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

Hint:

When the velocity components are given in terms of the coordinates, you can use the relationship \( v_x = \frac{dx}{dt} \) and \( v_y = \frac{dy}{dt} \) to derive the equation of motion.

Answer 1

The Correct Option is C

Step 1: The velocity components are \( v_x = K y \) and \( v_y = x \). 
Step 2: To find the equation of the path, we need to eliminate time \( t \). Using the fact that \( v_x = \frac{dx}{dt} \) and \( v_y = \frac{dy}{dt} \), we can write: \[ \frac{dx}{K y} = \frac{dy}{x}. \] 
Step 3: Cross multiplying and integrating both sides: \[ x^2 = y^2 + {constant}. \] Hence, the general equation for the path is: \[ y^2 = x^2 + {constant}. \]