From her second floor window $6$ meters above the ground, Xiomara throws a tennis ball at a rate of $11$ meters per second up towards her friend Itzagueri who is $12$ meters above the ground. Assume $a(t) =-9.8$ meters per sec^{$^2$} for the acceleration due to gravity.

1. Assuming Itzagueri does not catch the ball, describe the motion of the ball $2$ seconds after it was thrown.

   Hint (a):

   At $t = 2$, is the ball rising or falling? Is it's velocity increasing or decreasing?

2. When does the ball reach its highest point?

   Hint (b):

   The ball stops moving at its highest point.

3. Will the ball reach Itzagueri? Support your answer with calculations.

   Answer (c):

   The height is given by $h(t) =-4.9t^2 + 11t + 6$.
   $h$(answer from part (b)) $≈ 12.173$ m; so yes.

4. Determine how high the ball is when it is falling at the rate of $5$ meters per second.

   Hint (d):

   Before you evaluate the height, find the time when $v(t) =-5.$

5. Explain what $\int _ { 0 } ^ { 2 } a ( t ) d t$ represents physically.

   Hint (e):

   Be sure to explain the significance of BOTH the integrand and the bounds.