Chapter 11 - Vectors and Vector-Valued Functions - 11.9 Curvature and Normal Vectors - 11.9 Exercises - Page 852: 38

$${a_N} = \frac{2}{{\sqrt {1 + 4{t^2}} }}{\text{ and }}{a_T} = \frac{{4t}}{{\sqrt {1 + 4{t^2}} }}$$

$$\eqalign{ & {\bf{r}}\left( t \right) = \left\langle {t,{t^2} + 1} \right\rangle \cr & {\text{Calculate }}{\bf{v}}\left( t \right){\text{, }}\left| {{\bf{v}}\left( t \right)} \right|{\text{ and }}{\bf{a}}\left( t \right) \cr & {\bf{v}}\left( t \right) = {\bf{r}}'\left( t \right) \cr & {\bf{v}}\left( t \right) = \left\langle {1,2t} \right\rangle \cr & \left| {{\bf{v}}\left( t \right)} \right| = \sqrt {1 + 4{t^2}} \cr & \cr & {\bf{a}}\left( t \right) = {\bf{v}}'\left( t \right) \cr & {\bf{a}}\left( t \right) = \left\langle {0,2} \right\rangle \cr & \cr & {\text{Find the components of acceleration: }}{a_N}{\bf{N}} + {a_T}{\bf{T}}, \cr & {\text{Where }}{a_N} = \kappa {\left| {\bf{v}} \right|^2} = \frac{{\left| {{\bf{v}} \times {\bf{a}}} \right|}}{{\left| {\bf{v}} \right|}}{\text{ and }}{a_T} = \frac{{{d^2}s}}{{d{t^2}}} = \frac{{{\bf{v}} \cdot {\bf{a}}}}{{\left| {\bf{v}} \right|}} \cr & {\text{Then}}{\text{,}} \cr} $$ \[{\mathbf{v}} \times {\mathbf{a}} = \left| {\begin{array}{*{20}{c}} {\mathbf{i}}&{\mathbf{j}}&{\mathbf{k}} \\ 1&{2t}&0 \\ 0&2&0 \end{array}} \right|\] $$\eqalign{ & {\bf{v}} \times {\bf{a}} = 0{\bf{i}} - 0{\bf{j}}\, + 2{\bf{k}} \cr & \left| {{\bf{v}} \times {\bf{a}}} \right| = 2 \cr & \cr & {a_N} = \frac{{\left| {{\bf{v}} \times {\bf{a}}} \right|}}{{\left| {\bf{v}} \right|}} = \frac{2}{{\sqrt {1 + 4{t^2}} }} \cr & \cr & and \cr & \cr & {\bf{v}} \cdot {\bf{a}} = \left\langle {1,2t} \right\rangle \cdot \left\langle {0,2} \right\rangle \cr & {\bf{v}} \cdot {\bf{a}} = 4t \cr & {a_T} = \frac{{{\bf{v}} \cdot {\bf{a}}}}{{\left| {\bf{v}} \right|}} = \frac{{4t}}{{\sqrt {1 + 4{t^2}} }} \cr & \cr & {a_N} = \frac{2}{{\sqrt {1 + 4{t^2}} }}{\text{ and }}{a_T} = \frac{{4t}}{{\sqrt {1 + 4{t^2}} }} \cr} $$