Now suppose that we kick a ball on the ground up into the air with
some initial velocity. How far away does it land? Again take the
initial position to be at the origin of our coordinate system.
To find where it lands we ask when does it's y component equal
zero. Setting the left hand side of eq. 5.17 to zero,
we can solve for x, giving where the ball will land. We'll call this
point the ``range'' R. It is then 
.
Often we want to express the initial velocity in terms of its polar
represent ion (see 4.2). So 
and 
. Using the trigonometric identity
we can rewrite the range
as
Now you might suspect that some mistake could have crept into
this derivation. How can you tell if it seems reasonable? Well first
check the units. Yep they're right. OK, now lets look at various
limiting cases. Take 
to be zero. The range is zero. That
makes sense because if you kick a ball from the ground level in the
horizontal direction, it doesn't fly through the air at all. It
hits the ground immediately.
OK, how about when
is 90 degrees? Well you get zero also. That seems right too because
in that case it's going straight up, and it's liable smack you on the
way down if you don't get out of the way.
If you maximize the range over all angles you get that it's largest at 45 degrees.