Question

A force $\overrightarrow{F}=(40\hat{i}+10\hat{j})\text{N}$ acts on a body of mass $5\text{kg}$. If the body starts from rest, its position vector $\overrightarrow{r}$ at time $t=10\text{s}$, will be :

• A. $(100\hat{i}+400\hat{j})\text{m}$
• B. $(100\hat{i}+100\hat{j})\text{m}$
• C. $(400\hat{i}+100\hat{j})\text{m}$
• D. $(400\hat{i}+400\hat{j})\text{m}$

Answer 1

The correct option is C $(400\hat{i}+100\hat{j})\text{m}$

Given that,
$\overrightarrow{F}=(40\hat{i}+10\hat{j})\text{N}$
$m=5\text{kg}$

As we know that,
$\overrightarrow{a}=\frac{d\overrightarrow{v}}{dt}=\frac{d^2\overrightarrow{r}}{d{t}^2}=\frac{\overrightarrow{F}}{m}=(8\hat{i}+2\hat{j}){\text{ms}}^{-2}$

Doing double integrating both side w.r.t. $t$ from $t=0$ to $t=t$ and we get

$\Rightarrow \frac{d\overrightarrow{r}}{dt}=\overrightarrow{v}=(8t\hat{i}+2t\hat{j}){\text{ms}}^{-1}$

$\Rightarrow \overrightarrow{r}=(8\hat{i}+2\hat{j})\frac{t^2}{2}\text{m}$

$\therefore \text{At}t=10\text{s}$

$\overrightarrow{r}=(8\hat{i}+2\hat{j})50\text{m}$

$\Rightarrow \overrightarrow{r}=(400\hat{i}+100\hat{j})\text{m}$

Hence, option $\left(C\right)$ is correct.