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

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

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

Step-by-step 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):

`\frac{d\vec{r}}{dt} = (d)/(dt)(3\ \hat{i}+2\ \hat{j}-3t\ \hat{k})\\\Rightarrow \frac{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
`\frac{d\vec{r}}{dt}` is the instantaneous velocity of the object.