Question

The velocity of a particle is given as} \[ \vec{v}=-x\hat{i}+2y\hat{j}-z\hat{k}\ \text{m/s}. \] The magnitude of acceleration at the point \((1,2,4)\) is _____ m/s\(^2\).

• A. \(\sqrt{6}\)
• B. \(9\)
• C. \(\sqrt{33}\)
• D. \(0\)

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Acceleration in a steady velocity field \(\vec{v}(x, y, z)\) is calculated using the convective acceleration formula: \(\vec{a} = (\vec{v} \cdot \nabla) \vec{v}\). Each component \(a_x, a_y, a_z\) is derived by differentiating the velocity components with respect to coordinates.
Step 2: Key Formula or Approach:
1. \(a_x = v_x \frac{\partial v_x}{\partial x} + v_y \frac{\partial v_x}{\partial y} + v_z \frac{\partial v_x}{\partial z}\).
2. \(a_y = v_x \frac{\partial v_y}{\partial x} + v_y \frac{\partial v_y}{\partial y} + v_z \frac{\partial v_y}{\partial z}\).
3. \(a_z = v_x \frac{\partial v_z}{\partial x} + v_y \frac{\partial v_z}{\partial y} + v_z \frac{\partial v_z}{\partial z}\).
Step 3: Detailed Explanation:
Given \(\vec{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} = (-x) \hat{i} + (2y) \hat{j} + (-z) \hat{k}\).
Find the acceleration components:
\[ a_x = v_x \frac{\partial (-x)}{\partial x} = (-x)(-1) = x \]
\[ a_y = v_y \frac{\partial (2y)}{\partial y} = (2y)(2) = 4y \]
\[ a_z = v_z \frac{\partial (-z)}{\partial z} = (-z)(-1) = z \]
At the point \((1, 2, 4)\):
\(a_x = 1, a_y = 4(2) = 8, a_z = 4\).
Calculate the magnitude:
\[ |\vec{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2} = \sqrt{1^2 + 8^2 + 4^2} \]
\[ = \sqrt{1 + 64 + 16} = \sqrt{81} = 9 \text{ m/s}^2 \].
Step 4: Final Answer:
The magnitude of acceleration is 9 \(\text{m/s}^2\).