Answer
64.8k+ views
Hint: As seen, force in itself is dependent on the position of the particle hence the acceleration is not a constant. For such a changing force, the equation of motion can be used to solve the problem, with an integral sign before the acceleration.
Formula used: In this solution we will be using the following formulae;
\[{v^2} = {u^2} + 2\int {adx} \] where \[v\] is the final velocity (or instantaneous velocity at a particular position) of a body, \[u\] is the initial velocity at the start point, \[a\] is the acceleration of the object, \[x\] signifies the distance, and \[\int {adx} \] signifies an integral of a changing acceleration with respect to the position of the object.
Acceleration \[a = \dfrac{F}{m}\] where \[F\] is the force acting on a body of mass \[m\].
Complete Step-by-Step solution:
The force acting on a body is in itself proportional to the position of the body. Hence, the acceleration of the body is changing. For such a situation, we have the equation of motion to be given as
\[{v^2} = {u^2} + 2\int {adx} \] where \[v\] is the final velocity of a body, \[u\] is the initial velocity at the start point, \[a\] is the acceleration of the object, \[x\] signifies the distance, and \[\int {adx} \] signifies an integral of a changing acceleration with respect to the position of the object.
But, generally, acceleration is given by
\[a = \dfrac{F}{m}\] where \[F\] is the force acting on a body of mass \[m\].
Hence,
\[{v^2} = {u^2} + 2\int {\dfrac{F}{m}dx} \]
Assuming the body begins from rest, and inserting known values and limit of integration, we have
\[{v^2} = 2\int_2^{10} {\dfrac{{3x}}{8}dx} = \dfrac{3}{4}\int_2^{10} {xdx} \]
\[ \Rightarrow {v^2} = \dfrac{3}{4}\left( {\dfrac{{{{10}^2}}}{2} - \dfrac{{{2^2}}}{2}} \right) = \dfrac{3}{8}\left( {100 - 4} \right)\]
By computation,
\[{v^2} = 36\]
\[v = 6m/s\]
Hence, the correct option is A.
Note: For understanding, the equation, \[{v^2} = {u^2} + 2\int {adx} \] can be derived from the definition of instantaneous acceleration, which is
\[a = \dfrac{{dv}}{{dt}}\]
\[ \Rightarrow dv = adt\]
But also, we know that
\[v = \dfrac{{dx}}{{dt}}\], hence,
\[dt = \dfrac{{dx}}{v}\]
Hence, inserting into \[dv = adt\], we get
\[dv = a\dfrac{{dx}}{v}\]
\[ \Rightarrow vdv = adx\]
Integrating both sides from initial to final value, we have
\[\dfrac{{{v^2} - {u^2}}}{2} = \int_{{x_o}}^{{x_1}} {adx} \]
Rearranging we have,
\[{v^2} = {u^2} + 2\int_{{x_o}}^{{x_1}} {adx} \]
Formula used: In this solution we will be using the following formulae;
\[{v^2} = {u^2} + 2\int {adx} \] where \[v\] is the final velocity (or instantaneous velocity at a particular position) of a body, \[u\] is the initial velocity at the start point, \[a\] is the acceleration of the object, \[x\] signifies the distance, and \[\int {adx} \] signifies an integral of a changing acceleration with respect to the position of the object.
Acceleration \[a = \dfrac{F}{m}\] where \[F\] is the force acting on a body of mass \[m\].
Complete Step-by-Step solution:
The force acting on a body is in itself proportional to the position of the body. Hence, the acceleration of the body is changing. For such a situation, we have the equation of motion to be given as
\[{v^2} = {u^2} + 2\int {adx} \] where \[v\] is the final velocity of a body, \[u\] is the initial velocity at the start point, \[a\] is the acceleration of the object, \[x\] signifies the distance, and \[\int {adx} \] signifies an integral of a changing acceleration with respect to the position of the object.
But, generally, acceleration is given by
\[a = \dfrac{F}{m}\] where \[F\] is the force acting on a body of mass \[m\].
Hence,
\[{v^2} = {u^2} + 2\int {\dfrac{F}{m}dx} \]
Assuming the body begins from rest, and inserting known values and limit of integration, we have
\[{v^2} = 2\int_2^{10} {\dfrac{{3x}}{8}dx} = \dfrac{3}{4}\int_2^{10} {xdx} \]
\[ \Rightarrow {v^2} = \dfrac{3}{4}\left( {\dfrac{{{{10}^2}}}{2} - \dfrac{{{2^2}}}{2}} \right) = \dfrac{3}{8}\left( {100 - 4} \right)\]
By computation,
\[{v^2} = 36\]
\[v = 6m/s\]
Hence, the correct option is A.
Note: For understanding, the equation, \[{v^2} = {u^2} + 2\int {adx} \] can be derived from the definition of instantaneous acceleration, which is
\[a = \dfrac{{dv}}{{dt}}\]
\[ \Rightarrow dv = adt\]
But also, we know that
\[v = \dfrac{{dx}}{{dt}}\], hence,
\[dt = \dfrac{{dx}}{v}\]
Hence, inserting into \[dv = adt\], we get
\[dv = a\dfrac{{dx}}{v}\]
\[ \Rightarrow vdv = adx\]
Integrating both sides from initial to final value, we have
\[\dfrac{{{v^2} - {u^2}}}{2} = \int_{{x_o}}^{{x_1}} {adx} \]
Rearranging we have,
\[{v^2} = {u^2} + 2\int_{{x_o}}^{{x_1}} {adx} \]
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)