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

JEE Main · 2010 · 11 Apr 2010 · Offline · Kinematics

A particle is moving with velocity

v = K (y \hat{i} + x \hat{j})

, where

K

is a constant. The general equation for its path is

(A) $y = x^{2} +$ constant
(B) $y^{2} = x +$ constant
(C) $x y =$ constant
(D) $y^{2} = x^{2} +$ constant✓ Correct

Answer: (D)

Solution

The velocity components are

\frac{d x}{d t} = K y

and

\frac{d y}{d t} = K x

. Dividing one by the other removes both the time and the constant

K

\frac{d y}{d x} = \frac{x}{y}

, so

y d y = x d x

. Integrating both sides gives

\frac{y ^{2}}{2} = \frac{x ^{2}}{2} + C

, that is

y^{2} = x^{2} + constant

, which is the equation of the path.

JEE Main · 2010 · 11 Apr 2010 · Offline · Single correct option

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