Question

If \(v=3xy\), the magnitude of the velocity vector at \((2,-2)\) is ____

• A. \(4\sqrt{2}\)
• B. \(6\sqrt{2}\)
• C. \(12\)
• D. \(0\)

Hint:

For a velocity potential \(\phi\), \[ \boxed{ u=\frac{\partial\phi}{\partial x}, \qquad v=\frac{\partial\phi}{\partial y} } \] Velocity magnitude: \[ \boxed{ V=\sqrt{u^2+v^2}. } \]

Answer 1

The Correct Option is B

Step 1: Use the velocity potential relation. Given the velocity potential, \[ \phi = 3xy \] The velocity components are \[ u=\frac{\partial \phi}{\partial x}=3y, \] \[ v=\frac{\partial \phi}{\partial y}=3x. \]

Step 2: Evaluate at \((2,-2)\). \[ u=3(-2)=-6, \] \[ v=3(2)=6. \] Hence, the magnitude of velocity is \[ V=\sqrt{u^2+v^2} =\sqrt{(-6)^2+6^2} =\sqrt{72} =6\sqrt{2}. \] Therefore, \[ \boxed{6\sqrt{2}} \] Hence, \[ \boxed{(B)} \] is the correct answer.