question_answer

The position vector of a particle is determined by the expression\[\vec{r}=3{{t}^{2}}\hat{i}+4{{t}^{2}}\hat{j}+7\hat{k}\]. The distance traversed in first 10 s is :

A)  500 m        

B)  300 m

C)  150 m        

D)  100 m

Correct Answer: A

Solution :

The position vector is the vector from the origin of the coordinate system (0, 0, 0) to the point \[P\,(x,y,z)\]. The vector change in position, from origin gives the distance traversed. Given, \[\vec{r}=3\,{{t}^{2}}\hat{i}+4{{t}^{2}}\hat{j}+7\hat{k}\] At t = 0, we get \[{{\vec{r}}_{0}}=7\hat{k}\] At t = 10 s, we get \[\vec{r}=3\times {{(10)}^{2}}\hat{i}+4\,{{(10)}^{2}}\hat{j}+7\hat{k}\] \[=300\,\hat{i}+400\hat{j}+7\hat{k}\] Distance travelled, \[\vec{s}=\vec{r}-{{\vec{r}}_{0}}\] \[=300\hat{i}+400\hat{j}+7\hat{k}-7\hat{k}\] \[=300\hat{i}+400\hat{j}\] Hence, \[\left| {\vec{s}} \right|=\left| \vec{r}-{{{\vec{r}}}_{0}} \right|\] \[=\sqrt{{{(300)}^{2}}+{{(400)}^{2}}}=500\,\,m\].