GeoCitesSites.com

Ch 3 Velocity

Updated 10/09/03
Menu

Introduction

 1    a] Kinematics is the study of motion
       b] Dynamics is the study of why things move.  
       c] Motion at constant speed and direction needs no cause. 
          “An object in motion will stay in motion is a straight line”
          Motion that includes either a change in speed or direction is caused by a
          force.

 2    a] Motion is a change in position over time.
       b] Motion can be described by the concepts of position, displacement, speed, velocity, and acceleration.
           Position:  the location of an object on a coordinate system (7 mi [NW] of home)
           Distance--the total length of a path, without direction.
           Displacement (Δd)--straight line change in position from start to end.
           Speed (m/s):  the distance traveled per unit time
           Velocity (m/s):  the distance and direction traveled per unit time;
           V = displacement/elapsed time
           Acceleration (m/s/s):  the rate of change in velocity;
           a = (Vfinal – Vinitial)/elapsed time      It is more than a change in speed!
     

 3    a] To measure it you must set up a coordinate system or reference frame.
       b] A one dimensional example is:  a line with numbers (football field)
           A two dimensional example is graph paper.
           A three dimensional example is longitude & latitude lines on a globe.

 4    To simplify our study we will study objects that
       a] Are point objects and not three-dimensional bodies.
       b] Move in straight lines or in circles.

 5    a] Large & small objects have the same diagram:  a circle with a label
      Ex:  bacteria 0 or planet 0
      b] Large & small distances is a reference to the size of one object compared to the
      rest of the diagram and not to the real distances involved.
                                    || Bacteria
                                    |                        |
                                    |                        |
                                    |                        |
      Large Distance Ex:  |<----2 um --->|
      Small Distance Ex:  |
ßà| 2 um      used when you don’t have enough room.
                         
à||ß 2 um  
      c] Size:  use two headed arrow with reference lines (see above)
         Direction:  use one headed arrow without reference lines:  0 -----
à

 6    a] All motion is relative means that the speed of an object depends on the observers location or reference frame.
       b] If you stand on the side walk an look at a building it is not moving.
       You say its speed is zero with respect to the sidewalk.
       If you stand on the moon the building is moving with the earth’s rotation. 
      You can say its speed is about 720 mi/h with respect to the moon.

      Note:  People create reference frames naturally by using the background.
     c] Everyday motion is usually relative to the earth's surface.
         Planetary motion is relative to the sun and not to the earth’s surface.

1.1 OBJECTIVE:  Distinguish between vector and scalar quantities.

 7    a] A Scalar is described completely by its magnitude.
       b] A vector requires both direction & magnitude.
       c] A vector can be represented by an arrow.
      d]     Scalars    |                Vectors                             .
       Distance (m)   | Displacement = distance with direction (m, north)
       Speed (m/s)    | Velocity     = speed with direction ( 25 mi/h south)
          Time (s)       | Acceleration (mi/h/s)
        Mass (kg)      | Force (N, Newton, left)
       Temp (
oC)      | Momentum = mass x velocity = Scalar x vector (kg m/s, east)
      Density (g/mL) | Torque = distance x force = Scalar x vector (m Newton, up)
        Area (m
2)      |
      Volume (m
3)    |
       Pay Rate ($/h) |
      GNP (Billion $)|
      There are hundreds of scalars: Consumer Price Index, Dow Jones, GPA, IQ, conductivity, resistance, moles,
       moles/liter, etc..

 8    Under what conditions can an object accelerate at constant speed?
       An object moving in a circle at constant speed is constantly changing its direction, i.e. changing its Velocity 
      (= speed with direction)

 9    Use units to name the following vectors or scalars: 
       No Direction = Scalar, Direction = Vector
      a] 50 km/hr, S [b] 6 km [N], V  [c] 2000 kg/m
3, S  [d] 6 centuries S
      e] 800 kg,  S  [f] 1.0 kg/wk S  [g] 20 m/s [S] V   [h] 400 N [down] V

10    a] The letter d stands for displacement.  Diagram:  ------>
                                                                                        5 m
        b] The Greek letter delta,
Δ, means difference or change or Final - Initial
        Ex: 
ΔD = Dfinal - Dinitial, ΔT = Tfinal - Tinitial, ΔV = vfinal - Vinitial

OBJECTIVE:  Establish a strategy for solving word problems.

12  WHAT ARE THE 5 STEPS IN SOLVING A WORD PROBLEM?
    Ex:  How long will it take you to travel 170 mi at 50 mi/hr
    a] State the given and find as a set of equations.
      t = ?      D = 170 mi      S = 50 mi/hr
    b] Write the general equation:      S = D/t
       Do the needed algebra           t = D/S
    c] Substitute                      t = 170 mi/50 mi/hr
    d] Calculate                       t = 3.4 hr
    e] Use unit and size checks.
          Unit Check:  d = Vt + a t2/2
           Sub units m/s for V, s for t, m/s2 for a
                  m ?=? m s + m s2   
                             s     s2     
                  m ?=? m + m = m (it checks) 
Note:  2 is a constant and has no units.
Note:  If you add or subtract meters and meters you get meters.  This is not algebra but a unit check-->you don't get 2 meters.

     
Size Check #1:  Estimate t < 200 mi/50 mi/hr
                               t < 4 hr  Since t = 3.4 hr it checks  
      Size Check #2:  Min < Avg < Max

13    a] Circumference of a circle = 2pr      .
            Diagonal of a triangle =
a2 + b2
        b] Speed equations:  S = D/t, memorize
            Multiply both sides by t:  Sxt = D   derive
            Divide both sides by S:   t = D/S    derive

14    A CAR IS DRIVEN 40 km EAST AND THEN 60 km SOUTH.
        a] DRAW & LABEL THE DIAGRAM.
        b] d =
(40 km)2 + (60 km)2 = 72.1 km, south east
        c] SIZE CHECK:  Long Side
< DISPLACEMENT < Long + Short side
                                          60 km 
<         72.1 km          < 40 km + 60 km = 100 km
        d] AVERAGE SPEED = Total distance/total time = (40+60)km/(5+7)min = 8.3 km/min
            Note:  Avg Speed is not (Speed #1 + Speed #2)/2
        e] AVERAGE VELOCITY = Displacement/total time
                                                   = 72.1 km south east/12 min = 6.01 km south east/min

15    a] V = ?                     V = D/t
            D = 300 ft north      V = 300 ft north/11s
             t = 11 seconds.       V = 27.3 ft north/s

         b] V = 360 mi/h[NW]            V = D/t
             D = ?                            360 mi/h[NW] = D/30 s
             t = 30 sec.          Multiply both sides by 30 s:   30 s x 360 mi/h[NW] = D
            Note units
              D = 10,800 mi s/h x 1 h/3600 s = 3 mi
 
         c] V = 20 mi/h[SW]                   V = D/t
             D = 150 mi           20 mi/h[SW] = 150 mi/t
               t = ?                            t = D/V = 150 mi/20 mi/h = 7.5 h

         d] S = ?                                                                         S = D/t
             D = 2
pr = 2p93,000,000 mi = 584,040,000 mi         S = 584,040,000 mi = 66,671 mi/h
              t = 365 days x 24 h/day = 8760 h                                         8760 h  

    e] Earth is 93,000,000 mi from the sun.  (Light, c = 186,000 mi/s). 
       Diagram & find how long will it take light to reach earth (min & sec)?
       D = 93,000,000 mi    S = D/t ---
>t = D/S = 93,000,000 mi/186,000 mi/s
       S = 186,000 mi/s                 t = 500 sec x 1 Min/60 sec = 8.33333333 min
       t = ?                            t = 8 min + 0.3333333333 min x 60 sec/min  
                                        t = 8 min 20 sec (Note:  Don’t round off)

    f] Your ship has a 2,000 ft turning radius. 
       Find how long it takes to make a U-turn if it is moving at 12 mi/h.
       U turn D = half a circumference = 2
pR/2 = p 2000 ft
       S = 12 mi/h x 5,280 ft/mi = 63,360 ft/h
       t = D/S =
p 2000 ft/63,360 ft/h = 0.0991 h x 60 min/h = 5.95 min

16    A car is traveling 50 mi/h east, and a truck, 270 mi away, is moving at 
      40 mi/h west along the same road.  
      When and where do they meet?  Diagram your answer.
      Vcar --
> 50 mi/h       40 mi/h <--Vtruck
      |
<---------------270 mi--------------->|

      Time: Tcar = Ttruck = T      
      Distance:  Dcar + Dtruck = 270 mi
      Eq:  D = S x t  yields-->  Dcar = Scar x Tcar = Scar T
                                             Dtruck = Struck x Ttruck = Struck x T
      Substitute this into:
      Dcar + Dtruck  = 270 mi ---> Scar T + Struck T  = 270 mi
                                50 mi/h T + 40 mi/h T = 270 mi
                                            90 mi/h T = 270 mi


          T = 270 mi/90 mi/h = 3 hr---
> They meet 3 hr after starting.
          Dcar = Scar x T = 50 mi/h x 3 hr = 150 mi
          Dtruck = Struck x T = 40 mi/h x 3 hr = 120 mi
          Check:  Dcar + Dtruck ?=? 270 mi
          150 mi + 120 mi  =  270 mi (it checks)
         Answer
          Vcar --
> 50 mi/h       40 mi/h <--Vtruck
          |---150 mi-----------
>|<-----120 mi-----|
                                ^
                                                                 Meet Here

          |
<---------------270 mi--------------->|

PHY Ch 3b MOTION CALC

1    a] WHAT ARE THE 5 STEPS IN SOLVING A WORD PROBLEM?
Ex:  How long will it take you to travel 170 mi at 50 mi/hr
        a] State the given and find as a set of equations.
         t = ?     D = 170 mi    S = 50 mi/hr
        b] Write the general equation:    S = D/t
             Do the needed algebra          t = D/S
        c] Substitute                     t = 170 mi/50 mi/hr
        d] Calculate                      t = 3.4 hr
      
b] NAME & GIVE EXAMPLES OF 2 LOGIC CHECKS.
        Unit Check:  d = Vt + a t2/2
       Sub units:  m/s for V, s for t, m/s2 for a
                  m ?=? m s + m s2  
                        s     s2     
                  m ?=? m + m = m (it checks) 
Note:  2 is a constant and has no units.
Note:  If you add or subtract meters and meters you get meters. 
           This is not algebra but a unit check-->you don't get 2 meters.

Size Check:  Estimate t < 200 mi/50 mi/hr
                          t < 4 hr    Since t = 3.4 hr it checks

 2   A CAR IS DRIVEN 40 km EAST IN 5 MIN AND THEN 60 km SOUTH IN 7 MIN.
a] DRAW & LABEL THE DIAGRAM.
b] CALCULATE THE DISPLACEMENT.
         d = (40 km)2 + (60 km)2 = 72.1 km, south east
c] SHOW THE SIZE CHECK FOR DISPLACEMENT (Equation & Numbers).
   SIZE CHECK:  Long Side < DISPLACEMENT < Long + Short side
                   60 km  <   72.1 km    < 40 km + 60 km = 100 km
d] FIND THE AVERAGE SPEED.                                     
   AVERAGE SPEED = Total distance/total time
                 = (40+60)km/(5+7)min = 8.3 km/min
   Note:  Avg Speed is not (Speed #1 + Speed #2)/2
e] FIND THE AVERAGE VELOCITY.
   AVERAGE VELOCITY = Displacement/total time
                 = 72.1 km south east/12 min = 6.01 km south east/min

 3   What is the cause of motion?  (Hint:  2 answers)

 4   a] State the equation for velocity.
      b] Derive the equations for displacement and elapsed time.

 5   An airplane experiences a displacement of 680 m [NW] in 3.5 s.
      a] Draw & label the given & find.
      b] Find its velocity.
      V = ?                          V = D/t
      D = 680 m [NW]        V = 680 m [NW]/3.5 s
        t = 3.5 seconds.        V = 194.3 m [NW]/s

 6   You see lightning [SE], 3.5 s later you hear the thunder (sound = 1,116 ft/s). 
a] Draw & label the given & find.
b] Find the lightning's location (mi).
   V = 1,116 ft/s [NW]                                                          V      =  D/t
   D = ?                                          Substitute:   1,116 ft/s [NW] = D/3.5 s
   t = 3.5 sec.      
                         Multiply both sides by 3.5 s:   3.5 s x 1,116 ft/s [NW] = D
                                D = 3906 ft
Convert to mi (1 mi = 5,280 ft):  D = 3906 ft x 1 mi/5280 ft
                                  D = 0.740 mi

 7   A ship moving at 20 mph [SW] is 150 mi from port. 
      a] Draw & label the given & find.
      b] Find when will it arrive in port.

 8   A car is traveling 50 mi/h east, and a truck, 270 mi away, is moving at 40 mi/h west along the same road. 
a] Diagram the given & find.
b] Find when and where they meet. 
c] Show the check.
Vcar --> 50 mi/h       40 mi/h <--Vtruck
|<---------------270 mi--------------->|

Distance:  Dcar + Dtruck = 270 mi
Time: Tcar = Ttruck = T
Eq:  D = Sxt yields  Dcar = Scar x Tcar = Scar T
                     Dtruck = Struck x Ttruck = Struck x T
Substitute this into
Dcar + Dtruck = 270 mi -->Scar T + Struck T  = 270 mi
                       50 mi/h T + 40 mi/h T = 270 mi
                                   90 mi/h T = 270 mi
         T = 270 mi/90 mi/h = 3 hr--> They meet 3 hr after starting.
Where?  Dcar = Scar x T = 50 mi/h x 3 hr = 150 mi
      Dtruck = Struck x T = 40 mi/h x 3 hr = 120 mi
Check:  Dcar + Dtruck ?=? 270 mi
      150 mi + 120 mi  =  270 mi (it checks)
Answer
Vcar --> 50 mi/h       40 mi/h <--Vtruck
|---150 mi------------>|<----120 mi-----|
                                Meet here
|<-----------------270 mi----------------->|

 9   You have a reaction time of 1.0 s.  Your car is moving at 30 mi/hr.  a] How far (ft) between cars must you be to have a chance at avoiding a collision?  (5,280 ft = 1 mi)  [b] Show the size check.

10   a] Earth is 93,000,000 mi from the sun.  (Light, c = 186,000 mi/s). 
Diagram & find the time for light to reach earth (hr, min & sec)
D = 93,000,000 mi             S = D/t --->t = D/S = 93,000,000 mi/186,000 mi/s
S = 186,000 mi/s                 t = 500 sec
t = ?                  
The answer looks like:  _?_ h _?_ min _?_ sec
Since there are 3,600 sec in 1 hr, 500 sec is less than 1 hr
Thus:  t = _0_ h _?_ min _?_ sec
t = 500 sec x 1 Min/60 sec = 8.3333333333 min
Thus:  _0_ h _8_ min _?_ sec
Subtract the 8 and leave all other digits in the calculator: 
DO NOT ROUND OFF.
SEC = 0.3333333333 min x 60 sec/min
t = _0_ h _8_ min _20_ sec

Pract:  Neptune is 2,798,800,000 mi from the sun. 
t = D/S = 2,798,800,000 mi/186,000 mi/s
      t = 2,798,800 mi/186 mi/s
      t = 15047.31183 s
Step 1:  Find hrs:  t = 15047.311...s x 1h/3,600s =
                      = 4.179. . . h
Subtract 4 in your calculator (DO NOT ROUND OFF)
Enter it into the answer:
Ans:  t = 4 h _?_ min _?_ sec

Step 2:  Find min:  t = 0.179. . . h x 60 min/h
                    t = 10.788. . . min

Subtract 10 in your calculator (DO NOT ROUND OFF)
Enter it into the answer:
Ans:  t = 4 h _10_ min _?_ sec

Step 3:  Find sec:  t = 0.788. . . min x 60 sec/min
                    t = 47.31183 sec
Enter it into the answer:
Ans:  t = 4 h _10_ min _47.3_ sec

Step 4:  Check:  note the decimals in your seconds 0.31183 are the same as those at the start:  t = 15047.31183 s

b] Earth's diameter is 8,000 mi.  Diagram & find the velocity of a building at its equator (mi/h).
V = Speed + Direction.
Direction is tangent to the equator, East
Speed = D/t = 2pR/24 h = pDiam/24 = p8000 mi/hr = 1,047 mi/h
V = 1,047 mi/h tangent to the equator, East

c] Earth is 93,000,000 mi from the sun.  Diagram & find its velocity.
V = Speed + Direction.
Direction is tangent to the orbit, East
D = 2pR = 2p93,000,000 = 584,336,234 mi
t = 365 days 24 h/day  = 8760 hr
Speed = D/t = 584,336,234 mi/ 8760 h = 66,705 mi/h
V = 66,705 mi/h tangent to the orbit, East

Menu