Answer

Verified

386.7k+ views

**Hint:**To get the acceleration of a particle, we need to double differentiate the distance with respect to time, which will give us the value of acceleration of the particle. After first differentiation, we will get the velocity of the particle and when we again differentiate the obtained equation, we will get the acceleration of the particle with respect to time.

**Complete Step-by-Step solution:**

It is given that the distance $x$ covered by a particle varies with time $t$ as ${{x}^{2}}=2{{t}^{2}}+6t+1$.

Now,When we differentiate the above equation with respect to $t$, we will get the velocity of the particle.

So, after differentiating the particle with respect to $t$, we get:

$2x\dfrac{dx}{dt}=4t+6$

In the above equation, $\dfrac{dx}{dt}=v$, i.e., velocity, then

$\begin{align}

& 2xv=4t+6 \\

& \Rightarrow xv=2t+3 \\

\end{align}$

Again, differentiating the above equation with respect to $t$, we get:

$x\dfrac{dv}{dt}+v\dfrac{dx}{dt}=2$

In the above equation, $\dfrac{dx}{dt}=v$, i.e., velocity and $\dfrac{dv}{dt}=a$, i.e., acceleration, then

$xa+{{v}^{2}}=2$

Now, substituting the value of $v$ in the above equation from the equation $xv=2t+3$, we get:

$\begin{align}

& xa+{{\left( \dfrac{2t+3}{x} \right)}^{2}}=2 \\

& \Rightarrow a{{x}^{3}}+4{{t}^{2}}+12t+9=2{{x}^{2}} \\

& \Rightarrow a{{x}^{3}}+2(2{{t}^{2}}+6t+1)+7=2{{x}^{2}} \\

\end{align}$

We know that, ${{x}^{2}}=2{{t}^{2}}+6t+1$.

Then,

$\begin{align}

& a{{x}^{3}}+2({{x}^{2}})+7=2{{x}^{2}} \\

& \Rightarrow a{{x}^{3}}+7=0 \\

& \therefore a=\dfrac{-7}{{{x}^{3}}} \\

\end{align}$

Hence acceleration varies with ${{x}^{-3}}$.

**Therefore, the correct answer is Option (D).**

**Additional Information:**

In a compressible sound transmission medium - mainly air - air particles get an accelerated motion: the particle acceleration or sound acceleration with the symbol a in metre/second2. In acoustics or physics, acceleration (symbol: a) is defined as the rate of change (or time derivative) of velocity. It is thus a vector quantity with dimension length/time2. In SI units, this is $dfrac{m}{s^2}$.

**Note:**

This is a simple question of differentiation. We need to keep one thing in mind that what we will get after differentiating the equation. Just remember that when we differentiate distance, we get velocity and when we differentiate velocity, we get acceleration, this is a very important relationship to remember.

Recently Updated Pages

Cryolite and fluorspar are mixed with Al2O3 during class 11 chemistry CBSE

Select the smallest atom A F B Cl C Br D I class 11 chemistry CBSE

The best reagent to convert pent 3 en 2 ol and pent class 11 chemistry CBSE

Reverse process of sublimation is aFusion bCondensation class 11 chemistry CBSE

The best and latest technique for isolation purification class 11 chemistry CBSE

Hydrochloric acid is a Strong acid b Weak acid c Strong class 11 chemistry CBSE

Trending doubts

Give 10 examples for herbs , shrubs , climbers , creepers

Difference Between Plant Cell and Animal Cell

Name 10 Living and Non living things class 9 biology CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

List some examples of Rabi and Kharif crops class 8 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE