First of all let us notice the fact that the basis vectors of "\\mathbb{R}^3 (i, j, k)" are constant (as opposed to local vectors "e_r,e_\\theta, e_z" of the cylindrical coordinates base, for example). Therefore the derivation of "R(t)" comes down to deriving every coordinate. Secondly, let us write "R(t)" as a vector for more obviousness:
"R(t)=\\begin{pmatrix}\\sin(\\hat{a}t) \\\\ \\cos(\\hat{a}t) \\\\ t\\end{pmatrix}"
Now let's calculate the first and then the second derivative:
"\\frac{d}{dt} R(t) = \\begin{pmatrix} \\hat{a}\\cos(\\hat{a}t) \\\\ -\\hat{a}\\sin(\\hat{a}t) \\\\ 1 \\end{pmatrix}"
"\\frac{d^2}{dt^2} R(t) = \\begin{pmatrix} -\\hat{a}^2 \\sin(\\hat{a}t) \\\\ -\\hat{a}^2 \\cos(\\hat{a}t) \\\\ 0 \\end{pmatrix}"
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