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### Section 12.11 : Velocity and Acceleration

1. An objects acceleration is given by $$\vec a = \cos \left( {2t} \right)\,\vec i + 4{t^3}\,\vec j + 6\sin \left( {3t} \right)\vec k$$. The objects initial velocity is $$\vec v\left( 0 \right) = 6\,\vec i + 2\vec j + 7\vec k$$ and the objects initial position is $$\vec r\left( 0 \right) = \vec i - 9\vec j + 6\vec k$$. Determine the objects velocity and position functions.
2. An objects acceleration is given by $$\vec a = 10t\,\vec i - 6\,\vec j + t\cos \left( t \right)\vec k$$. The objects initial velocity is $$\vec v\left( 0 \right) = - \,\vec i + 11\vec j - \vec k$$ and the objects initial position is $$\vec r\left( 0 \right) = 4\vec i + \vec j + 10\vec k$$. Determine the objects velocity and position functions.
3. Determine the tangential and normal components of acceleration for the object whose position is given by $$\vec r\left( t \right) = \left\langle {5t,1 - 2t,4{t^{\frac{3}{2}}}} \right\rangle$$.
4. Determine the tangential and normal components of acceleration for the object whose position is given by $$\vec r\left( t \right) = \left\langle {6,{{\bf{e}}^{ - 5t}},3t{{\bf{e}}^{ - 5t}}} \right\rangle$$.