Question 133644
Ok, remember up is positive, down is negative.  (it is that convention that causes us to put the negative sign on the lead coefficient in {{{h=-16t^2+v[0]t+s[0]}}})


Starting at 100 feet ({{{s[0]=100}}} and starting at a velocity of 220 feet per second downward ({{{v[0]=-220}}}) with an acceleration downward due to gravity of 16 ft per second per second ({{{-16t^2}}}), what is the time required for the falcon to reach 0 height (the ground).  I use {{{v[0]}}} and {{{s[0]}}} for initial velocity and distance because that is the value of those quantities at time {{{t[0]}}}.


Just substitute the values to produce the following quadratic equation:
{{{-16t^2-220t+100=0}}})


Just to make the numbers more manageable, divide the whole thing by -4 producing this equivalent quadratic equation:
{{{4t^2+55t-25=0}}})


Using the quadratic formula, {{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} with {{{a=4}}}, {{{b=55}}}, and {{{c=-25}}} we get:


{{{x = (-55 +- sqrt(55^2-4*4*(-25) ))/(2*4) }}} 
{{{x = (-55 +- sqrt(3025+400) )/(8) }}} 
{{{x = (-55 +- sqrt(3425) )/(8) }}} 


{{{3425=5*5*137}}}, so {{{sqrt(3425)=5*sqrt(137)}}}


{{{x = (-55 + 5*sqrt(137) )/(8) }}} or {{{x = (-55 - 5*sqrt(137) )/(8) }}}


A little calculator work shows that {{{x = (-55 - 5*sqrt(137) )/(8) }}} is a little less than -14.  A negative result for a time measurement doesn't make any sense, so let's exclude this result as an extraneous root.


The calculator says that {{{x = (-55 + 5*sqrt(137) )/(8) }}} is a little over 0.44 seconds.  The pigeon has a little less than half a second to avoid becoming the falcon's lunch.


Of course, this calculation doesn't tell the whole story.  We would have to know how far a very frightened pigeon on the ground can move in 0.44 seconds, and then we would have to know something of the aerodynamics of a diving falcon -- specifically: can the falcon alter his trajectory in that time to compensate for the movement of the escaping pigeon?