Question

A particle moves in the x-y plane with a velocity, $\overrightarrow{v}=2y\widehat{i}+4\widehat{j}$ . Equation of the path followed by the particle is:
$\begin{align}
  & (A)x=\sqrt{y} \\
 & (B)y=\sqrt{x} \\
 & (C)4x={{y}^{2}} \\
 & (D)4y={{x}^{2}} \\
\end{align}$

Answer 1

Hint: We will first find the X and Y-components of the velocity using the given velocity equation in the problem. Once we get these terms, they can be written as a differential of displacement with respect to time in X and Y directions respectively. After this, we shall proceed to create a differential equation in X and Y so as to get our required path that the particle is following.

Complete answer:
The velocity vector has been given as:
$\Rightarrow \overrightarrow{v}=2y\widehat{i}+4\widehat{j}$
Now, let the X-component of the velocity vector be denoted by $\overrightarrow{{{v}_{x}}}$ .
And, let the Y-component of the velocity vector be denoted by $\overrightarrow{{{v}_{y}}}$ .
Then, these will be equal to:
$\Rightarrow {{v}_{x}}=2y$ And,
$\Rightarrow {{v}_{y}}=4$
Now these components of velocity in the X and Y direction could be written as the differentials of the displacement with respect to time in the respective directions. That is:
$\Rightarrow \dfrac{dx}{dt}=2y$ [Let this expression be equation number (1)]
$\Rightarrow \dfrac{dy}{dt}=4$ [Let this expression be equation number (2)]
On dividing equation number (2) by (1), we get:
$\begin{align}
  & \Rightarrow \dfrac{\dfrac{dy}{dt}}{\dfrac{dx}{dt}}=\dfrac{4}{2y} \\
 & \Rightarrow \dfrac{dy}{dx}=\dfrac{4}{2y} \\
 & \Rightarrow 2y(dy)=4(dx) \\
\end{align}$
Thus, we get a linear differential in X and Y. This could be easily solved as follows:
Integrating both sides of the equation, we get:
$\Rightarrow \int{2y.dy}=\int{4.dx}$
$\Rightarrow 2\left[ \dfrac{{{y}^{2}}}{2} \right]=4\left[ x \right]$ $\left[ \text{Using: }\int{{{a}^{n}}dn=\dfrac{{{a}^{n+1}}}{n+1}} \right]$
$\therefore {{y}^{2}}=4ax$
Hence, the equation of the path followed by the particle is given by ${{y}^{2}}=4ax$.

Hence, option (C) is the correct option.

Note:
In this problem we saw the use of basic definition of velocity and how we used it to calculate the path of the particle. Also, whenever dealing with questions like these we should have a basic knowledge of integral and differential calculus as they too played a key part in solving the problem. Formulas for all the basic integrals should be remembered thoroughly.