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  • Committer: Hilaire Fernandes
  • Date: 2016-02-15 08:53:02 UTC
  • Revision ID: hilaire.fernandes@gmail.com-20160215085302-sky2a9tz123dgs68
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<TeXmacs|1.99.4>
 
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<style|article>
 
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<\body>
 
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    \;
 
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    <assign|figure-text|<macro|<localize|Figure>>>
 
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  <doc-data|<doc-title|Drawing the Parabolic Trajectory of an Object under
 
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  Gravity>|<doc-author|<\author-data|<author-name|Chao-Kuei
 
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  Hung>|<author-email|ckhung@drgeo.eu>>
 
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    \;
 
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  <|author-data>
 
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    \;
 
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  <|author-data>
 
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    \;
 
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  </author-data>>>
 
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  <abstract-data|<\abstract>
 
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    When you throw an object into the air, its trajectory is a parabola. How
 
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    do its instantaneous velocity and the vertical and horizontal components
 
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    of the instantaneous velocity change over the course of its trajectory?
 
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    What happens when the initial speed is larger? Smaller? What happens if
 
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    the gravity were larger? Smaller? Zero? Negative? Holding gravity and the
 
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    initial speed constant, in which direction can you throw to make the
 
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    object reach the farthest point along the horizon?
 
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  </abstract>>
 
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  <section|Introduction>
 
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  In this article we will show you how to construct a model of a gravity
 
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  sketch. The reader is assumed to be familiar with the basics of Dr. Geo
 
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  such as drawing parallel lines and changing the colors/names/styles/... of
 
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  the points and lines.
 
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  <hlink|This sketch|http://user.frdm.info/ckhung/b/dg2/grav/step7.fgeo>
 
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  created using <hlink|Dr. Geo|http://www.drgeo.eu/> can help a high school
 
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  physics teacher explore all these questions with her/his students. Figure 1
 
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  illustrates the constructed sketch after reading this article.
 
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|png>|500px|||>|The
 
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  final gravity sketch.>
 
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  <section|The initial plot>
 
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  <\enumerate-alpha>
 
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    <item>Enable <samp|axes> and <samp|magnetic grid> in <samp|edit> menu.
 
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    <item>Create points <with|font-shape|italic|H<rsub|a>> and
 
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    <with|font-shape|italic|H<rsub|b>> far out along the x-axis, say (-40,0)
 
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    and (40,0). Create the straight line passing
 
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    <with|font-shape|italic|H<rsub|a>> and <with|font-shape|italic|H<rsub|b>>
 
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    as the horizon. Hide and lock <with|font-shape|italic|H<rsub|a>> and
 
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    <with|font-shape|italic|H<rsub|b>>.
 
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    <item>Create a circle centered at the origin <math|O> and passing a point
 
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    <with|font-shape|italic|Speed >along the x-axis. Lock
 
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    <with|font-shape|italic|O>.
 
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    <item>Create a straight line passing <with|font-shape|italic|O>
 
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    perpendicular to line <with|font-shape|italic|(H<rsub|a>H<rsub|b>)>. This
 
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    is just the y-axis.
 
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    <item>Disable <samp|axes> and <samp|magnetic grid> in <samp|edit> menu.
 
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    <item>Create a line segment from <with|font-shape|italic|O> to a free
 
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    point <with|font-shape|italic|V<rsub|0>> on the circle. This is the
 
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    initial velocity vector.
 
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    <item>From <with|font-shape|italic|V<rsub|0>> draw lines perpendicular or
 
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    horizontal to the axes and intersect the x-axis/y-axis at
 
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    <with|font-shape|italic|V<rsub|ox>>/<with|font-shape|italic|V<rsub|oy>>.
 
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    Hide the two auxiliary lines. Draw line segments
 
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    <with|font-shape|italic|(OV<rsub|ox>)> and
 
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    <with|font-shape|italic|(OV<rsub|oy>)>. These are the initial velocities
 
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    along the two axes.
 
80
  </enumerate-alpha>
 
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  We also need the gravity vector. But later you would find that it's
 
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  difficult to drag the gravity vector directly because it's too close to the
 
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  origin (if we want to see the interesting portion of the parabola). So we
 
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  create a point <with|font-shape|italic|Gravity Handle> along the negative
 
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  y-axis at a comfortable distance, and create the point
 
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  <with|font-shape|italic|g> as the end point of the real gravity vector at
 
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  roughly 0.2 or 0.25 times distance. To do this you need to create a number
 
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  <with|font-shape|italic|g-scale>, set it to 0.2 or 0.25, and use the
 
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  Homothety transform (i.e. homogeneous dilation) to scale
 
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  <with|font-shape|italic|Gravity Handle> down towards
 
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  <with|font-shape|italic|O>. The figure 2 bellow illustrates these
 
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  constructions.
 
94
 
 
95
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|png>|332px|407px||>|The
 
96
  initial velocity and the gravity vector.>
 
97
 
 
98
  Download the resulting construction of this step:
 
99
  <hlink|step2.fgeo|http://user.frdm.info/ckhung/b/dg2/grav/step2.fgeo>.
 
100
 
 
101
  <section|The focus and the directrix of the parabola>
 
102
 
 
103
  To draw the parabola, we need to find its focus and directrix. Obviously
 
104
  the axis of the parabola is the vertical line at
 
105
  <math|d=V<rsub|ox>*\<times\>t<rsub|x>> distance away from the y-axis where
 
106
  <math|t<rsub|x>> is the time for the vertical component of the velocity to
 
107
  reach zero. Therefore <math|<frac|d|V<rsub|ox>>=<frac|V<rsub|oy>|-g>>.<\footnote>
 
108
    <math|V<rsub|y><around*|(|t|)>=g\<times\>t+V<rsub|oy>> as
 
109
    <math|V<rsub|y><around*|(|t<rsub|x>|)>=0> therefore
 
110
    <math|t<rsub|x>=<frac|V<rsub|oy>|-g>>\ 
 
111
  </footnote>
 
112
 
 
113
  At time <math|t<rsub|x>> the object also reaches the hightest point of its
 
114
  trajectory at the height of:
 
115
 
 
116
  <\equation*>
 
117
    <math|h<rsub|x>=-<frac|1|2>g\<times\>t<rsup|2><rsub|x><rsup|>=<frac|d\<times\>V<rsub|oy>|V<rsub|ox>\<times\>2>>,<space|1em>therefore
 
118
    <math|<frac|2*\<times\>h<rsub|x>|V<rsub|oy>>=<frac|d|V<rsub|ox>>>.
 
119
  </equation*>
 
120
 
 
121
  <\enumerate-alpha>
 
122
    <item>Use mirror or symmetry to find the point
 
123
    <with|font-shape|italic|-g> on the positive y-axis.
 
124
 
 
125
    <item>Draw straight line <with|font-shape|italic|(-gV<rsub|ox>)>. Draw a
 
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    straight line parallel to <with|font-shape|italic|(-gV<rsub|ox>)> and
 
127
    passing <with|font-shape|italic|V<rsub|oy>>. Intersect the latter line
 
128
    with the x-axis at point <with|font-shape|italic|d>. Draw x-axis's
 
129
    perpendicular line at <with|font-shape|italic|d>. This is the
 
130
    <with|font-shape|italic|axis> of symmetry of the parabola.
 
131
 
 
132
    <item>Draw straight line <with|font-shape|italic|(V<rsub|ox>V<rsub|oy>)>.
 
133
    Draw a straight line parallel to <with|font-shape|italic|(V<rsub|ox>V<rsub|oy>)>
 
134
    and passing <with|font-shape|italic|d>. Intersect the latter line with
 
135
    the y-axis at point <with|font-shape|italic|2h>. Find the midpoint of
 
136
    <with|font-shape|italic|2h> and <with|font-shape|italic|O> as
 
137
    <with|font-shape|italic|h>.
 
138
 
 
139
    <item>Draw x-axis's parallel line passing <with|font-shape|italic|h> and
 
140
    intersect it with <with|font-shape|italic|axis> at point
 
141
    <with|font-shape|italic|vertex>. This is the top of the trajectory, also
 
142
    called the vertex of the parabola.
 
143
 
 
144
    <item>Hide the two pairs of auxiliary parallel lines. Hide
 
145
    <with|font-shape|italic|2h> and <with|font-shape|italic|h>.
 
146
  </enumerate-alpha>
 
147
 
 
148
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3E6926336F84F9823B4BAD0F4DBED4BD8CE735268DB76EEC240D12139A5070CAF5C8DE6C89D0760ED6A91F6D5EF28315CFB17441982FD8DBA34B34F872CC3212CBAAEE32DD923A01695AF56DA97359B831741406D908C860D9E04103272B8CCC002FA259641E45F9C2514C0485335FDE1EB38CDB282616654B15645935C7BB5965FD6D428220B4B3C0CD855A0AF7E0C99DC1E6256F1682F1A2F703BD76F71227BE54420C359292B8E131EE7FF906571642304B03B1A1EE62686ECB2C328FA2A7C85D1B228F135F5E1EFEE4BC37AE5D8C86BDEB7713C4FAC3529C12E5FE5CC11C641B44F9A23EFC72E2CB9787FB9FD59F1E96A12EEA79B7225E544DF1C24D102A9E8209C0F4A037213983E22951BE20BD7863A8F9F2A2705EC238A55F866E10C4A1EDBBC17ADFB9F7976EC2487FD830C433D4D3E8E990AC7DAC05C6EBD487EB6ABE54420C47D9B1306CFED5C245DD9248880377D92D5B1F9A2755BB300BE40B862338818A2F758994CFE28E74DE60AE9978F511B0DDFA4EF352E5962F902FB1FAC909255F4E420C6C2AF4A59BF0A7C7EFF88BA71CB800EF205A09C9D22D5F205FE0031393A07C39974831C7DF0A23F3CD9F3D2DBFE42CAE7E67DB7774DBF50EBBE7831A99635F0175CB17C817F5E900CA97B70D2106F631170FBEF3FDC7FFFCCFE9575C218E5705356951FBDB7BC981272C38252485DC727598175F9A420CEC1EF7B3DDEE37BF7EFCCDBFB0697F187B88E33DB565C47E777F78A5A2A9CC49D7310D02989D7914E58BFAA2F6D583DDBF6CF632B26F750BBE94CD4B8FFF95636F7EFAECD9A78CBF470943285D545D621569CC82E70FBE2D7F301CCC872F9FED76FFFAF06FCEDFB367242ABEEC769F7FFDF8DFFEFAF77F65373D87A0464118A2902E76992B887FE3E71FEFFAA5050C80958B812482BF288A7F573CF05FFCE2C95F9CF097BFF3176CF8CB27BB130A23437EB0DBB17EE8EF1DC8E1DF2BBED0EC3F7CEFBBDFD9ED923F57FCB96CB0B6238887FF03E1F979DE7FEFFBEFFF71F79FFEF37FD9C9E37B3F27DFFDCE5FFD94ED737F56F0E5473FEB3DCF78F29759FF53DCFEF2A7FF75F7F9E3CF77FFADF86A2B09A8E7D1F8F7D9AC470F8BD5E8BFEF1E7D24802F1A74296CCDD7DFFF39A3E872E1BB10E161B3C2F8B430838FBFF7186B86A41C66C317FB51C1173137F0D9892DCFA9D7BCFBE67FB0892E535797BC3FDCAB26CCF3DDEEE3BF02906CC580305FD46FFABE6C54BD70A1E6CB9347E5774F1F9682F1D33DCE6F0BBA84A4DA56ABC5FFFCEB159DE3C438B7452966147F91E3CB938F1A175DB27DEA0DA63B235A5DF35E7EE22C27BEF54B7542D5F319E6B62885305FD4E70364F8F2C5D3D6BF64FBFAEAC77A9C5FBCAC454BDE7F65AB05CD071C0B029DE9E6259510CE37CE882FDF7E7E51621DA5F517134646439972816FFDB5DD946D809F21290ED1C7AEBEFD489C2F3D081B99C7313701BD9EA11755BEB1511068418E7793C28CEAA520F9D2CED278FED01BCCDAE088F1FAD7B20DF5A101C7BBC940982FEAEB311B7CB13691AC85EE28533AFDA2CBE7FA3A67F42351BAB14A4E7BA743C38D77933927C1DF432B841EC6992F09C9F28C4812A6DB1FB6EA7313CE494DFA917783071369E69F062D625F35A90835DE4D1CA29F8E59083D80335F4A99555FD615BD6C607352D2691A3B95B5D78EAFEF5592C07E9A9FFE2CDE1D07A9DB731B679D96DCBB6C0D30DE4D1C33EA3F3AF1A5BA85D2BDCD777DC5531D37E1E4A685951470C1A74D503CC78D5FFCF2364AE821824DB247E24BE06EE28B4373A9294243942F1A360E27BED46FB3F4349DFE9EC7524DF1F8CD791B98B8B464992E82950A511ED03F93A0E42D927D0989E5E63D87D6982B10E58BA7BEDFBE635F00A62F0D29539E255B9B616CBF3031F4335DCB216EE0BA1659AFD76EB9520989114DC3726D7FDD7F68763545E05312FD45F509C786FF423F1BE29DAEDADE7A70EC716E3D2C3FAEF8E254CE4CEDD2D0BF90C4FCE8A1A3A143533545F546465CFF45B99F7EE64BB45DD96100B0A31F138CA71D92C9C9A7A60F26DAD8C42FB81196CBD2F14FFA7D0ECB97537F4071E8840C47BAA4C6BB8941F8A9AB6F586BC45F22B29691493D231FCBB387BE4B8E6F70B1F8B85BCF26514C3581F3623DCAEA3F8B752ADE20F1A53CF4584F83A73A5720CC17F50DD4CDF8AE076588C705E3FDEDD94DB8A31F496A7910DB2917AAD0033D9923CEFD47F4D0135224E2E3DD4420CC17F51B6AD07CC009A33D8FC5763AF4F78D29B0387E6D179D7EB5CD545242A19111E68BA53C738BC39751C1F8AAEBB7E126B84A44E6BAFD8DD3D258CA1292E25EABF20D350E5F46FBC34E2B41A2B56E295C336C9D93C8DDE09FA2385F9457C020F16544B6A159FDE22875133AC8DD3B869FBAA39146640328FED01187DBF7038B2FC3CA94ED6A064F634901EB1CA75C60862417C40F8E9F7174B2D46FACDC687CB18794292F463C4D882E2382594D8D73BC1B2FC4F902A87D3D003AF0F3BC043CFC5FBFF8DF0F713E6848B6A14FE426E11CD008849043307E50B2150012972E9FBF1907D55E39AFDBABB2ABF5A9C4F14690F48738DC5EC7A69C02AB5C8C9B4BB6012F2129C117EC125E8F388D1CD127650FD123A4CFEACD3C0EE7E069DD92EA6C9FBFE7FAF11E35450848F0055D52358D1B961F972FBD218EB10B1C125D4604AF603C4AF392045FD025BE89D37820AB1DE27A4417BFCB977162FF8C5752D0AF472630AA1A3E2129E3BA613B30FB7573E0E747FF67F77F3FC1FBB09E2CCD64C906969B30A05F2724180F9C2B90E10B728A9ABE19CD8819DE7EBAC4850BC3643F518CCC90DE61B21671B341238D327CF171538EDD819FC87CB90871300ED80EA99B006B6406F5314505E3693A0326082073941037E5E86C83FDA6F14261F3A51BE260AF0883761386F5548586F250144686DFFDE9BB54995F565B6387CD974E8883AF755A514981E38A2F7E20B902A903305A6C20A0F3C5F69B216BDE708193AA288E149CE354A18C34CA2D9D527C515B0383CF9756F1D49EFF4D2E8C0C7AAE20951B592BDB212977794A1724057C698438C4943C14242485E6383540FBABC48300720F5CE982A4802F8D108770F41ABBA4209470616A480401E4AE4CA99C8E0ABE9C05E3254C27B290028460BC70A451F24D1058E485A1842FC72C8DA46B8659526067209524D4C8F0FBE7927C410ED9B5A0862FF6BE0C716C52C9C360D62DF1CF71EA3F453ADEED8EEF7724F91262E7901A50C497CA858168F7157613A6E61F8DF634F05D2C6F1040D6338BD4B5AD29E24B19E248E476AD4708BA0993F3B22CC051D57C9146D9CFC5AFCA3C41155FE84C3CC004DD8670BB09D3F3D512FE519423A7C8919094E6A9BA108C32BED8DB0DE4BE8FBF6E89611E1F6FF1D404922D616B5E927EDA3E8CE566803ABE8404D88DE72C296099DF08923D6C9FE29E2108206F1D944D1250C917F08B5A412724434017A6064BAE40FE435355CA87EAF892A6ACFD613C2895DDE1DC8E0441B1299C8C34CAF32554A534A58E2F8575010A7174C0EC26B04032F3388009354500A3261DDA628432BE64116488A38DE20DDE43352F0DAAA9C99EE2481000802F8E2203A38C2FA5678037D6174C741920F33880E14823C4892B3230AAF892559B239F63543527EEB235CB5E641288A3AA87228D107C51941450C417EFF810448BAB990023BA8CE1969FD01B6904318C6AB6488AF8929E68021DE268C381685EC271CB8F08FD38E80401601652D9D90F4C50C39790DC1DBF14EA0FE3C1508F334BBCAE3E5B24B7FC7C8A9D4823CC1DC9549435A8E14BD319932AAE6642E8EF7B4497D9F982E9C2D468471A81DE20EC379142095FDA9BBD54C17BD033DE8D832FED9E06AC9B5218993A0800F49C1305E32794F065DBF65984FBC378107613923C7C81CE3CF6E3A4C5066517627C992E157C493A6FAB4C7F180FDA754B5C7C4176CB4FA87205507CB1F083762AFA492E844BF15D9823841392E86EF9F96644701F85BFD42BE04BCF45A08638DAA00949115B91E04516BB08C1F8E205D8961B9F2FBD4652B63F8C0B3421C9FF482605E3E10067CAF23532CBF1F9B2EF0B66F0285302E02E0BF6DC3DCE4ADCF212804BDF69663C12D0F992F5BFA6F8218E0EF813927899C78B9B017728EC93C6E6CB202F548438DAA0550F5CB902656E39A46B9DE34EEF40E6CB6A38B4AE24C4D101678764AAC82D07DD8AA16674B1F93276F28A421C6DF0352F2972CB61B7EEA85A16B87CC9D7230529EA421C254EF13A8E0E49F4CC637D46B047C3740D51F97237CE0885210EBB15DF2D8D0C5B6710D3CC1B5900BF3798F715952F533B52DCA5B683763E80554D51496411DACE22DE574CBE4C9FB6BA10474FFE882D57805B3C55017C5DC6BBAF887C61308B2A5D989E7C234B8FB380603C37C06F025E14068F2F4C6E97BACCE300A667483A049DD3F01F9063F9BC687CB9638BF8E3F487F1C0D94CA8292601F65C260442264889242CBE78AC4BA8D2CCE300267205E899470C0396C628627F587CD9B2BA898A338F0370463B24B1DD7294050F277E8EC49788FD95549E791CC088642B76120FC741DA621006872F5C891718654A000CF73823BBE5480E758C100A40E14BC617605411E260C4504212D72D47E28B87A01D80C117DE266915210E660C242451DD72AC0D7B18803B5E087CF1D7BC0E09A432E520D8FB03FA1292A82E0CDAD587410A7C4478BEF0D3855A243CA5F72378FA494AD1E54E8F33A25B8EF7B68013069C2F2274519279E4EB3FBA4C48667CA3AAB93E0BEFAA4360A7179A2FD9FA4EE8F7F05D184EBED81709493CB71C75F24A0CFA2A02F325152DBDC0CF3CF2F3A523D9EAA1154FE15EFA760BB8D8C3F2258A85E314DA338F0338F638DB882E0CF2ABB2117F2A1780E44B61FA2498AC3FF33800272541952BC05253C336AD199CF106E48B23E98B23295342E098904472CBD1A309095834008E2FB92BE9B2AAEB0F1339399A900C57388A19F8D127CB4D610E04C6175FFE69A315F9C0801A191FC52D5710AD0CB7304E0C105FBC4D00F0AC15CA360821CCD62E86AAA092A2D41464E80D0C5F9C0066613728F338803C86906CED404D1173E202649340F8E2CBBA2E4728EA0F93421C0148B6B6A1A8E83D8CB7D2E70DC09730821BEF8598A51189D7F55EAE9B4B8C9AEEBF6AAC6BEE22932E6694E74B0EE57A97F063ACE26A28BE946E79E807809397D435D5586E24676264F9126E8077C168B20D607CA9DD729E018D1350D9479ECAF90E927CC90349BE5E00AD780A8E2F271127A8F16E4A7507ACBD8C1723C597C273810FB161651E01F912AE8F86C5A2E3DDA48FA7942F857D24E21B2519BE646E8A51E6942B956D1042D32D6F2424850FA7F8F4C348785112E74B1E6C91F6324A651BC490350B029D34901BEFA69A2FC527C6B1D8878AF2C5D9224ABDAA946D1044C72D971BEFA69E2F346616897CAC185F9C082256387C7893338F35BA6E7948C7BB092EA43AF85232867F8110E18BB371535C17C3D4E2A9E64DB874CB85C7BBE9E18BED65FC36869F2FF86C2990E2155743A14FDE864F4DF1044D7C29AB34B67C0CE7E58B1529608B6D866CC304FADD72915C8136BED8B466902BE0C5C7972486EF98EBC739C4612E06DCF2C2C8ACF98C8C4EBE14CB284F4080832F0E486C8A15E09947C0785D8D61F9CFDE29B023D70A7D669C28A730B1319C952FDEB4701730A005E3E1F932E6967B3EC7FDD2CD17BBF4D5B72C9461E34B1271BD2F30002E9E42E0CB784F83B3614D481AC097C2C824911B4D52669A2FC571D8A8070DE0CC23065FA6DC72C684A4117CB1A9E715B9FB74F46C26F86265B1AB852C14B0994714BE4CAAA9318D7733852F14791A9028CB87D21B237CB1B2C80D363A23ADF8CA94D26070CB0BCF6F62FF61125F6CBAA2A4318953BFEFB4FAF962D1DFD86F12DD51568533F144913114043AE514D8E1FF6E185F4AE459B4274154B0A6F54274F8E2E4491A05C5CF65B96EAE94D8CF20F3C8E4968FA8299AC997EAC4927413BB2488B7519AFA7E92E4FFEFF0FFF324F1FD6C136DE380B8719425069DBEE299372260956DA06A8A039EA0E9D7E8587992A59B4D146DFFE6F0C7DB28DA6CD22CC92D03DB0B6790796457531B4A489ACE970654CCB79782AA997812EDA21CF386FA139237BE004251E651C651E2EA69E84948DEF802083CD9062B4B73EAD31516C2A27CC9D34CF0C9F1B9E51709C91B5F20019479BC8CD745BE4F42DB236158FC55FCE9FA99A0950979DDF2C2C83472B737BE800246B6A137BE4BF736494C1F1D092D57FCE0FC6EF9CA3FE70A6E7C810548E6F1922F7E1C97E1FCF5BAF88378F646A2B357444DEDA4A678E30B2C4094292FF8621536619FD032966239B2C98A96B18BC793C5D4D4AA84E43CF8F2E1FD870FEF5F1DDE147F7E309C341082F197F33E838225055FD679F9951DD2F0A0F0E145DDF2C2C8ECE7C197378733EE759FCC0430328FA11BEFD3A4F48EE2ACB02F4910CBC85D88ABA99914501FC1570DBE7CA9FB64A68022DBD0F1513DB97D98B85B3E0FBED8F767BEE83E95497838CA94A038CA367063267CF9F24417C3DD170A05637D6521ACA666FC95D538F1057F9A8C3C169579ECFC9EEE1367C4FB9A2EAF749F08138C158C3F43B0A7612E7CF16ABEBCD57D226C98816C839814C95CF862BF2AE9F246F76930C268C1F80A623D0D33E18BF7EE6DC997176FE7E0BFD8922E0C4A7FC005840A0267C29797F72F68CCEEFEDDFD6BDDA7C20899CCA31ABED8C97A46FA0C7C28DC5DBAA5FE707F7839130323937954C417111766267CF1DE1F68C8EEFEF06A2E7491118C57C51701B77C267C396EA8E7B19DAE209E7954C6177E35B5D9F0A524CC7CAC0B05B46C0302B8DDF2F9F0E51DDD1EE93E093EA009C6C381D72D9F0D5FDEDD1F7E78B89F1961F047554B83B3A7612E7C29E8F2EA27EFE7666166D0F3C8A9A666FCF554785BD0C526E1FBB95998640E99479E5A9A79F0C52B37D22E757AEF67E5F2CE42309EC78599075FEC772F6DDA47617BEFBFD27D2A9C9841E691C72D9F095F4A10E325797AC0DD1FA61E3C9947E32FA60162BC2FD00781CCA3B2785D0D0EB77C567C013C9653FE4F09F8651B54F385A3A761567C39BDA86116457D7E41DC71864B3DF83E51F890946D826AC02DDBA09C2FEC6A6AB3E2CBF1013B6493277DE9DF621DDE34D9516E157BF78B44E1F2C62D18AF9E2F1EAB5B3E2BBE1C1FF071A291EFA579257541FFC1F18BBF7292FA54AF2F2C5DB8235F2A390C3B0BB3529C224B9D827B7479F353157570BCB20DEAF9C2EC96CF8A2FC76B3B7E41E2CCAAA42EA86C41F17F1226C44FE852B52D2955BA9AC5C3AAE4308A9FF753D7B6834D1257F6651B254A7A3F5A33F1CC04A35B3E2BBED45744F962A5A9651F9F75905841F99F8B1D77F167E81E7FD62514E5CB4D67F2D1AFA8848A6DDF51BEACE85789922432B0603C06D8DCF259F1A5FE9B3E662773938A147E1C93E24BFB4899909A95BA1BFDB81EF9F19624158948E8ACE92128B31CB25E0712522A1C584AE671567C395ACCD2620449E90053A98B62258AF36D72E24BB6D9541E66CD97FA672ABE78944B1E397EA50810B20DC860EA6930FE2A1A38ED8F3654E591F265458B964BA90B2B2EC970A448BD18D77C39CA61D09F27E1AAF8C9A4F45F3C4A3C45BB240E654A5D6071CB67C597535069435CB7786129456AA90B9B503FA4F887D4DD165CAA9D928A2F61E80671F9335EF9233E893795BF9BBB81F0386C5ECC21F3383D8A72567C69BCA177E7AF3BEF6D589022EE5AD6BBD677DEF957EED46D5CE690799C74CBE7CA97513890990330CCA0786A3AF368FC2534C0CA8218719EBD0C98B7EE1AE27535ACA928CCDF0215BA304B2B1223A50000002574455874646174653A63726561746500323031362D30322D30335431303A32393A30382B30383A30309AD9A2B90000002574455874646174653A6D6F6469667900323031362D30322D30335431303A32383A32372B30383A3030B02B06AF0000000049454E44AE426082>|png>|500px|||>|Finding
 
149
  the vertex of the parabola.>
 
150
 
 
151
  Download the resulting construction of this step:
 
152
  <hlink|step3.fgeo|http://user.frdm.info/ckhung/b/dg2/grav/step3.fgeo>.
 
153
 
 
154
  \;
 
155
 
 
156
  Next we find the focus of the parabola. It lies somewhere on the axis of
 
157
  symmetry. Recall that any light coming from the focus will be reflected --
 
158
  if the parabola represents amirror -- into the direction parallel to the
 
159
  axis of symmetry, and vice versa. Also recall that the instantaneous
 
160
  velocity is tangent to the parabola.
 
161
 
 
162
  <\enumerate-alpha>
 
163
    <item>Draw the straight line <with|font-shape|italic|(OV<rsub|oy>)> and
 
164
    use it to mirror the <with|font-shape|italic|Gravity handle> (or
 
165
    <with|font-shape|italic|g> or <with|font-shape|italic|V<rsub|oy>>, either
 
166
    will work) into the point <with|font-shape|italic|G<rsub|2>>.
 
167
 
 
168
    <item>Draw straightline <with|font-shape|italic|<with|font-shape|italic|(G<rsub|2>>O)>
 
169
    and intersect it with <with|font-shape|italic|axis> to obtain
 
170
    <with|font-shape|italic|Focus>.
 
171
 
 
172
    <item>Apply Homothety or Reflection to Focus and draw a line parallel to
 
173
    the x-axis. This is the directrix of the parabola.
 
174
 
 
175
    <item>Hide the auxiliary lines and points.
 
176
  </enumerate-alpha>
 
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|png>|500px|||>|Finding
 
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  the focus and the directrix of the parabola.>
 
180
 
 
181
  Download the resulting construction of this step:
 
182
  <hlink|step4.fgeo|http://user.frdm.info/ckhung/b/dg2/grav/step4.fgeo>.
 
183
 
 
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  <section|The parabola itself>
 
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186
  Now we can draw the parabola. Recall that it is the locus of all points
 
187
  that are equally distant from the focus as from the directrix.
 
188
 
 
189
  <\enumerate-alpha>
 
190
    <item>Draw a free point <with|font-shape|italic|p> on the x-axis. This
 
191
    point will provide user interaction to control the position of the
 
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    instantaneous velocity drawing.
 
193
 
 
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    <item>Draw the line vertical to the x-axis and passing
 
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    <with|font-shape|italic|p>, then intersect the line with the directrix at
 
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    point <with|font-shape|italic|q>.
 
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    <item>Draw line segment <with|font-shape|italic|(Focus q)>. Find its
 
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    midpoint <with|font-shape|italic|m> and draw the straight line
 
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    <with|font-shape|italic|(T)> perpendicular to the line segment. Intersect
 
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    <with|font-shape|italic|(T)> with <with|font-shape|italic|(pq)>
 
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    <with|font-shape|italic|r>.
 
203
 
 
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    <item>Use the locus tool in line menu to draw the parabola, using
 
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    <with|font-shape|italic|p> as the free (independent) point and
 
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    <with|font-shape|italic|r> as the relative (dependent) point.
 
207
 
 
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    <item>Hide the auxiliary lines and points except line
 
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    <with|font-shape|italic|(T)>.
 
210
  </enumerate-alpha>
 
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|png>|500px|||>|Drawing
 
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  the parabola using the locus tool.>
 
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  Download the resulting construction of this step:
 
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  <hlink|step5.fgeo|http://user.frdm.info/ckhung/b/dg2/grav/step5.fgeo>.
 
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  <section|The instantaneous velocity>
 
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  Finally we will draw the instantaneous velocity at point
 
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  <with|font-shape|italic|r>.
 
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  <\enumerate-alpha>
 
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    <item>Draw vector <with|font-shape|italic|(Or)>. Use it to translate the
 
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    point <with|font-shape|italic|V<rsub|ox>> to
 
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    <with|font-shape|italic|V<rsub|px>>. Draw line segment
 
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    <with|font-shape|italic|(rV<rsub|px>)>. This horizontal component of the
 
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    instantaneous velocity remains constant at all times.
 
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    <item>The instantaneous velocity lies on the tangent line
 
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    <with|font-shape|italic|(T)>. So from <with|font-shape|italic|V<rsub|px>>
 
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    draw a vertical line to intersect with <with|font-shape|italic|(T)> at
 
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    <with|font-shape|italic|V<rsub|p>>. This is the end point of the
 
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    instantaneous velocity.
 
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    <item>From <with|font-shape|italic|V<rsub|p>> draw a horizontal line to
 
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    intersect with straight line <with|font-shape|italic|(pr)> at
 
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    <with|font-shape|italic|V<rsub|py>>. This is the end point of the
 
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    vertical component of the instantaneous velocity.
 
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    <item>Clean up the drawing and add some textual explanations.
 
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  </enumerate-alpha>
 
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FCA83C6CB1F89AC8910F6965C78DC5EBB43432986127CE9AFB07E2E72D364F9FC30E26AC12DB9F0B8759B1CA886123C15561ACB40D355941B24E209E4C2E2D65D94E1CC53C250FB06EB27233FCD3F1E110C13F6965C58DC3A43434758F394D2EB6D79143472A2D1E213DAD195825B72E170EBD6D3AC2987123CF5B527583F1BF969B3FCE411A5825B726170EBEEE470BECAEB5C5A990D2E8F79075717CB1F3D2CA114DC920B835BA7C9E122875082272DEBD3CE31B7E0FCC1F307D7086EC9959EDB9776BB4FA1CFBB4FE77732465683CB33B760F69983220CC12DB952737B7DD3AEA7C9694DE64AD3B2FD3DD7DC82D567C3C130110723576A6E9FB5EB6968A7770F94D5E0F2CDED68F9C9D0C313795E72A5E5D66972C86F4DE68A523D23A7FAF4E5ABA1072BFC0462A5E5F645343434E3888F142AAC499C8DCE3F174C660B6EC995925BBBC96193477AF7404DA6831473D7E7BF100886096EC9958E5B6768E88A42D3F92955B6B7D7D1E8C19FDE5F10DC922B1DB7769303ADEEDD64A9798C5DE247E3CB1FF17FCE89DB2E5AFCC9955E28A4E4D6AEA799D0EADE4D5685EB706D6D3FE74F39C9895B1D21CB767075FE4AC52D6A7228C2B9B5457B433EDEF4F92F7AC1B09CE2B7B233D00F0C8D4EA70F1AF04E94EAF592A4E1F6FEEE2AEF844340958F9D3FF884FB435EFE2D6416B16BEA8AA6A290B86132DDD22D17A5E0D66E7258FF99A2CE284B288CEFBC83ABCDE58F393FECB91DF63B54DC7A87DBE6C00ECBA01669F87FA0EB454DF22B5029B845F53499668A632AC3D35C0A6EC19FBDFC73F6F73DB7A64E67D709875B686C5D66D11C6020553144B3B44EF52EDC868BB24DEE09878014F2C1CAE5E0167CDC717103DC527240BFD4750FE7BAB84092A1A3406736054F1A2D2DCB3AE1B8DEDCDD1F2D7269CC8913F9CAAC24DC8217FE3CFA1AE0969203EAAE0D9A4DD486690E00F26CD5B3EAC5C4671652E24D5093435E1D0E31222FC32D0BB79BCB1F07216E29594437EF20DBEF0353D3CD21E8E8D0F31A663B2C775AD9DC26F900682787EDA2D8B3AA7267AFAB8F5D7E24580F466E6FE7F3E0AB17CA979986513907C1D5697BFBE2EEB9429D5B5B6AF5E28D87FA0B7F918E9FB0B0ACC57E505B98DBEA45BF3C6D907F9B5475F0D2EEEA5385456E7D553E840BF5B92F5389DF2E90E1597A2F6188DB2A7F6AC185591295D757BBA767445B466752C5AB196D7DE4F2639FF8F1EC87713C3DCBB25720A39FF84BC5852BD92A397EFBCCEE5E4165096155DC51686AD0F1FC3CA46D91D9035B589EE6A3F1F2B39F2D2A1BCF5A89DCC245D94F3279223A1AABE7A31049A66936BF88605B6D336A6905F5D99FB256AC1F5C314AE4F6DEEE99226A6E8F5571474137911EBBB4680B727BC9FAC115A3246E9FDE5D4D0BA9B93D56451D05431E74B5A66A636B7E00390AB38C3AB2B705C72C5929815BB828FBCB390EF94814614481D7BC83212B1DADA99B92DAEE0E15B90BA9953EEC2FA7B268EFDF2EB71788DB42339BEC94C0ED9DDD478BCDEF06D420AB1EE78D5B9BD736E455B779F502AA0DD3EC003041B4655D3D78DC2ED081C63FF1576AB22C4BE0F6D66E57707E3728B2CC272FDC0678D53A7B5EFD5FF7D0351BEBC1DA7A90F1036DE1C6129C4B75087C3B8AE7F6E6EE06435F896C26066B6E3D5ECD685EC382DC82A5F557B3DDE322E46C742A3CD132AC586EEFEFFE1ACB58E080687F1D56DC425E87DDBD7D4DF7471B6B09FF67744857CB60DE48F4F35EEF765F611202736594221286ECABC36BF3B47D3D14B2B7609B5CD67452613743707B67F7D32BA627C67524CC90CF06DDB60A79EDE1F3EAC9E616ACA0D5A5A5FE8DBA737B7BF73385578185A5F15833DAB779D525536D6A43525E3D6D6C624797D694DAF93D51776E6FEE7EB6F02AB0B0941EDBFB0FC908F00AED2B9DB256C7DE8271F66098AF2FD59CDB67777FBDF82AB0B0B870708D566360C7B368F2EA69E37A0833EB925676A7E6F6F6EEEEE75825CAF65219CEFFE8CB0D645F4DAAF6F5501EB76069D1CAA6D7DCBF7DF1B59FA7B9D93999E84F354EA15688D77C1B065C3F01BAB8C9B5FB18AAB7BD7D69F73ABB4499AF8156E09DD9BCAAA629E5685F0FE5739B3918E6ABD6DC5E5F7DF567997B0900C804DDD3D87907D77F45BC0E8BE2D5D3661F015BF9086753ADFD84E75E7BC44575064189C22FA4E5B65580FF7A4A7B7B8BDAA5A87CC2D539EF706BF7FA2FB23E2B5B4DFC915927ED2DE2B5A74B85FA03710A700B2E2C8BC68AA2CEDCDEFBEACF73E02500A2CC432CB78EFFEAF1CAC7A4D24D3053B6A6120CAB31B74FBFF628C79DF57044B0303BE2D6F65F39E3D5D326E4D42E69742AD497DBEBABD7FF06EB737225934F1EEF3BFEABCA25AF9EC2DC6E6804C3EA5B7F7BE797DE607D4A9EFA26FEDF78BCA2F5D6A0C127AF9E36E18A9A1FA690EFADADBDBDF5DAA35F667D4ABE30E632DABC361D7FE015CE79F5B439087E2D32373FD497DB9B5FFB9BACCF68AFDEE939440EAFE5B0AF873AB0B7286D96B1F9A1B6DCDEFFA537381A86D68DDFCB28C06B17F2CAD149A7D721B7286D96B10FB5A6DCDE7DEDD1DF627D420145F4EA205E35E80FA87AF9ECEBA1364749B29575992DDF5B536E5F7CFD57589F4F508DFDC07C9FD766BB5B765E3D1D730B2CEB61A643D6339E70FBAB6F7C8AF5F940F954B624C46BC7E155D3AAC2ABA7CD7187CE24E324907ADADB5FFDFADF667D3A4828FCD56AA0012F26F407B0EC2BEB3E742C45D85B30CF96EFAD25B7CF7CE3579CC7AEA1319786668F153E2BB40C16E50B3AA6EDBF225EB137280973CBB9798EE27664650A86D591DBBB5F7DE327D177A93318A2DDAF861F521560E840A3FCF2CB5DB45717DA84135D1A7451F54CBFD5ED6968E09B2AA99A79E6DDA38EDBD31BE696206F51A436519DBC5B2BCB24D03AFAB71F7DE386FDDDD916A0A5DAFEA5A10185EE06E586391876BC4D3881DED3CC1E1A5028E9A606FD015553F43D6ED8B63EC42D49BEAD486D228B6E5797A736E64A500DEDEDD3DF78FDDAFEC1949195B3ABB6E10BDF6E52EEAB7536CB699970BD65AA6820515B6AB40C784F8A8AA076FEBBC2361F016E3B5DA394DC6EAC0CCD0FF5E3F6FAD71EFD1DE79A66BB69CA901ED9D0E8BFF0860CED29AACF327BDD81A4C8F06749421B6EA92AFA6ECF6BDEDFEB5023BE23DB70F39D8ED8444FFC385F5A0F495DDCFA71FBDC9B7F777FA502F1396B36157ADC3A036025645FBB1D5D7737E16C7973C583DF036652211A1266DF9F896C7A81CF23813631CD390BF2E687DA717BEBD7DFFC7BFB2B5D7428EC8F19E4D58967190E9DF6269C86BD312DEA3A70BFF7E15581774B8378A7DE03C3CDA562EC2D182F89F3BDB5E3F6EF3FFA0792BB78871FAE1DBBF2B52F3D457C588757D51EB01D8EBF2A922E2990A9B62EC1E559435225D4D6E07D1F4A7AC03B21AFC03D30DC5C2A8E5B305D91363FD42D9E70FF6BBF0181B203070674357568FA9ABA4430A1CBB3AF926D5FA30710D8149B7DC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|png>|500px|||>>
 
245
    The instantaneous velocity <with|font-shape|italic|V<rsub|p>> and its
 
246
    components <with|font-shape|italic|V<rsub|px>> and
 
247
    <with|font-shape|italic|V<rsub|py>>.
 
248
  </big-figure>
 
249
 
 
250
  Download the resulting construction of this step:
 
251
  <hlink|step7.fgeo|http://user.frdm.info/ckhung/b/dg2/grav/step7.fgeo>.
 
252
 
 
253
  <section|Conclusion>
 
254
 
 
255
  My first attempt created a sketch that became broken when the initial
 
256
  velocity moves out of the first quadrant
 
257
  <math|<around*|(|x\<geqslant\>0,y\<geqslant\>0|)>>. I guess it was because
 
258
  at some step an absolute value (length) was taken. It's probably better to
 
259
  think in terms of make use of parallel lines instead of sign-less lengths
 
260
  when you construct a sketch.
 
261
 
 
262
  In fact Dr. Geo has very powerful smalltalk scripting capabilities.
 
263
  However, drawing this sketch using only the basic drawing primitives makes
 
264
  it so... classic and so much more fun. I also learned one or two properties
 
265
  about parabola that I never knew as a high school kid (or as an adult).
 
266
</body>
 
267
 
 
268
<\initial>
 
269
  <\collection>
 
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    <associate|font-base-size|11>
 
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    <associate|page-medium|papyrus>
 
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  </collection>
 
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</initial>
 
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275
<\references>
 
276
  <\collection>
 
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    <associate|auto-1|<tuple|1|1>>
 
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    <associate|auto-10|<tuple|5|5>>
 
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    <associate|auto-11|<tuple|6|5>>
 
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    <associate|auto-12|<tuple|6|5>>
 
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    <associate|auto-2|<tuple|1|1>>
 
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    <associate|auto-3|<tuple|2|1>>
 
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    <associate|auto-4|<tuple|2|2>>
 
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    <associate|auto-5|<tuple|3|2>>
 
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    <associate|auto-6|<tuple|3|3>>
 
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    <associate|auto-7|<tuple|4|4>>
 
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    <associate|auto-8|<tuple|4|4>>
 
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    <associate|auto-9|<tuple|5|5>>
 
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    <associate|footnote-1|<tuple|1|3>>
 
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    <associate|footnr-1|<tuple|1|2>>
 
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    <associate|p843774|<tuple|?|?>>
 
292
  </collection>
 
293
</references>
 
294
 
 
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<\auxiliary>
 
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  <\collection>
 
297
    <\associate|figure>
 
298
      <tuple|normal|The final gravity sketch.|<pageref|auto-2>>
 
299
 
 
300
      <tuple|normal|The initial velocity and the gravity
 
301
      vector.|<pageref|auto-4>>
 
302
 
 
303
      <tuple|normal|Finding the vertex of the parabola.|<pageref|auto-6>>
 
304
 
 
305
      <tuple|normal|Finding the focus and the directrix of the
 
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      parabola.|<pageref|auto-7>>
 
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      <tuple|normal|Drawing the parabola using the locus
 
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      tool.|<pageref|auto-9>>
 
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      <\tuple|normal>
 
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        The instantaneous velocity <with|font-shape|<quote|italic>|V<rsub|p>>
 
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        and its components <with|font-shape|<quote|italic>|V<rsub|px>> and
 
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        <with|font-shape|<quote|italic>|V<rsub|py>>.
 
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      </tuple|<pageref|auto-11>>
 
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    </associate>
 
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    <\associate|toc>
 
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      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|1<space|2spc>Introduction>
 
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      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
 
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      <no-break><pageref|auto-1><vspace|0.5fn>
 
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      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|2<space|2spc>The
 
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      initial plot> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
 
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      <no-break><pageref|auto-3><vspace|0.5fn>
 
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      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|3<space|2spc>The
 
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      focus and the directrix of the parabola>
 
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      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
 
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      <no-break><pageref|auto-5><vspace|0.5fn>
 
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      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|4<space|2spc>The
 
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      parabola itself> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
 
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      <no-break><pageref|auto-8><vspace|0.5fn>
 
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      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|5<space|2spc>The
 
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      instantaneous velocity> <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
 
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      <no-break><pageref|auto-10><vspace|0.5fn>
 
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      <vspace*|1fn><with|font-series|<quote|bold>|math-font-series|<quote|bold>|6<space|2spc>Conclusion>
 
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      <datoms|<macro|x|<repeat|<arg|x>|<with|font-series|medium|<with|font-size|1|<space|0.2fn>.<space|0.2fn>>>>>|<htab|5mm>>
 
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      <no-break><pageref|auto-12><vspace|0.5fn>
 
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    </associate>
 
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  </collection>
 
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</auxiliary>
 
 
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