91Ó°ÊÓ

It is given that,

$\stackrel{â‡\P Ä}{\mathrm{r}}=\hat{\mathrm{i}}+4{\mathrm{t}}^2\hat{\mathrm{j}}+\mathrm{t}\hat{\mathrm{k}}$

This problem deals with simple algebraic operation that involves calculation of velocity and the acceleration at given time. Velocity is the rate of change of its position and the acceleration is the rate of change of velocity with respect to time. Thus the velocity vector and acceleration can be expressed using the standard equation for the velocity and acceleration.

Formulae:

$$\begin{gathered} \stackrel{â‡\P Ä}{\mathrm{v}}=\frac{\mathrm{d}\stackrel{â‡\P Ä}{\mathrm{r}}}{\mathrm{dt}}\left(i\right) \\ \mathrm{Where},\stackrel{â‡\P Ä}{\mathrm{r}}\mathrm{is}\mathrm{the}\mathrm{position}\mathrm{vector} \\ \stackrel{â‡\P Ä}{\mathrm{a}}=\frac{\mathrm{d}\stackrel{â‡\P Ä}{\mathrm{v}}}{\mathrm{dt}}\left(ii\right) \end{gathered}$$

Velocity can be expressed by substituting the value of $\stackrel{â‡\P Ä}{r}$in equation (i)

$\stackrel{â‡\P Ä}{\mathrm{v}}=\frac{\mathrm{d}\left(\hat{\mathrm{i}}+4{\mathrm{t}}^2\hat{\mathrm{j}}+\mathrm{t}\hat{\mathrm{k}}\right)}{\mathrm{dt}}$

Thus,

$\stackrel{â‡\P Ä}{\mathrm{v}}=\left(\left(8\mathrm{t}\right)\hat{\mathrm{j}}+\hat{\mathrm{k}}\right)\mathrm{m}/\mathrm{s}$

Acceleration can be expressed by substituting the value of $\stackrel{â‡\P Ä}{v}$in equation (ii)

$\stackrel{â‡\P Ä}{\mathrm{a}}=\frac{\mathrm{d}(\left(8\mathrm{t}\right)\hat{\mathrm{i}}+\hat{\mathrm{k}})}{\mathrm{dt}}$

Thus, the acceleration is

$\stackrel{â‡\P Ä}{\mathrm{a}}=8\hat{\mathrm{j}}\mathrm{m}/{\mathrm{s}}^2$