91Ó°ÊÓ

The acceleration of an object is given by \(\mathbf{a}(t)=t \mathbf{j}+t \mathbf{k} .\) The velocity at \(t=1 \sec\) is \(\mathbf{v}(1)=5 \mathbf{j}\) and the position of the object at \(t=1\) sec is \(\mathbf{r}(1)=0 \mathbf{i}+0 \mathbf{j}+0 \mathbf{k}\). Find the object's position at any time.

\(\begin{array}{ll}& \text { Find }\end{array}\) \(\mathbf{r}(t) \quad\) given \(\quad\) that \(\quad \mathbf{a}(t)=-32 \mathbf{j}\) \(\mathbf{v}(0)=600 \sqrt{3} \mathbf{i}+600 \mathbf{j},\) and \(\mathbf{r}(0)=\mathbf{0} .\)

Find the tangential and normal components of acceleration for \(\mathbf{r}(t)=a \cos (\omega t) \mathbf{i}+b \sin (\omega t) \mathbf{j}\) at \(t=0\).

Given \(\mathbf{r}(t)=t^{2} \mathbf{i}+2 t \mathbf{j}\) and \(t=1,\) find the tangential and normal components of acceleration.

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\left\langle e^{t} \cos t, e^{t} \sin t, e^{t}\right\rangle\). The graph is shown here:

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\langle\cos (2 t), \sin (2 t), 1\rangle\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\left\langle 2 t, t^{2}, \frac{t^{3}}{3}\right\rangle\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\left\langle\frac{2}{3}(1+t)^{3 / 2}, \frac{2}{3}(1-t)^{3 / 2}, \sqrt{2} t\right\rangle\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=\left\langle 6 t, 3 t^{2}, 2 t^{3}\right\rangle\)

Find the tangential and normal components of acceleration. \(\mathbf{r}(t)=t^{2} \mathbf{i}+t^{2} \mathbf{j}+t^{3} \mathbf{k}\)