A science project studying catapults sent a projectile into the air with an initial velocity of $30\frac{m}{s}.$ The formula for height $($s$)$ in meters with respect to time in seconds is $s=-4\mathrm{.}9t^{2}30t.$ Calculate the average rate of change of the height over the intervals listed.

a$.$ from $t=1$ to $t=3$

b$.$ from $t=2$ to $t=3$

c$.$ from $t=2\mathrm{.}5$ to $t=3$

d$.$ from $t=2\mathrm{.}9$ to $t=3$

e$.$ What do you think might be happening close to $t=3?$