» physicsforums
  (human.hw.help)
 » hyperphysics
 (detailed ebook)
 » Minds-On-Physics
  (practice like hw)
  press "continue",
  below red words

Topic 2 (Rates - especially momentum & velocity)

Quiz 2 was Feb.06 ... here's my grading key as jpg.

A. time "flow" , duration , and change

calendar: split each Moon cycle into 4 pieces (called a week)
. . . east crescent to half-lit, east gibbous to full, west gibbous to half-lit, west crescent to new
      the synodic month of phases is actually 29½ days, not 28 days ... details later.
. . . Moon is called "waxing" as it grows to full, then "waning" as it shrinks to none
      the lit-up side faces the Sun ... the Sun is what lights the Moon.
. . . the Moon's lit-up longitude is the angle from Sun to Moon, as seen by us
      increases about 1° in 2 hours ... (<= 360° in 720hr, really in 709hr)

year: Earth takes 365¼ days to orbit the Sun ... 1 year
. . . Earth Westerners use 12 months × 30½ days/month(average) -1 day = 365 days
      then inject an extra leap day every 4 years.
. . . one old calendar had 13 months × 28 days/month +1 year day = 365 days
      but that can't be split into 4 seasons neatly
. . . seasons begin Mar.20, Jun.21, Sep.22, Dec.21 in our calendar
      on ascending Equinox, Northmost Solsice, descending Equinox, Southmost Solstice

Notice the presumption of constancy in these definitions: constant spin, identical repeats . . . doesn't really happen like that - each day is not identical to the others . . . the Sun appears to move about 1° among the stars, every day . . . but some days its path is tilted a bit North-or-South, not straight Westward . . . so around the equinoxes it is not keeping up with how far West it usually goes       (Sun crosses the meridian moving due West, so only its East-West motion is relevant for time-of-day) . . . but around the solstices the Sun seems to move Westward faster than average => an analemma shows where the actual Sun is, relative to the "mean Sun" during a year Notice that the Sunrise path starts at a different horizon point each day . . . the sunrise point (on the horizon) is directly East on the Equinoxes . . . Northmost on the Summer Solstice, Southmost on the Winter Solstice Notice that the Sun does not rise along a "straight up" path . . . it is inclined from vertical by the observer's latitude . . . horizontal at the North pole sunrise all day long (on the equinox) Sunset follows the same inclined path . . . shifted N or S from due West by the same amount as sunrise . . . so summer has a few extra hours of daylight ; winter has a few less than 12 hrs.

longer: planets and other astronomy events form longer cycles
. . . Jupiter takes about 12 years to cycle thru the zodiac
      some call each year by the zodiac constellation that Jupiter is in front of
      ... year of the snake, or dog, etc.
. . . Saturn's zodiac cycle takes about 30 years
      so Jupiter passes Saturn every 20 years ; 3 times in 60 years
      (May,2000 Ari/Tau ; next Dec.2020 Cap, Nov.2040 Vir, May.2060 Tau
      ... Jul.2119 Tau ... May.2179 Tau ... Mar.2239 Tau/Gem)

. . . after 3 of those 60 year full cycles, (180 years)
      their meeting-place has shifted 30° to the next zodiac sign
. . . after all 12 zodiac signs have been the meeting-place (2160 years)
      the meeting-place is at the (new) sign of the Spring Equinox

. . . Earth's spin axis shifts ever-so-slightly as the Moon pulls our water-bulge
      so solstices and equinoxes shift very slowly (1° every 71½ years)
      after 30 of those lifetimes (2148 yr), they've moved a whole constellation
      ... Spring Equinox was in Gemini, in myth; in Taurus, in legend
      ... ancients named it "first point of Aries"; antiquity in Pisces; modern age Aquarius

process duration, or start-to-stop interval
Physical processes don't care what time it is on the clock, or what day it is on the calendar, or what year it is on the long-count
. . . except for Astronomy, which founded the original definitions for clocks, calendars, years, etc.
A process will endure for a while, from when it starts until when it ends.
. . . we suppose something will change during the process, while time had been going.

"Change" is so important in science that we will usually abbreviate it.
. . . it is a difference time-wise : we use
      Δ   Greek "D" , named "Delta" ; read it as "change in ____"
. . . your Height is not the same as it was 20 years ago, it has changed !
      "difference" means subtract one Height from the other Height ... but in which order ?
      . . . the order makes a difference in subtraction ... (unlike addition)
      did it change upward, or did it change downward?
      which subtracts from the other, in order to get an upward (positive) Change in Height ?
      . . . ΔH = later H − earlier H = H_late − H_early   <=   modify with preceding adjectives or with subscripts
=> Change always means : "later" minus "earlier" . . . "after" you grew minus "before" you grew

Notice that , because time continues forward (whether we like it or not) , time change is always positive
=> Δt > 0

D. Directional quantities are called vectors

Displacement = change in place , in some direction
Δx = place change = "after place" − "before place" = x_after − x_before
. . . as you travel from Huntington to Charleston , your (East-West) displacement is :
      Δx = 52 mi (E) − 8 mi (E) = 44 mi (E) .
. . . as you travel from Charleston to Milton , your displacement is   Δx = 28 mi (E) − 52 mi (E) = −24 mi (E) . . . = 24 mi Westward .
      as you travel from Huntington to Milton , your displacement is 20 mi Eastward .
      . . . no matter what route you use (even Hunt → Charls → Milton)

travel distance is Length along your own path
      a path does not need to be in an objectively constant direction (like "East") so direction is irrelevant.
. . . you always travel along your path , never against your path
      so travel distance can never be negative , never cancels previous distance
. . . Hunt → Milt is 20 mi , if travel is directly there
      but Hunt → Charls → Milt would be   44 mi + 24 mi = (44 + 24) mi = 68 mi .

any change per time is a "rate" ... change / time
. . . production rate tells how much is produced ... per hour or per year
. . . interest rate tells how much money it will cost to borrow $1 ... per year

travel rate is called speed ... travel distance / time duration
=> speed·Δt = distance . . . having speed for a while moves it thru a distance

Displacement rate is called velocity   v . . . Displacement / time span
. . . speed, but also including direction info .
  (1)   keep track of + and − signs !
  (2)   upward is totally distinct from horizontal (independent!)
  (3)   forward is independent from sideways (independent!)
. . . velocity is computed using just the end points (displacement and time-span)
      so what we compute is the average velocity within that time-span.
      an objects' velocity tends to be constant (for short time-spans)
=> v_avg = Δx / Δt   . . . notice: it is a ratio of changes ... (definition)

Displacement occurs when a velocity is kept for a while
      (this is the previous sentence with different word order)
. . . the longer is the duration that the object has velocity, the larger its displacement
. . . the faster is the velocity that the object has, the farther its place changes
. . . place change results from time change , if velocity is non-zero
=> Δx = v_avg Δt   . . . easiest to remember and to work with

How long must the velocity be retained, to achieve some displacement?
. . . dividing the last equation by v   - both sides, of course!
=> Δt = Δx / v   . . . if velocity is zero , it does not move (so Δx = 0 for any duration)

velocity is relative ... to the velocity of the meter-stick carrier (measurer)
. . . to figure out their velocity relative to you, subtract your velocity from theirs
      this should make sense, because you are not moving along the meter-stick you carry
      so whatever your "real" velocity is, (say, along the road),
      ... you have to subtract that to get your zero velocity relative to yourself
. . . if you're traveling 70 mi/hr Eastward (toward Charleston),
      then the trees seem to be moving 31 m/s Westward , as seen by you.
. . . the slow driver ahead of you (28 m/s along road) is becoming 3 m closer to you each second,
      "moving" toward the zero on your meter-stick ... that is, −3 m/s relative to you (negative meaning Westward)
. . . for a barge to go 3 m/s upstream , as the water goes 1 m/s downstream,
      it must travel 3 m/s − ( −1 m/s) = 3 m/s + 1 m/s = 4 m/s relative to the water.
that is, its velocity relative to the water is added to the velocity of the water relative to you,
      to get its velocity relative to you ... add them as arrows !

Earth spins Eastward once each day
. . . which direction do all the stars seem to be moving, as seen by us?

Momentum . . . m v . . . one key to explanation
total momentum is always the same . . . it is a conserved quantity (like mass is)
      (mo-ment-um , latin for "thing of the moment") a conditional property that objects have
. . . recall that velocity has direction , so momentum "flows with" the object that has it
      (symbol is p, for im-pet-us , latin for "thing put into" ...
      ... Newton likened it to the pimento, which stays within the pickled olive)
. . . there is no guarantee that the product of 2 quantities must be physically meaningful (and additive)
      momentum is the rate that mass moment changes :   Δ(mx) / Δt   =   m Δx / Δt .
. . . when the 1 kg spring-cart pushed the 3 kg loaded cart,
      (1 kg)(2 m/s) + (3 kg)(−½ m/s) = ½ kg m/s . . . almost zero , like total motion was before the push

momentum must be put into an object ... by an external active subject
. . . the subject does the pushing . . . the object gets pushed
. . . the subject must be outside the object , distinct from it (can't pull yourself up)
. . . the loaded cart was pushed to momentum   p₃ = (3 kg)(−½ m/s) = −1.5 kg m/s , by the spring-cart.
      the spring cart was pushed to momentum   p₁ = (1 kg)(2 m/s) = 2 kg m/s . . . pushed by the loaded cart.
      ... we know that the loaded cart did this push, because the spring cart did not get pushed when the loaded cart wasn't there!

total momentum tends to be constant   Σ p_later ≈ Σ p_earlier
This is Newton's first Law . . . in the absence of Force , an object's motion stays the same
. . . by "motion" Newton meant momentum ... m v
      is a quantity because mass and velocity are both quantities
      has the same direction as velocity because mass has no direction (scalar) and is always positive
. . . if the mass is constant ... it usually is ... the velocity is also constant.
=> if something's velocity is not constant , there is a reason for it !

there is no guarantee that the product of 2 quantities must be physically meaningful (and additive)
. . . momentum is the rate that mass moment changes :   Δ(mx) / Δt   =   m Δx / Δt .

being Pushed for a while , changes its motion   =>   F·Δt = Δp
. . . the motion (total) is momentum   p = mv
. . . the Push is called a Force ... weight (pull by gravity) is not the same as mass (amount of matter)

if something's velocity changes . . . then it accelerated
acceleration is the velocity change rate ... it is directional ...
      (you can not literally push yourself to go faster)
. . . We use the letter Δ , capital (greek) "Delta" , to mean "change in ____" something.
      so , a = Δv / Δt ,   where this Δt is the time interval that the velocity changed during.
=> a·Δt = Δv . . . accelerating for a while changes its velocity ... that's what acceleration does

What might cause acceleration? ... the same thing that causes momentum to change ... Force !
. . . must be an external Force , applied to our object (by an active subject)
. . . since mass doesn't change, F Δ t = Δ mv   becomes   F Δ t = m Δ v  
. . . Force causes mass to accelerate
=> F = m a   . . . famous equation called "Newton's 2nd Law" .

External Force causes a mass to accelerate   =>   Σ F = m·a . . . = m Δ v / Δ t

the mass is the mass in the system you're watching.
      inside the object (in contrast with "outside the system" or "outside the subject")
. . . it is the total mass in the system
      add up all the masses , if there's more than one in there.
. . . the mass of the system is what gets accelerated
      the acceleration of the center-of-mass ... is the average acceleration of the system.
. . . this is meaningful and not complicated, since mass does not depend on the environment
      Σ m   has the same value everywhere.

Force is exerted by something Outside the system       the active subject in the sentence describing what happened. . . . applied to the (passive) object inside the system       so each Force vector (arrow) must pierce the system boundary. . . . it is the total external Force , added directionally , that is important       net Force means what remains after they've all been summed ;       ... Forces tend to oppose one another , canceling . . . . represent each Force that is applied to the object as an arrow       pointing in the direction that the Force is pulling (or pushing)       and with length that shows how strong that Force is       ... decide which is positive and which is negative! . . . the special situation of zero net Force is called "equilibrium"       Forces cancel => acceleration is zero => velocity is constant

each Force has a nameable cause
      which depends on a particular property.
. . . gravity pulls any mass that is immersed in the gravity field downward
      around here, the gravity field has 9.8 N/kg intensity.
      ... each kilogram is pulled with a 9.8 Newton Force.
      ... (a few percent different at different places)
      on our Moon's surface, gravity's intensity averages 1.62 N/kg
      ... (a few percent different at different places)
. . . Surface Forces push outward, strong enough to keep the object from penetrating it
      (they do compress slightly, depending on how hard the surface is - spring-like!).
      the surface Area pushes straight out thru it (perpendicular = "Normal")
. . . a spring pulls inward depending on how far it has been stretched
      stiffer springs pull harder, for the same stretch distance.
      to stretch a spring, you need to pull outward on both ends
      ... this Tension makes the spring material deform
. . . compression springs (in click pens) push outward (stretch is negative)
      with strength that increases with how far they've been compressed
. . . Tension Forces pull inward, hard enough to keep from breaking
      (they do stretch slightly, depending on their stiffness - spring-like!).
      ... ropes and cords usually have much less mass than the object they are tied to, and stretch very little (neglect it)
      ... so Tension is essentially the same all along that rope.

most of the time, one Force (gravity's Force) pulls an object down onto the table
. . . the table deformed a wee bit as the object was placed onto it, until it pushes outward just strong enough
      that the surface Force upward (+) just cancels gravity's Force downward (−)
the table will deform much farther if the object is thrown onto its surface
. . . so the table pushes it up much harder than gravity pulls it down
      and the ball accelerates upward until the table stops pushing it so hard.

except for gravity's Force, these can all be identified by contact
      the subject pushing on the object (which is being watched).
. . . each side of the contact surface pushes thru it into the other side
      equal strength Forces, in opposite directions
. . . these pairs of Forces act on different things (object c.f. subject)
      so you do not add them up to get a net Force.
. . . (if both are inside your system , neither pierces the system boundary
      so neither accelerates the system's c.o.m. ... make your system smaller!)

Friction Force often "Accidentally" Cancels another Force
friction Forces tend to resist sliding
      that is, they oppose the slide.
. . . if the pushing subject can only move so fast, then
      the pusher will often push at constant velocity for a while
      ... zero acceleration implies zero total ("net") Force.
      they started it moving by pushing stronger (so, +'ve net Force) for a little while, to give it some momentum (mv),
      ... but let it stop by pushing weaker than friction (so, −'ve net Force) until is slows to a stop.
      from very beginning to very end (stopped to stopped) is zero average net Force.
. . . wind resistance & water resistance Forces (both viscous and drag)
      are stronger for faster speed thru the fluid
      ... speed increases until the other (imposed) Force gradually becomes cancelled.
      bigger size objects encounter more resistance, if at the same speed
      ... because they hit more wind (or water) as they travel
      so fluffy objects achieve slower terminal (equilibrium) velocity.

Curved Path implies acceleration toward the Center of Curve

=> a_centrip.   =   v ² /r   . . .   centripetal = "center-pointing"

recall : velocity includes direction . . . represent as arrows . . . change in velocity . . . Δ v = vafter − vbefore       even though the speed is the same, Southward is not the same as Eastward!       the diagram is top view of an object moving clockwise around a circle . . . "subtract" means "add the opposite" . . . the "additive inverse" is the negative       ... (that's where negative numbers come from!)       the opposite of Eastward is Westward       so we should add Westward to Southward . . . . add vectors tail-to-tip, like road-trip "directions" or connect-the-numbers drawing game       the change in velocity during this ¼ circle is 1.41 |v| to the South-West       (check that by using Pythagoras: c² = a² + b² )       it needs to remove the original Eastwardness       as well as have all the Southwardness leftover. = Δv points toward the circle's center during that time span.

Large circles take a long time to travel Around
      Time for one complete go-round is called the (repeat time) Period ... usual abbreviation   T .
. . . total distance around is circumference   = 2 π r = π·diameter .
      from v = d / Δt ,   Δt = d / v
      ¼ circle will take   Δt = π r / 2 v . . . (<=right?)
. . . acceleration   a = Δ v / Δ t . . . substitute formulas to get
      a ≈ 1.41 |v| 2 v / π r
      . . . the 2·1.41/π = 0.9 is not exactly 1 here only because ¼ circle is very coarse to calculate Δv from
=> a_centripetal = v ² / r . . .

example: bucket of water overhead
suppose you try to swing a bucket filled with water all the way around in 1 second.
. . . using your shoulder as a pivot, the water is about 1 meter from that center.
speed ≈ d/Δt = 2 π r / T ≈ 6 (1m) / (1s) ≈ 6 m/s .
      (1 gal ~ 4 liter would have KE = ½ (4kg) (6m/s)² = 72 J   + PE = (4·10 N)(1m), so be careful!)
a = v² /r = (6m/s)² /(1m) = 36 m²/s²m = 36 m/s² .
. . . it does not need to go this fast to stay in the bucket, because gravity only provides 9.8 m/s²
      your shoulder will need to provide the other 26.2 m/s² acceleration.
      at ½ this speed,   a is ¼ as quick ... 9 m/s² ... water falls quicker than bucket
. . . or you can use a slower speed with a much shorter radius (notice: speed counts twice)
      ½ the speed (3m/s) with ¼ the radius again makes a = 36 m/s² .
      ... but takes ½ as long because the path is so short.

Force Applied 90° to velocity changes the object's path (deflects) , not its speed
Consider swinging a bucket of water around you, in a horizontal circle
      hold one end of rope above your head, the bucket on the other end moving at wist level.
. . . The rope Tension does no Work . . . so can not change the bucket's KE .
      change the speed of a bucket by pulling along the curved path
. . . caused Force (e.g. string) does not need to point to the center
      it is the net Force that causes acceleration
      so string + gravity will point inward to circle center (horizontal circle)
. . . "centripetal Force" IS NOT ANY FORCE'S NAME !
      name each Force by its cause ...
      even if centripetal acceleration is its effect .
. . . triangles of string Tension along the diagonal
      show that part cancels weight (Force by gravity downward)
      and some causes centripetal acceleration (horizontal).
      if you draw a Force triangle to scale you can verify each portion.

the Gravity Field is Caused by Mass ; g Intensity Decreases w/ Distance ² (as g spreads)

=> g = G M_subject / d ²   . . . G = 6.67 E−11 [ Nm²/kg² ] everywhere (Universal)

the property that gravity influences   (mass)   is the same property that causes it.
      but the cause M is the subject while the the influenced m is the object.
. . . the full formula for the Force that Earth's gravity applies to an object ... replace "g" with its formula above
=> F = m   G M / d ² .

Every Mass (even You) Causes a Gravity Field
. . . but only humungous masses have noticeable gravity .
example: 70 kg student ... what gravity do they cause at 1m distance
. . . find, on your calculator, the EE or Exp button
      I get 467E−11 [N/kg] = 4.67E−9 [N/kg]
      really small compared to Earth's 9.8 [N/kg].
=> but not zero!

Force Increases with Proximity ... 2× as Close , 4× as strong
. . . ½ as far means d = ½ m ... so d² = ¼ m² .
      4.67E−9 [Nm²/kg] / (¼ m²) = 18.68E−9 [N/kg]
      closeness counts twice!
. . . we should check the Sun's gravity out here at Earth distance:
      (66.7E−12) (2E30 [kg]) / (150E9 m )²
=> g_{by Sun} = 5.93E−3 N/kg . . . 0.006 N/kg ... compared to Earth's 9.8 N/kg here (we're close to it!)

Use the Average Distance to the Mass ... Center-to-Center
How to convince ourselves that this is the correct value of   G ? Try it for Earth!
. . . We're 6.37E6 meter from the middle of Earth's 5.98E24 kg
      g = (66.7E−12 ) ( 5.98E24 kg) / ( 6.37E6 m) ²
=> g_{by Earth} = 9.83 N/kg ... if you've punched it in right (use the EE button, not these 4 keystrokes "× 1 0 ^" )
. . . actually , this agreement is how we know Earth's mass ... in terms of laboratory-scale masses [kilogram]

very useful to use ratios if you know one g or F
      essentially using "G" in different units
. . . G = 9.8 Newton·EarthRadius²/(kg·EarthMass) . . . to find surface gravity for different planets.
. . . example: Moon's mass is 1/81 × Earth's mass ... implies weaker ... only 1/81 as strong ( M = 0.012345 EarthMass)
      but Moon's surface is only 1/3.66 × as far from its center as Earth's surface ( d = 0.273 EarthRadius)
      . . . that implies stronger by factor 3.66² = 13.4 ×
=> g_{on Moon} = g_{on Earth} ( 13.4 / 81 ) = (9.8 N/kg) × ( .012345 / .0273² ) = 1.62 N/kg . . . ~ 1/6 of Earth's.

. . . example: Sun's mass is 333,333 × Earth's mass ... implies intense gravity ...
      but Earth is 150 Million km from the Sun ... that is d = 150,000,000/6400 = 23,440 EarthRadius)
. . . g _{by Sun, here} = 9.8 Newton·EarthRadius²/(kg·EarthMass) × 333,333 EarthMass / (23440 EarthRadius)² = 0.00595 N/kg
=> G _{for planet orbits} = 0.00595 Newton·EarthOrbitRadius²/(kg·SunMass) . . . to find the Sun's gravity at different planet distances.
. . . the Earth orbit distance (from Sun) is called the "Astronomical Unit" , abbreviated AU . Used to describe solar-system sized distances

use the conceptual ratio technique above ... think of distance squared as an Area

Kepler #1 : Orbits are Ellipses (not circles) - Sun is at one Focus (not center) We all orbit the Sun counter-clockwise, as seen from Draco (from above the North Pole) . . . that's what we mean by "North" : forward along the axis, as a right-hand screw turns       Earth's spin North (toward star Polaris) is tilted 23½° toward Taurus/Gemini from Earth's orbit North       ... straight up from the ecliptic plane (as seen from Earth, Draco "winds around" the N.ecliptic pole) an ellipse is a stretched-out circle . . . cut the circle's center in half (each half becomes a focus), move the two foci apart       the farther they're spread, the more eccentric the ellipse becomes. . . . not an oval (semicircles joined by straight lines) - a smooth curve! . . . pin string to the foci, pull tight with a pencil, trace around the ellipse. this orbit shows eccentricity of about 60% . . . the aphelion is 4× the perihelion . . . eccentricity is difference / sum : e = (ap - peri)/(ap + peri) ≈ (4-1)/(4+1) = 3/5       more like a comet orbit here . . . real planet orbits are only slightly eccentric ... 1.7% for Earth, 9% for Mars => gravity source Pulls from one focus   . . . planet's average location is toward the "empty" focus

Planet gains speed as it falls "toward" Sun . . . of course it misses, but it still gains speed as it gets closer (pulled by gravity).       it is fastest where it is closest to the Sun (peri helion , "around" Sun as in "peri"scope) . . . and loses speed as it "rises" away from (farther from) the Sun       it is slowest where they are farthest from the Sun (ap helion , "high above" Sun as in "apex") . . . until it starts falling inward again (gaining KE). Kepler #2 : sideways speed × radial distance = constant for one orbit . . . it is moving 2× as fast at the closest , than it is at the farthest.       (for orbits around Earth, these are called peri gee and ap ogee) . . . on the illustrator linked below, set the sweep time to 1/16 of an orbit ...       each sweep segment has the same Area (half as far moves twice as fast) => this is because Angular (Rotational) Momentum (d×mv) is Conserved (more later...)

Kepler #3 : Average Distance Cubed / Orbit Time Squared = ... Mass of Gravity Source Compare orbits to Earth's orbit, get Gravity Source compared to the Sun's Mass.       Kepler left the comparison in terms of base units distance and time . . . Newton's re-wording: a planet's acceleration a, is caused by gravity g       . . . v ² /r = GM / r ² . . . Leibniz re-wording: its Kinetic Energy is half-way out of its Potential-Energy hole       . . . ½mv² = ½ m(−GM/r)   ... but that's topic 3 . . . example : Mars' orbit distance 1.52× Earth's , Mars orbit time 1.88 yrs.       1.52 ³ / 1.88 ² = 3.52 / 3.53 = 1 "Solar Mass".       trivial, since we know Mars orbits our Sun (but it's a check) . . . example : Jupiter's Europa : orbit distance 671,000 km (from Jup) = .671/150 = .00447 AU       orbit time 3.55 days = 3.55/365 = .00973 . . .       . . . (.00447)³ / (.00973)² = .0000000895/ .0000946 = .00095 => Mass of jupiter = .00095 × Mass of Sun (about 1/1000 ... 2E27 kg .

Kepler #3 is the way we figure out the masses of other stars ... and extrasolar planets !
here's a link to a pretty good illustrator for ellipse orbit vocab and Kepler #2.
. . . you need to move their planet from aphelion so you can see r₁ separate from r₂ .