# 1.7 Worked Example - Derivatives in Kinematics

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A home-built model rocket is launched straight up into the air. At time $$\displaystyle t = 0$$ the rocket is at rest, about to be launched. The position of the rocket as a function of time is given by:

$$\displaystyle y(t) = \frac{1}{2}(a_0-g)t^2 - \frac{1}{30}\frac{a_0 }{t_{0}^{4}}t^{6}$$ for $$\displaystyle 0 \lt t \lt t_0$$

where $$\displaystyle a_0$$ is a positive constant, $$\displaystyle g$$ is the acceleration of gravity and $$\displaystyle a_0 > g$$. The contant $$\displaystyle t_0$$ is the amount of time that the fuel takes to burn out. Express your answer in terms of g, t,$$\displaystyle a_0$$, and $$\displaystyle t_0$$.

Find $$\displaystyle a$$, the $$\displaystyle y$$-component of the acceleration as a function of time.

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