Two Stops, One Acceleration

A test sled runs along a straight rail. Its position, in metres, $t$ seconds after release is

$s(t) = 2t^{3} - 15t^{2} + 24t + 30, \qquad t \ge 0$

Quantity	Notation	Meaning
Position	$s(t)$	metres along the rail
Velocity	$\dfrac{ds}{dt}$	how fast the position is changing
Acceleration	$\dfrac{d^{2}s}{dt^{2}}$	how fast the velocity is changing

During the run the sled comes momentarily to rest — velocity exactly zero — at two separate instants. Call the later of those two instants $T$ .

Find

$\left. \frac{d^{2}s}{dt^{2}} \right|_{t = T}$

Both derivatives must be built from the limit definition. Give an exact value.

Two Stops, One Acceleration

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