Answer
64.8k+ views
Hint The modulus of the mean velocity vector can be determined by using the formula of the mean velocity vector and the modulus of the final result is taken, then the modulus of the mean velocity vector can be determined. The formula of the mean velocity vector gives the relation between the velocity vector and the time.
Useful formula
The mean velocity vector is given by,
$\vec m = \dfrac{{\int {\vec V.dt} }}{{\int {dt} }}$
Where, $\vec m$ is the mean velocity vector, $\vec V$ is the velocity vector and $t$ is the time taken.
Complete step by step solution
The unit vector of the $\vec r$ is given by,
$\vec r = at\hat i - b{t^2}\hat j$
The velocity vector is given by,
$\vec V = \dfrac{{d\vec r}}{{dt}}$
By differentiating the $\vec r$ with respect to the time, then the velocity vector is written as,
$\vec V = a\hat i - 2bt\hat j$
Now,
The mean velocity vector is given by,
$\vec m = \dfrac{{\int {\vec V.dt} }}{{\int {dt} }}\,.................\left( 1 \right)$
By substituting the velocity vector in the above equation (1), then the above equation (1) is written as,
$\vec m = \dfrac{{\int {\left( {a\hat i - 2bt\hat j} \right)dt} }}{{\int {dt} }}$
By integrating the above equation, then the above equation is written as,
$\vec m = \dfrac{{at\hat i - b{t^2}\hat j}}{t}$
By cancelling the terms in the above equation, then the above equation is written as,
$\vec m = a\hat i - bt\hat j$
By taking modulus on the both sides, then the above equation is written as,
$\left| {\vec m} \right| = \left| {a\hat i - bt\hat j} \right|$
The modulus is the square root of the sum of the individual squares of the coefficient of the $\hat i$ and $\hat j$, then the above equation is written as,
$\left| {\vec m} \right| = \sqrt {{a^2} + {b^2}{t^2}} $
Thus, the above equation shows the modulus of the mean velocity vector.
Hence, the option (B) is the correct answer.
Note The mean velocity vector is directly proportional to the integration of the velocity vector and the mean velocity vector is inversely proportional to the time. As the velocity vector increases, then the mean velocity vector also increases.
Useful formula
The mean velocity vector is given by,
$\vec m = \dfrac{{\int {\vec V.dt} }}{{\int {dt} }}$
Where, $\vec m$ is the mean velocity vector, $\vec V$ is the velocity vector and $t$ is the time taken.
Complete step by step solution
The unit vector of the $\vec r$ is given by,
$\vec r = at\hat i - b{t^2}\hat j$
The velocity vector is given by,
$\vec V = \dfrac{{d\vec r}}{{dt}}$
By differentiating the $\vec r$ with respect to the time, then the velocity vector is written as,
$\vec V = a\hat i - 2bt\hat j$
Now,
The mean velocity vector is given by,
$\vec m = \dfrac{{\int {\vec V.dt} }}{{\int {dt} }}\,.................\left( 1 \right)$
By substituting the velocity vector in the above equation (1), then the above equation (1) is written as,
$\vec m = \dfrac{{\int {\left( {a\hat i - 2bt\hat j} \right)dt} }}{{\int {dt} }}$
By integrating the above equation, then the above equation is written as,
$\vec m = \dfrac{{at\hat i - b{t^2}\hat j}}{t}$
By cancelling the terms in the above equation, then the above equation is written as,
$\vec m = a\hat i - bt\hat j$
By taking modulus on the both sides, then the above equation is written as,
$\left| {\vec m} \right| = \left| {a\hat i - bt\hat j} \right|$
The modulus is the square root of the sum of the individual squares of the coefficient of the $\hat i$ and $\hat j$, then the above equation is written as,
$\left| {\vec m} \right| = \sqrt {{a^2} + {b^2}{t^2}} $
Thus, the above equation shows the modulus of the mean velocity vector.
Hence, the option (B) is the correct answer.
Note The mean velocity vector is directly proportional to the integration of the velocity vector and the mean velocity vector is inversely proportional to the time. As the velocity vector increases, then the mean velocity vector also increases.
Recently Updated Pages
Write a composition in approximately 450 500 words class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Arrange the sentences P Q R between S1 and S5 such class 10 english JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What is the common property of the oxides CONO and class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
What happens when dilute hydrochloric acid is added class 10 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
If four points A63B 35C4 2 and Dx3x are given in such class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
The area of square inscribed in a circle of diameter class 10 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Other Pages
Excluding stoppages the speed of a bus is 54 kmph and class 11 maths JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
In the ground state an element has 13 electrons in class 11 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Electric field due to uniformly charged sphere class 12 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
A boat takes 2 hours to go 8 km and come back to a class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
According to classical free electron theory A There class 11 physics JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Differentiate between homogeneous and heterogeneous class 12 chemistry JEE_Main
![arrow-right](/cdn/images/seo-templates/arrow-right.png)