Question

A particle is moving with a velocity \(\vec{v}=K(y\hat{i}+x\hat{j})\), where K is a constant. The general equation for its path is :

• A. \(y=x^2+\text{constant}\)
• B. \(y^2=x+\text{constant}\)
• C. \(y^2=x^2+\text{constant}\)
• D. \(xy=\text{constant}\)

Hint:

Write \(\frac{dx}{dt}=Ky\) and \(\frac{dy}{dt}=Kx\), divide to get \(\frac{dy}{dx}\) and integrate.

Answer 1

The Correct Option is C

\(\frac{dx}{dt}=Ky,\ \frac{dy}{dt}=Kx\Rightarrow\frac{dy}{dx}=\frac{x}{y}\).
\(y\,dy=x\,dx\Rightarrow\frac{y^2}{2}=\frac{x^2}{2}+C\).
So \(y^2=x^2+\text{constant}\).
Answer: (3) \(y^2=x^2+\text{constant}\)