Question

A particle is moving with velocity $\overrightarrow{v}=K(y\overrightarrow{i}+x\overrightarrow{j})$, where K is a constant. The general equation for its path is:
$\begin{array}{rl}& (A)y=x^2+constant\\ & (B)y^2=x+constant\\ & (C)xy=constant\\ & (D)y^2=x^2+constant\end{array}$

Answer 1

Hint: First consider the general equation of $\overrightarrow{v}$ and compare it with the given equation of $\overrightarrow{v}$. Then we will get a value for the x and y component of $\overrightarrow{v}$. Then differentiating the x and y component of $\overrightarrow{v}$ with respect to time t. Then we get two equations. And by dividing them we will get ${\displaystyle \frac{dy}{dx}}$. Then by using a variable separable method and integrating we will get the final answer.

Complete answer:
Given that,
$\overrightarrow{v}=K(y\overrightarrow{i}+x\overrightarrow{j})$
$\overrightarrow{v}=Ky\overrightarrow{i}+Kx\overrightarrow{j}$ ………….(1)
Consider the general equation of $\overrightarrow{v}$,
$v=v_x\overrightarrow{i}+v_{{}_y}\overrightarrow{j}$ ………….(2)
Comparing equation (1) and (2),
We will get like this,
$v_x=Ky$
and
$v_y=Kx$
Then by differentiating the x and y component of v we get,
${\displaystyle \frac{dx}{dt}}=Ky$ ………….(3)
${\displaystyle \frac{dy}{dt}}=Kx$ …………(4)
Now by dividing equation(4) by (3),
${\displaystyle \frac{dy}{dx}}={\displaystyle \frac{{\displaystyle \frac{dy}{dt}}}{{\displaystyle \frac{dx}{dt}}}}={\displaystyle \frac{x}{y}}$
Then by using variable separable method and rearranging we get,
$ydy=xdx$
Integrating on both sides we get,
$\int ydy=\int xdx$
We know that in general,
$\int xdx={\displaystyle \frac{x^2}{2}}$
By using this concept it becomes,
${\displaystyle \frac{y^2}{2}}={\displaystyle \frac{x^2}{2}}+c$ ………………..(5)
Then multiplying equation (5) by 2 we get,
$y^2=x^2+2c$
where 2c is the constant of integration.
Then it becomes,
$y^2=x^2+$ constant.
This is the general equation for its path.

Hence, option(D) is correct.

Note:
The general equation for $\overrightarrow{v}$ is $v_x\overrightarrow{i}+v_y\overrightarrow{j}$ and compare it with the given equation of $\overrightarrow{v}$. Then we will get a value for the x and y component of $\overrightarrow{v}$. While using Variable separation method for integration, always bring x components to one side and y components to the other side. The name itself shows that. Thus we get the general equation of path .