Question

The instantaneous position of an object is specified by its position vector leading from a fixed origin to the location of the object, modeled as a particle. Suppose for a certain object the position vector is a function of time given by $\overrightarrow{\mathbf{r}}\mathbf{=}\mathbf{4}\hat{\mathbf{i}}\mathbf{+}\mathbf{3}\hat{\mathbf{j}}\mathbf{-}\mathbf{2}\mathit{t}\hat{\mathbf{k}}$, where$\overrightarrow{\mathbf{r}}$is in meters and is t in seconds. (a) Evaluaterole="math" localid="1663653936011" $\mathit{d}\overrightarrow{\mathbf{r}}\mathbf{/}\mathit{d}\mathit{t}$. (b) What physical quantity does$\mathit{d}\overrightarrow{\mathbf{r}}\mathbf{/}\mathit{d}\mathit{t}$represent about the object?

Short answer

(a)The value of$\frac{d\overrightarrow{\mathrm{r}}}{dt}$ is$-2\hat{\mathrm{k}}$

(b) The velocity of the object in space is represented by$\frac{d\overrightarrow{\mathrm{r}}}{dt}$ , although the object is travelling in z-dimension only.

Step-by-step solution

The given expression in the question

The position vector is $\overrightarrow{\mathrm{r}}=4\hat{\mathrm{i}}+3\hat{\mathrm{j}}-2t\hat{\mathrm{k}}$.

Here t is in seconds and r is in meters.

Step 2(a) : The value of dr→/dt

For the evaluation, the value is:

$\begin{array}{rcl}\frac{d\overrightarrow{\mathrm{r}}}{dt}& =& \frac{d}{dt}(4\hat{\mathrm{i}}+3\hat{\mathrm{j}}-2t\hat{\mathrm{k}})\\ & =& \frac{d}{dt}\left(4\hat{\mathrm{i}}\right)+\frac{d}{dt}\left(3\hat{\mathrm{j}}\right)-\frac{d}{dt}\left(2t\hat{\mathrm{k}}\right)\\ & =& 0\hat{\mathrm{i}}+0\hat{\mathrm{j}}-2\hat{\mathrm{k}}\\ & =& -2\hat{\mathrm{k}}\end{array}$

The value of $\frac{\mathrm{d}\overrightarrow{\mathrm{r}}}{\mathrm{dt}}$is$-2\hat{\mathrm{k}}$

Step 3(b) : The physical quantity of the object

The velocity of the object in space is represented by $\frac{\mathrm{d}\overrightarrow{\mathrm{r}}}{\mathrm{dt}}$, although the object is travelling in z- dimension only.