we describe the change in Volume as they are compressed :
. . . each atom itself is a little bit compressible , with non-infinite elastic stiffness .
=> P = −B ΔV/V , where B is called the Bulk Modulus for the material.

Pressure : more intense Pressure at deeper depths within a non-accelerating fluid ;
. . . ΣF_z=m a_z = 0 => P_bottom·A_up = P_top·A_down + m_fluidg ;   m = ρ A×δh
      Pressure at bottom of fluid sample = Pressure at top of fluid sample + ρ g δh ...
=> δP = ρ g δh .

Pressure Units are [N/m²] ; this set of units is also called a "Pascal" , abbreviated Pa .
. . . but other pressure units are also used : besides the "psi" = pound-Force per square inch , there are also
      torr : pressure caused by 1[mm] deep mercury column in standard Earth gravity (9.81[N/kg) . . . and
      "inches of water" , P caused by 1" deep column of water in 9.81[N/kg] gravity . . . and
      "atmosphere" , the pressure caused by "standard" depth (to "sea level") dry Earth atmosphere in standard Earth gravity

. . . Force difference : F_{on bottom} − F_{on top} is called "Buoyant Force" by the fluid on object ;
=> δF = − ρ_fluid g V_object . . . upward, as it used to cancel the weight of fluid that isn't there anymore ... the "displaced fluid"
      typical Bouyant Force for condensed matter (solid or liquid) immersed in sparse matter (gas, plasma) is −0.001 × mg ;
      but condensed matter can float in other condensed matter ; and some gases float in other gases .
=> for many situations bouyant Forces can NOT be ignored .

Important statement #5 about how the Universe works:

Gravity : every mass in the universe pulls on every other mass in the universe

each source mass M_s contributes to the gravity field g at each location.
. . . the total gravitational influence emanating from source mass M_s ~ GM_s , pointed toward the source mass ,
      with G = 6.67E−11[Nm²/kg²] = 66.7[N/kg · (Mkm)²/(1E30kg)] ,
      or . . . G = 5.931E−3[N/kg · (au)²/M_Solar] <= astronomical units
. . . this influence spreads out, 2 ways in 3-d space, so the contribution to g is weaker far from the source ;
      so strength contributed goes as 1/"distance squared" = 1/r² from that source
      . . . { its contributed g piercing thru the whole surface of any surrounding shell is the same => g·A = 4π GM_s }
=> g_{by M} = G M / r²_M-to-point (toward M) . . . then use   F_{gravity, on m} = mg ; NOTICE : m ≠ M !

The contribution to g from every source mass (reasonably nearby) must be added as a vector
      to get the total   g   at that place (the field point) .
. . . so there IS only ONE gravity field vector at any location , after adding all the contributions ... (3 components, one vector)

Since the gravitational influence spreads with distance from the souce Mass ,
      the gravity field is more intense on the "near side" of an object than it is on the "far side" of the object.
. . . relative to the average field at the object , these "excess" and "deficit" Forces (per unit object mass) seem to pull outward
      Tidal Tension (per unit mass in the object) = g_near − g_average   and   g_average − g_near   .
=> Δg = g / (1 ± Δr/R)²
. . . this is almost the same on each side , and if the stressed object is small (compared with source Mass distance ; Δr/R << 1 )
=> Δg ≈ ( ^d/_dr{g(r)}|_R ) · Δr   . . . ≈ 2gΔr/R   .

For a "field point" anywhere inside a hollow shell of mass, the gravity field contributions from all the shell source Mass will cancel !
. . . only M located non-isotropically around the field point contribute to g .
=> gravity is weaker at the bottom of a deep hole into Earth ... only the "inside" Mass counts .