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Date: Sat, 14 Jun 2003 15:08:53 -0400
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      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html">Index</A>=
</TD></TR>
  <TR>
    <TD height=3D17>&nbsp;<A=20
      href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html">=20
      HyperPhysics</A>*****<A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html#mechcon">=20
      Mechanics </A></TD>
    <TD><A href=3D"javascript:history.go(-1)">Go=20
Back</A></TD></TR></TBODY></TABLE><BR><BR><BR><BR><BR><A =
name=3Dgrav></A>
<TABLE border=3D1 cellPadding=3D0 cellSpacing=3D2 height=3D370 =
width=3D580>
  <TBODY>
  <TR>
    <TD height=3D309 width=3D514>
      <H1 align=3Dcenter>Gravity</H1>
      <P>Gravity is the weakest of the <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html#c1"=
>four=20
      fundamental forces</A>, yet it is the dominant force in the =
universe for=20
      shaping the large scale structure of galaxies, stars, etc. The=20
      gravitational force between two masses m1 and m2 is given by the=20
      relationship: </P><BR><IMG=20
      =
src=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/images/gravf.gif">=20
      <P>This is often called the "universal law of gravitation" and G =
the=20
      universal gravitation constant. It is an example of an <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/forces/isq.html#isqg">=
inverse=20
      square law</A> force. The force is always attractive and acts =
along the=20
      line joining the centers of mass of the two masses. The forces on =
the two=20
      masses are equal in size but opposite in direction, obeying <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/newt.html#nt3">Newton'=
s=20
      third law</A>. Viewed as an <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/forces/exchg.html#c1">=
exchange=20
      force</A>, the massless exchange particle is called the <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/particles/expar.html#c=
6">graviton</A>.=20
      </P>
      <P>The gravity force has the same form as <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefor.html#c=
1">Coulomb's=20
      law</A> for the forces between electric charges, i.e., it is an =
inverse=20
      square law force which depends upon the product of the two =
interacting=20
      sources. This led Einstein to start with the <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/forces/funfor.html#c3"=
>electromagnetic=20
      force</A> and gravity as the <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/forces/einun.html#c1">=
first=20
      attempt</A> to demonstrate the <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/astro/unify.html#c1">u=
nification</A>=20
      of the fundamental forces. It turns out that this was the wrong =
place to=20
      start, and that gravity will be the last of the forces to unify =
with the=20
      other three forces. <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/forces/unify.html#c1">=
Electroweak=20
      unification</A> (unification of the electromagnetic and weak =
forces) was=20
      demonstrated in 1983, a result which could not be anticipated in =
the time=20
      of Einstein's search. It now appears that the common form of the =
gravity=20
      and electromagnetic forces arises from the fact that each of them =
involves=20
      an exchange particle of zero mass, not because of an inherent =
symmetry=20
      which would make them easy to unify. </P></TD>
    <TD align=3Dmiddle width=3D66><A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html">Index</A>=
<BR><BR><A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/grav.html#grvcon">Grav=
ity=20
      Concepts</A></TD></TR>
  <TR>
    <TD height=3D17>&nbsp;<A=20
      href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html">=20
      HyperPhysics</A>*****<A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html#mechcon">=20
      Mechanics </A></TD>
    <TD><A href=3D"javascript:history.go(-1)">Go=20
Back</A></TD></TR></TBODY></TABLE><BR><BR><BR><BR><BR><A =
name=3Dtraex></A>
<TABLE border=3D1 cellPadding=3D0 cellSpacing=3D2 height=3D370 =
width=3D580>
  <TBODY>
  <TR>
    <TD height=3D309 width=3D514>
      <H1 align=3Dcenter>Examples of Trajectories</H1>
      <H2>Common misconceptions about guns:</H2>
      <H2 align=3Dleft>A dropped bullet will hit the ground before one =
which is=20
      fired from a gun.</H2>As shown in the illustration of a <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html#tra2">horizo=
ntal=20
      launch</A>, gravity acts the same way on both bullets, giving them =
the=20
      same downward acceleration and making them strike the ground at =
the same=20
      time if the bullet is fired horizontally over level ground. =
<BR><BR>
      <H2 align=3Dleft>Bullets fired from high-powered rifles drop only =
a few=20
      inches in hundreds of yards.</H2>Fired at twice the speed of =
sound, a=20
      bullet will drop over 3 inches in 100 yards, and at 300 yards =
downrange=20
      will have dropped about 30 inches. Plug in numbers into the <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/grav.html#bul">bullet =

      drop calculation</A> to see for yourself. Ammunition manufacturers =

      contribute to this misconception by stating the drop of their =
projectiles=20
      as just the extra drop caused by frictional drag compared to an =
ideal=20
      frictionless projectile. </TD>
    <TD align=3Dmiddle width=3D66><A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html">Index</A>=
<BR><BR><A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html#tracon">Traj=
ectory=20
      concepts</A><BR><BR></TD></TR>
  <TR>
    <TD height=3D17>&nbsp;<A=20
      href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html">=20
      HyperPhysics</A>*****<A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html#mechcon">=20
      Mechanics </A></TD>
    <TD><A href=3D"javascript:history.go(-1)">Go=20
Back</A></TD></TR></TBODY></TABLE><BR><BR><BR><BR><BR><A name=3Dbul></A>
<TABLE border=3D1 cellPadding=3D2 cellSpacing=3D2 height=3D370 =
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  <TBODY>
  <TR>
    <TD height=3D309 width=3D514>
      <H1 align=3Dcenter>Drop of a Bullet</H1><IMG=20
      src=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/images/gun.gif"> =

      <P>If air friction is neglected, then the drop of a bullet fired=20
      horizontally can be treated as an ordinary <A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html#tra11">horiz=
ontal=20
      trajectory</A>. The air friction is significant, so this is an=20
      idealization. </P>
      <FORM>
      <SCRIPT ?language=3D'javascript"'>
function drop(r,v){return .5*9.8*r*r/(v*v)}
function rang(v,d){return v*Math.sqrt(2*d/9.8)}
function dtime(h){return Math.sqrt(2*h/9.8)}
function vmuz(r,d){return Math.sqrt(r*r*9.8/(2*d))}
function =
vunit(v){fh=3Ddocument.forms[0];fh.vm.value=3Dv;fh.vmc.value=3Dv/.3048;fh=
.vmm.value=3Dv*2.24;fh.vmk.value=3Dv/.278}
function =
runit(r){fh=3Ddocument.forms[0];fh.rc.value=3Dr/.3048;fh.ry.value=3Dr/(.3=
048*3);fh.r.value=3Dr}
function =
dunit(d){fh=3Ddocument.forms[0];fh.dc.value=3Dd*100;fh.di.value=3Dd*100/2=
.54;fh.d.value=3Dd}
</SCRIPT>

      <P>If the muzzle velocity is<BR>v =3D<INPUT name=3Dvm=20
      =
onchange=3Dvb=3Ddocument.forms[0].vm.value;vunit(vb);fh=3Ddocument.forms[=
0];rb=3Dfh.r.value;fh.d.value=3Ddb=3Ddrop(rb,vb);dunit(db);hb=3Ddocument.=
forms[0].h.value;fh.rd.value=3Dvb*dtime(hb);fh.rdc.value=3Dvb*dtime(hb)/.=
3048=20
      size=3D8> m/s =3D <INPUT name=3Dvmc=20
      =
onchange=3Dvb=3Ddocument.forms[0].vmc.value*.3048;vunit(vb);fh=3Ddocument=
.forms[0];rb=3Dfh.r.value;fh.d.value=3Ddb=3Ddrop(rb,vb);dunit(db);hb=3Ddo=
cument.forms[0].h.value;fh.rd.value=3Dvb*dtime(hb);fh.rdc.value=3Dvb*dtim=
e(hb)/.3048=20
      size=3D8> ft/s =3D <INPUT name=3Dvmm=20
      =
onchange=3Dvb=3Ddocument.forms[0].vmm.value/2.24;vunit(vb);fh=3Ddocument.=
forms[0];rb=3Dfh.r.value;fh.d.value=3Ddb=3Ddrop(rb,vb);dunit(db);hb=3Ddoc=
ument.forms[0].h.value;fh.rd.value=3Dvb*dtime(hb);fh.rdc.value=3Dvb*dtime=
(hb)/.3048=20
      size=3D8> mi/hr =3D <INPUT name=3Dvmk=20
      =
onchange=3Dvb=3Ddocument.forms[0].vmk.value*.278;vunit(vb);fh=3Ddocument.=
forms[0];rb=3Dfh.r.value;fh.d.value=3Ddb=3Ddrop(rb,vb);dunit(db);hb=3Ddoc=
ument.forms[0].h.value;fh.rd.value=3Dvb*dtime(hb);fh.rdc.value=3Dvb*dtime=
(hb)/.3048=20
      size=3D8> km/hr <BR><BR>and the distance downrange is<BR>R =
=3D<INPUT name=3Dr=20
      =
onchange=3Dvb=3Ddocument.forms[0].vm.value;fh=3Ddocument.forms[0];rb=3Dfh=
.r.value;runit(rb);fh.d.value=3Ddb=3Ddrop(rb,vb);dunit(db);pdb=3Dfh.pd.va=
lue/100;fh.pvm.value=3Dvmuz(rb,pdb);fh.pvmc.value=3Dvmuz(rb,pdb)/.3048=20
      size=3D8> m =3D <INPUT name=3Drc=20
      =
onchange=3Dvb=3Ddocument.forms[0].vm.value;fh=3Ddocument.forms[0];rb=3Dfh=
.rc.value*.3048;runit(rb);fh.d.value=3Ddb=3Ddrop(rb,vb);dunit(db);pdb=3Df=
h.pd.value/100;fh.pvm.value=3Dvmuz(rb,pdb);fh.pvmc.value=3Dvmuz(rb,pdb)/.=
3048=20
      size=3D8> ft =3D <INPUT name=3Dry=20
      =
onchange=3Dvb=3Ddocument.forms[0].vm.value;fh=3Ddocument.forms[0];rb=3Dfh=
.ry.value*.3048*3;runit(rb);fh.d.value=3Ddb=3Ddrop(rb,vb);dunit(db);pdb=3D=
fh.pd.value/100;fh.pvm.value=3Dvmuz(rb,pdb);fh.pvmc.value=3Dvmuz(rb,pdb)/=
.3048=20
      size=3D8> yards,<BR><BR>Then the amount of drop of the bullet =
below the=20
      horizontal would be<BR>d =3D <INPUT name=3Dd=20
      =
onchange=3Ddb=3Ddocument.forms[0].d.value;fh=3Ddocument.forms[0];dunit(db=
);vb=3Dfh.vm.value;fh.r.value=3Drb=3Drang(vb,db);runit(rb);pdb=3Dfh.pd.va=
lue/100;fh.pvm.value=3Dvmuz(rb,pdb);fh.pvmc.value=3Dvmuz(rb,pdb)/.3048=20
      size=3D8> m =3D <INPUT name=3Ddc=20
      =
onchange=3Ddb=3Ddocument.forms[0].dc.value/100;fh=3Ddocument.forms[0];dun=
it(db);vb=3Dfh.vm.value;fh.r.value=3Drb=3Drang(vb,db);runit(rb);pdb=3Dfh.=
pd.value/100;fh.pvm.value=3Dvmuz(rb,pdb);fh.pvmc.value=3Dvmuz(rb,pdb)/.30=
48=20
      size=3D8> cm =3D <INPUT name=3Ddi=20
      =
onchange=3Ddb=3Ddocument.forms[0].di.value*2.54/100;fh=3Ddocument.forms[0=
];dunit(db);vb=3Dfh.vm.value;fh.r.value=3Drb=3Drang(vb,db);runit(rb);pdb=3D=
fh.pd.value/100;fh.pvm.value=3Dvmuz(rb,pdb);fh.pvmc.value=3Dvmuz(rb,pdb)/=
.3048=20
      size=3D8> inches<BR><BR>If the gun is fired on level ground at a =
height of=20
      <INPUT name=3Dh=20
      =
onchange=3Dhb=3Ddocument.forms[0].h.value;fh=3Ddocument.forms[0];fh.td.va=
lue=3Ddtime(hb);vb=3Dfh.vm.value;fh.rd.value=3Dvb*dtime(hb);fh.rdc.value=3D=
vb*dtime(hb)/.3048;fh.hc.value=3Dhb/.3048=20
      size=3D8>m =3D <INPUT name=3Dhc=20
      =
onchange=3Dhb=3Ddocument.forms[0].hc.value*.3048;fh=3Ddocument.forms[0];f=
h.td.value=3Ddtime(hb);vb=3Dfh.vm.value;fh.rd.value=3Dvb*dtime(hb);fh.rdc=
.value=3Dvb*dtime(hb)/.3048;fh.h.value=3Dhb=20
      size=3D8>ft, then the bullet will hit the ground in <INPUT =
name=3Dtd size=3D8>=20
      seconds, having traveled a distance of <INPUT name=3Drd size=3D8> =
meters =3D=20
      <INPUT name=3Drdc size=3D8> feet.<BR><BR>To hold the drop to =
<INPUT name=3Dpd=20
      =
onchange=3Dpdb=3Ddocument.forms[0].pd.value/100;fh=3Ddocument.forms[0];rb=
=3Dfh.r.value;fh.pvm.value=3Dvmuz(rb,pdb);fh.pvmc.value=3Dvmuz(rb,pdb)/.3=
048;fh.pdi.value=3Dpdb*100/2.54=20
      size=3D8>cm =3D <INPUT name=3Dpdi=20
      =
onchange=3Dpdb=3Ddocument.forms[0].pdi.value*2.54/100;fh=3Ddocument.forms=
[0];rb=3Dfh.r.value;fh.pvm.value=3Dvmuz(rb,pdb);fh.pvmc.value=3Dvmuz(rb,p=
db)/.3048;fh.pd.value=3Dpdb*100=20
      size=3D8>inches at the downrange distance R above would require a =
muzzle=20
      velocity of <INPUT name=3Dpvm size=3D8>m/s =3D <INPUT name=3Dpvmc =
size=3D8>ft/s.=20
      </FORM></P></TD>
    <TD align=3Dmiddle width=3D66><A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html">Index</A>=
<BR><BR><A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/traj.html#tracon">Traj=
ectory=20
      concepts</A><BR><BR></TD></TR>
  <TR>
    <TD height=3D17>&nbsp;<A=20
      href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html">=20
      HyperPhysics</A>*****<A=20
      =
href=3D"http://hyperphysics.phy-astr.gsu.edu/hbase/hph.html#mechcon">=20
      Mechanics </A></TD>
    <TD><A href=3D"javascript:history.go(-1)">Go=20
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