Let's solve the problem step by step: --- ## **Given** - Velocity components: - \( u = 5y^2 - x \) (in m/s) - \( v = 4x^2 \) (in m/s) - Point: \( (x, y) = (2~\text{m}, 1~\text{m}) \) --- ## **Step 1: Find the Magnitude of Velocity** The magnitude of velocity (\( V \)) is: \[ V = \sqrt{u^2 + v^2} \] ### **Calculate \( u \) and \( v \) at (2, 1):** \[ u = 5y^2 - x = 5(1)^2 - 2 = 5 - 2 = 3~\text{m/s} \] \[ v = 4x^2 = 4 \times (2)^2 = 4 \times 4 = 16~\text{m/s} \] ### **Magnitude:** \[ V = \sqrt{3^2 + 16^2} = \sqrt{9 + 256} = \sqrt{265} \approx 16.3~\text{m/s} \] --- ## **Step 2: Find the Acceleration Components** Total acceleration in a velocity field is: \[ \vec{a} = \frac{\partial \vec{V}}{\partial t} + (\vec{V} \cdot \nabla)\vec{V} \] Since the velocity field is steady (no explicit time dependence), only the convective term remains: \[ \vec{a} = (\vec{V} \cdot \nabla)\vec{V} \] ### **Calculate \( a_x \) and \( a_y \):** \[ a_x = u \frac{\partial u}{\partial x} + v \frac{\partial u}{\partial y} \] \[ a_y = u \frac{\partial v}{\partial x} + v \frac{\partial v}{\partial y} \] #### **Partial Derivatives:** - \( \frac{\partial u}{\partial x} = \frac{\partial}{\partial x}(5y^2 - x) = -1 \) - \( \frac{\partial u}{\partial y} = \frac{\partial}{\partial y}(5y^2 - x) = 10y \) - \( \frac{\partial v}{\partial x} = \frac{\partial}{\partial x}(4x^2) = 8x \) - \( \frac{\partial v}{\partial y} = \frac{\partial}{\partial y}(4x^2) = \) #### **Evaluate at (2, 1):** - \( u = 3 \) - \( v = 16 \) - \( \frac{\partial u}{\partial x} = -1 \) - \( \frac{\partial u}{\partial y} = 10 \times 1 = 10 \) - \( \frac{\partial v}{\partial x} = 8 \times 2 = 16 \) - \( \frac{\partial v}{\partial y} = \) #### **Plug in values:** \[ a_x = 3 \cdot (-1) + 16 \cdot 10 = -3 + 160 = 157~\text{m/s}^2 \] \[ a_y = 3 \cdot 16 + 16 \cdot = 48 + = 48~\text{m/s}^2 \] --- ## **Step 3: Magnitude of Total Acceleration** \[ a = \sqrt{a_x^2 + a_y^2} \] \[ a = \sqrt{157^2 + 48^2} = \sqrt{24649 + 2304} = \sqrt{26953} \approx 164~\text{m/s}^2 \] --- ## **Final Answers (Rounded to Three Significant Figures)** ### **Magnitude of Velocity:** \[ \boxed{16.3~\text{m/s}} \] ### **Magnitude of Total Acceleration:** \[ \boxed{164~\text{m/s}^2} \] --- ### **Summary Table** | Quantity | Value | Units | |-------------------|--------------|-----------| | Velocity | 16.3 | m/s | | Acceleration | 164 | m/s\(^2\) | If you have any questions about any steps, feel free to ask!