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 r with arrow = 3 î + 2 ĵ − 3t k, where r with arrow is in meters and t is in seconds. (a) Evaluate dr with arrow/dt. ( î + ĵ + k) m/s (b) What physical quantity does dr with arrow/dt represent about the object?

Answer 1

Answer:

(a) $\dfrac{d\vec{r}}{dt}=-3\ \hat{k}\ m/s$

(b) Instantaneous velocity of the object is represented by $\dfrac{d\vec{r}}{dt}$.

Explanation:

Given:

• $\vec{r}$ = the position vector of the object at any time instant = $3\ \hat{i}+2\ \hat{j}-3t\ \hat{k}$

        where $\vec{r}$ is in meters and $t$ is in seconds.

• $d\vec{r}$ = small change in position vector

• $dt$ = small change in time

Part (a):

$\dfrac{d\vec{r}}{dt} = \dfrac{d}{dt}(3\ \hat{i}+2\ \hat{j}-3t\ \hat{k})\\\Rightarrow \dfrac{d\vec{r}}{dt} =-3\ \hat{k}\ m/s$

Part (b):

As we know that the rate of change of position is the velocity of a particle which is calculated by the differential of the position vector of the particle at with respect to time. This differential gives us a unit vector along negative z-axis having unit as m/s. So, the physical quantity represented by $\dfrac{d\vec{r}}{dt}$ is the instantaneous velocity of the object.