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Projectile Motion;SPMlt;/A;SPMgt;
;SPMlt;/H1;SPMgt;
;SPMlt;P;SPMgt;
;SPMlt;DIV class=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;B;SPMgt;Brian Palmer and Steve Nava;SPMlt;/B;SPMgt;
;SPMlt;/DIV;SPMgt;
;SPMlt;P;SPMgt;
;SPMlt;DIV class=;SPMquot;CENTER;SPMquot;;SPMgt;Wednesday, January 1, 1997
;SPMlt;/DIV;SPMgt;
;SPMlt;P;SPMgt;
Working together, we completed Lab #5 as given to us in class. Summarizing,
Steve held the ball at what was determined by measurement to be about 1.895
meters from the ground. Then he threw it, releasing the ball from his hand
at that spot. It struck the ground 2.56 seconds later, at a point 43.92
meters from the initial throwing position.
;SPMlt;P;SPMgt;
The data was then available for analysis. Although it was performed in three
dimensions, there are only two colinear points which were recorded; thus the
third dimension (along the ;SPMlt;I;SPMgt;z;SPMlt;/I;SPMgt;-axis) can be ignored (this lab assumes that
the force along the other axes applied by the wind blowing at the time was
negligible). Then each of the two remaining dimensions, along the ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt; and ;SPMlt;I;SPMgt;y;SPMlt;/I;SPMgt;
axes, to use traditional terminology, may be analyzed as if the other were
not present. For the initial analysis, the kinematics equation, where ;SPMlt;!-- MATH
#math17#1Δx
--;SPMgt;
;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt; is the position displacement, ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt; is the initial velocity, ;SPMlt;I;SPMgt;a;SPMlt;/I;SPMgt;
is the acceleration undergone by the object, and ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; is the time through
which the analysis is concerned,
;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;!-- MATH
#math18#
|
Δx = v0t + #tex2html_wrap_indisplay549#at2
|
(1) |
--;SPMgt;
;SPMlt;TABLE WIDTH=;SPMquot;100;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;
;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0;SPMlt;/SUB;SPMgt;;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; + ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img1.png;SPMquot;
ALT=;SPMquot;#math19##tex2html_wrap_indisplay551#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;at;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;
;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
();SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;/TABLE;SPMgt;
;SPMlt;/DIV;SPMgt;
shall be used extensively.
;SPMlt;P;SPMgt;
When dealing with the ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;-component of the system, the relationship given in
equation 1 can be explicated as follows
;SPMlt;P;SPMgt;;SPMlt;!-- MATH
#math20#
Δxx = v0xt + #tex2html_wrap_indisplay553#axt2
--;SPMgt;
;SPMlt;/P;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt; = ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0x;SPMlt;/SUB;SPMgt;;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; + ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img1.png;SPMquot;
ALT=;SPMquot;#math21##tex2html_wrap_indisplay555#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;a;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt;;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;
;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;
Since after the initial force applied by the throwing action, there is no
force acting on the tennis ball, the relationship ;SPMlt;!-- MATH
#math22#a = #tex2html_wrap_inline557#
--;SPMgt;
;SPMlt;I;SPMgt;a;SPMlt;/I;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img2.png;SPMquot;
ALT=;SPMquot;#math23##tex2html_wrap_inline559#;SPMquot;;SPMgt; for a
force ;SPMlt;I;SPMgt;F;SPMlt;/I;SPMgt; acting upon a mass ;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt; shows that ;SPMlt;!-- MATH
#math24#a = #tex2html_wrap_inline561# = 0
--;SPMgt;
;SPMlt;I;SPMgt;a;SPMlt;/I;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img3.png;SPMquot;
ALT=;SPMquot;#math25##tex2html_wrap_inline563#;SPMquot;;SPMgt; = 0 ;SPMlt;!-- MATH
#math26##tex2html_wrap_inline565#m/3#tex2html_wrap_inline566#s2
--;SPMgt;
;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline570#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline572#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;. Simplifying the relationship further,
;SPMlt;P;SPMgt;;SPMlt;!-- MATH
#math27#
Δxx = v0xt
--;SPMgt;
;SPMlt;/P;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;x;SPMlt;/SUB;SPMgt; = ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0x;SPMlt;/SUB;SPMgt;;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;
;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;
The displacement and time are known, allowing the initial velocity to be
discovered
;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;!-- MATH
#math28#
|
v0x = #tex2html_wrap_indisplay575# = #tex2html_wrap_indisplay576# = 17.156#tex2html_wrap_indisplay577#m/#tex2html_wrap_indisplay578#s #tex2html_wrap_indisplay579# 17.16#tex2html_wrap_indisplay580#m/#tex2html_wrap_indisplay581#s
|
(2) |
--;SPMgt;
;SPMlt;TABLE WIDTH=;SPMquot;100;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;
;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0x;SPMlt;/SUB;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img5.png;SPMquot;
ALT=;SPMquot;#math29##tex2html_wrap_indisplay583#;SPMquot;;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img6.png;SPMquot;
ALT=;SPMquot;#math30##tex2html_wrap_indisplay585#;SPMquot;;SPMgt; = 17.156;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math31##tex2html_wrap_indisplay587#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math32##tex2html_wrap_indisplay589#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt; ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img8.png;SPMquot;
ALT=;SPMquot;#math33##tex2html_wrap_indisplay591#;SPMquot;;SPMgt; 17.16;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math34##tex2html_wrap_indisplay593#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math35##tex2html_wrap_indisplay595#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt;
;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
(1);SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;/TABLE;SPMgt;
;SPMlt;/DIV;SPMgt;
;SPMlt;P;SPMgt;
The unknown value of the initial velocity's ;SPMlt;I;SPMgt;y;SPMlt;/I;SPMgt; component can be discovered
similarly. However, for this, a force is known to be acting upon the tennis
ball: gravity. Gravity can, for any mass ;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt; such that the mass of the Earth
;SPMlt;I;SPMgt;M;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;E;SPMlt;/SUB;SPMgt; ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img9.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline597#;SPMquot;;SPMgt; ;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;, the acceleration due to gravity on the object may be
approximated as the constant ;SPMlt;I;SPMgt;g;SPMlt;/I;SPMgt;, about ;SPMlt;!-- MATH
#math36#9.8006#tex2html_wrap_inline599#m/#tex2html_wrap_inline600#s2
--;SPMgt;
9.8006;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline604#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline606#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;.
Thus,
;SPMlt;BR;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;;SPMlt;A ID=;SPMquot;yexpress;SPMquot;;SPMgt;;SPMlt;/A;SPMgt;;SPMlt;!-- MATH
#math37#
| Δxy |
= |
v0yt + #tex2html_wrap_indisplay611#at2 |
(3) |
| voyt |
= |
Δxy - #tex2html_wrap_indisplay615#at2 |
(4) |
| voy |
= |
#tex2html_wrap_indisplay619# - #tex2html_wrap_indisplay620#at |
|
| voy |
#tex2html_wrap_indisplay623# |
#tex2html_wrap_indisplay625# - #tex2html_wrap_indisplay626#gt |
(5) |
--;SPMgt;
;SPMlt;TABLE CELLPADDING=;SPMquot;0;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; WIDTH=;SPMquot;100;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;=;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0y;SPMlt;/SUB;SPMgt;;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; + ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img10.png;SPMquot;
ALT=;SPMquot;#math38##tex2html_wrap_inline628#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;at;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
(2);SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;oy;SPMlt;/SUB;SPMgt;;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;=;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt; - ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img10.png;SPMquot;
ALT=;SPMquot;#math39##tex2html_wrap_inline630#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;at;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
(3);SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;oy;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;=;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img11.png;SPMquot;
ALT=;SPMquot;#math40##tex2html_wrap_indisplay632#;SPMquot;;SPMgt; - ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img10.png;SPMquot;
ALT=;SPMquot;#math41##tex2html_wrap_inline634#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;at;SPMlt;/I;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
;SPMamp;nbsp;;SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;oy;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img8.png;SPMquot;
ALT=;SPMquot;#math42##tex2html_wrap_indisplay636#;SPMquot;;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img11.png;SPMquot;
ALT=;SPMquot;#math43##tex2html_wrap_indisplay638#;SPMquot;;SPMgt; - ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img10.png;SPMquot;
ALT=;SPMquot;#math44##tex2html_wrap_inline640#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;gt;SPMlt;/I;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
(4);SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;/TABLE;SPMgt;;SPMlt;/DIV;SPMgt;
;SPMlt;BR CLEAR=;SPMquot;ALL;SPMquot;;SPMgt;
The time ;SPMlt;I;SPMgt;t;SPMlt;/I;SPMgt; is the same as that for the ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;-component; that is, it is the
time until the ball hit the ground. Substituting in the values, and using
the definition that ;SPMlt;!-- MATH
#math45#Δh = hf - hi
--;SPMgt;
;SPMlt;I;SPMgt;;SPMamp;#916;h;SPMlt;/I;SPMgt; = ;SPMlt;I;SPMgt;h;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;f;SPMlt;/SUB;SPMgt; - ;SPMlt;I;SPMgt;h;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;i;SPMlt;/SUB;SPMgt;,
;SPMlt;P;SPMgt;;SPMlt;!-- MATH
#math46#
v0y = #tex2html_wrap_indisplay643# - #tex2html_wrap_indisplay644#gt = #tex2html_wrap_indisplay645# - #tex2html_wrap_indisplay646##tex2html_wrap_indisplay647#9.8#tex2html_wrap_indisplay648#m/#tex2html_wrap_indisplay649#s2#tex2html_wrap_indisplay650##tex2html_wrap_indisplay651#2.56#tex2html_wrap_indisplay652#s#tex2html_wrap_indisplay653# #tex2html_wrap_indisplay654# -0.74#tex2html_wrap_indisplay655#m/#tex2html_wrap_indisplay656#s +12.54#tex2html_wrap_indisplay657#m/6#tex2html_wrap_indisplay658#s = 13.284#tex2html_wrap_indisplay659#m/#tex2html_wrap_indisplay660#s
--;SPMgt;
;SPMlt;/P;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0y;SPMlt;/SUB;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img12.png;SPMquot;
ALT=;SPMquot;#math47##tex2html_wrap_indisplay662#;SPMquot;;SPMgt; - ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img10.png;SPMquot;
ALT=;SPMquot;#math48##tex2html_wrap_inline664#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;gt;SPMlt;/I;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img13.png;SPMquot;
ALT=;SPMquot;#math49##tex2html_wrap_indisplay666#;SPMquot;;SPMgt; - ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img10.png;SPMquot;
ALT=;SPMquot;#math50##tex2html_wrap_inline668#;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img14.png;SPMquot;
ALT=;SPMquot;#math51##tex2html_wrap_indisplay670##tex2html_wrap_indisplay671#9.8#tex2html_wrap_indisplay672#m/#tex2html_wrap_indisplay673#s2;SPMquot;;SPMgt;9.8;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math52##tex2html_wrap_indisplay675#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math53##tex2html_wrap_indisplay677#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img15.png;SPMquot;
ALT=;SPMquot;#math54##tex2html_wrap_indisplay679##tex2html_wrap_indisplay680#9.8#tex2html_wrap_indisplay681#m/#tex2html_wrap_indisplay682#s2#tex2html_wrap_indisplay683#;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img16.png;SPMquot;
ALT=;SPMquot;#math55##tex2html_wrap_indisplay685##tex2html_wrap_indisplay686#2.56#tex2html_wrap_indisplay687#s;SPMquot;;SPMgt;2.56;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math56##tex2html_wrap_indisplay689#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img17.png;SPMquot;
ALT=;SPMquot;#math57##tex2html_wrap_indisplay691##tex2html_wrap_indisplay692#2.56#tex2html_wrap_indisplay693#s#tex2html_wrap_indisplay694#;SPMquot;;SPMgt; ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img8.png;SPMquot;
ALT=;SPMquot;#math58##tex2html_wrap_indisplay696#;SPMquot;;SPMgt; -0.74;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math59##tex2html_wrap_indisplay698#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math60##tex2html_wrap_indisplay700#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt; +12.54;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math61##tex2html_wrap_indisplay702#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math62##tex2html_wrap_indisplay704#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt; = 13.284;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math63##tex2html_wrap_indisplay706#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math64##tex2html_wrap_indisplay708#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt;
;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;
;SPMlt;P;SPMgt;
At last, the ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt; and ;SPMlt;I;SPMgt;y;SPMlt;/I;SPMgt; components of the initial velocity have been
calculated. Using some of the definitions and theorems associated with
vectors, the characteristics of the vector as a whole can be calculated.
First, the magnitude of the initial velocity vector, ;SPMlt;!-- MATH
#math65##tex2html_wrap_inline710##tex2html_wrap_inline711##tex2html_wrap_inline712#
--;SPMgt;
;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img18.png;SPMquot;
ALT=;SPMquot;#math66##tex2html_wrap_inline714##tex2html_wrap_inline715##tex2html_wrap_inline716#;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img19.png;SPMquot;
ALT=;SPMquot;#math67##tex2html_wrap_inline718# ;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img20.png;SPMquot;
ALT=;SPMquot;#math68##tex2html_wrap_inline720##tex2html_wrap_inline721##tex2html_wrap_inline722##tex2html_wrap_inline723#;SPMquot;;SPMgt;, can be calculated as
;SPMlt;P;SPMgt;;SPMlt;!-- MATH
#math69#
#tex2html_wrap_indisplay725##tex2html_wrap_indisplay726##tex2html_wrap_indisplay727# = #tex2html_wrap_indisplay728# #tex2html_wrap_indisplay729# #tex2html_wrap_indisplay730# #tex2html_wrap_indisplay731# 21.7#tex2html_wrap_indisplay732#m/#tex2html_wrap_indisplay733#s
--;SPMgt;
;SPMlt;/P;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img21.png;SPMquot;
ALT=;SPMquot;#math70##tex2html_wrap_indisplay735##tex2html_wrap_indisplay736##tex2html_wrap_indisplay737#;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img22.png;SPMquot;
ALT=;SPMquot;#math71##tex2html_wrap_indisplay739#;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img23.png;SPMquot;
ALT=;SPMquot;#math72##tex2html_wrap_indisplay741##tex2html_wrap_indisplay742##tex2html_wrap_indisplay743##tex2html_wrap_indisplay744#;SPMquot;;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img24.png;SPMquot;
ALT=;SPMquot;#math73##tex2html_wrap_indisplay746#;SPMquot;;SPMgt; ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img8.png;SPMquot;
ALT=;SPMquot;#math74##tex2html_wrap_indisplay748#;SPMquot;;SPMgt; ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img25.png;SPMquot;
ALT=;SPMquot;#math75##tex2html_wrap_indisplay750#;SPMquot;;SPMgt; ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img8.png;SPMquot;
ALT=;SPMquot;#math76##tex2html_wrap_indisplay752#;SPMquot;;SPMgt; 21.7;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math77##tex2html_wrap_indisplay754#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math78##tex2html_wrap_indisplay756#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt;
;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;
The direction ;SPMlt;I;SPMgt;;SPMamp;#952;;SPMlt;/I;SPMgt; can also be determined, using the definition
;SPMlt;P;SPMgt;;SPMlt;!-- MATH
#math79#
θ = arctan#tex2html_wrap_indisplay758##tex2html_wrap_indisplay759##tex2html_wrap_indisplay760# #tex2html_wrap_indisplay761# 37.74#tex2html_wrap_indisplay762#9#tex2html_wrap_indisplay763#0xb0
--;SPMgt;
;SPMlt;/P;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;I;SPMgt;;SPMamp;#952;;SPMlt;/I;SPMgt; = arctan;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img26.png;SPMquot;
ALT=;SPMquot;#math80##tex2html_wrap_indisplay765##tex2html_wrap_indisplay766##tex2html_wrap_indisplay767#;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img27.png;SPMquot;
ALT=;SPMquot;#math81##tex2html_wrap_indisplay769#;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img28.png;SPMquot;
ALT=;SPMquot;#math82##tex2html_wrap_indisplay771##tex2html_wrap_indisplay772##tex2html_wrap_indisplay773##tex2html_wrap_indisplay774#;SPMquot;;SPMgt; ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img8.png;SPMquot;
ALT=;SPMquot;#math83##tex2html_wrap_indisplay776#;SPMquot;;SPMgt; 37.74;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img29.png;SPMquot;
ALT=;SPMquot;#math84##tex2html_wrap_indisplay778#;SPMquot;;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img30.png;SPMquot;
ALT=;SPMquot;#math85##tex2html_wrap_indisplay780#;SPMquot;;SPMgt;0;SPMlt;I;SPMgt;xb;SPMlt;/I;SPMgt;0
;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;
;SPMlt;P;SPMgt;
In general form, the calculations above for the vertical velocity can be
used to derive the equation of vertical velocity expressed as a function of
time, as well as the fact that the velocity ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt; = 0 ;SPMlt;!-- MATH
#math86##tex2html_wrap_inline782#m/#tex2html_wrap_inline783#s
--;SPMgt;
;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline785#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline787#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt; at the vertex of the flight. Using another fundamental equation of
kinematics,
;SPMlt;BR;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;!-- MATH
#math87#
| vy |
= |
v0y2 +2#tex2html_wrap_indisplay792#Δxy |
|
| vy |
= |
#tex2html_wrap_indisplay796# |
|
--;SPMgt;
;SPMlt;TABLE CELLPADDING=;SPMquot;0;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; WIDTH=;SPMquot;100;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;=;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;0y;SPMlt;/SUB;SPMgt;;SPMlt;SUP;SPMgt;2;SPMlt;/SUP;SPMgt; +2;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img31.png;SPMquot;
ALT=;SPMquot;#math88##tex2html_wrap_indisplay798# ;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
;SPMamp;nbsp;;SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;=;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img32.png;SPMquot;
ALT=;SPMquot;#math89##tex2html_wrap_indisplay800#;SPMquot;;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
;SPMamp;nbsp;;SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;/TABLE;SPMgt;;SPMlt;/DIV;SPMgt;
;SPMlt;BR CLEAR=;SPMquot;ALL;SPMquot;;SPMgt;
can be figured as describing the vertical velocity ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;. The highest
point (or rather, the maximum ;SPMlt;!-- MATH
#math90#Δxy
--;SPMgt;
;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;) that shall be encountered in
the trajectory can be discovered using the principle derived from min-max
functions of calculus that at the vertex of the parabola, the velocity ;SPMlt;I;SPMgt;v;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt; = 0 ;SPMlt;!-- MATH
#math91##tex2html_wrap_inline803#m/#tex2html_wrap_inline804#s
--;SPMgt;
;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline806#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;/;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline808#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;s;SPMlt;/I;SPMgt;. Rearranging the equation,
;SPMlt;P;SPMgt;;SPMlt;!-- MATH
#math92#
Δxy = #tex2html_wrap_indisplay810#
--;SPMgt;
;SPMlt;/P;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img33.png;SPMquot;
ALT=;SPMquot;#math93##tex2html_wrap_indisplay812#;SPMquot;;SPMgt;
;SPMlt;/DIV;SPMgt;;SPMlt;P;SPMgt;;SPMlt;/P;SPMgt;
Substiting in known values,
;SPMlt;BR;SPMgt;
;SPMlt;DIV ALIGN=;SPMquot;CENTER;SPMquot;;SPMgt;
;SPMlt;!-- MATH
#math94#
| Δxy |
= |
#tex2html_wrap_indisplay817# |
|
| Δxy |
= |
#tex2html_wrap_indisplay821# = #tex2html_wrap_indisplay822# |
|
| Δxy |
= |
8.998#tex2html_wrap_indisplay826#m |
|
--;SPMgt;
;SPMlt;TABLE CELLPADDING=;SPMquot;0;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; WIDTH=;SPMquot;100;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;=;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img34.png;SPMquot;
ALT=;SPMquot;#math95##tex2html_wrap_indisplay828#2#tex2html_wrap_indisplay829# -9.8#tex2html_wrap_indisplay830#m/#tex2html_wrap_indisplay831#s#tex2html_wrap_indisplay832#;SPMquot;;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
;SPMamp;nbsp;;SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;=;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img35.png;SPMquot;
ALT=;SPMquot;#math96##tex2html_wrap_indisplay834#;SPMquot;;SPMgt; = ;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img36.png;SPMquot;
ALT=;SPMquot;#math97##tex2html_wrap_indisplay836#;SPMquot;;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
;SPMamp;nbsp;;SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;TR VALIGN=;SPMquot;MIDDLE;SPMquot;;SPMgt;;SPMlt;TD NOWRAP ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;;SPMlt;I;SPMgt;;SPMamp;#916;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=;SPMquot;10;SPMquot; ALIGN=;SPMquot;CENTER;SPMquot; NOWRAP;SPMgt;=;SPMlt;/TD;SPMgt;
;SPMlt;TD ALIGN=;SPMquot;LEFT;SPMquot; NOWRAP;SPMgt;8.998;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img7.png;SPMquot;
ALT=;SPMquot;#math98##tex2html_wrap_indisplay838#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;;SPMlt;/TD;SPMgt;
;SPMlt;TD WIDTH=10 ALIGN=;SPMquot;RIGHT;SPMquot;;SPMgt;
;SPMamp;nbsp;;SPMlt;/TD;SPMgt;;SPMlt;/TR;SPMgt;
;SPMlt;/TABLE;SPMgt;;SPMlt;/DIV;SPMgt;
;SPMlt;BR CLEAR=;SPMquot;ALL;SPMquot;;SPMgt;
Since this is the difference in the ;SPMlt;I;SPMgt;x;SPMlt;/I;SPMgt;;SPMlt;SUB;SPMgt;y;SPMlt;/SUB;SPMgt; values, the maximum height of
the tennis ball is this value plus its initial value, or ;SPMlt;!-- MATH
#math99#8.998#tex2html_wrap_inline840#m +1.895#tex2html_wrap_inline841#m = 10.893#tex2html_wrap_inline842#m
--;SPMgt;
8.998;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline844#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt; +1.895;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline846#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt; = 10.893;SPMlt;IMG
STYLE=;SPMquot;;SPMquot; SRC=;SPMquot;img4.png;SPMquot;
ALT=;SPMquot;#tex2html_wrap_inline848#;SPMquot;;SPMgt;;SPMlt;I;SPMgt;m;SPMlt;/I;SPMgt;.
;SPMlt;P;SPMgt;
;SPMlt;HR;SPMgt;
;SPMlt;/BODY;SPMgt;
;SPMlt;/HTML;SPMgt;