# Acceleration of B Relative to A Let's break down the problem **step by step**: --- ## **Given Data** - **Airplane A** - \( v_A = 430\ \text{km/h} = 119.44\ \text{m/s} \) - \( a_A = 8\ \text{m/s}^2 \) (increasing speed in a straight line) - **Airplane B** - \( v_B = 540\ \text{km/h} = 150.00\ \text{m/s} \) - Tangential acceleration \( a_{B,t} = -3\ \text{m/s}^2 \) (decreasing speed) - Circular path radius \( r = 300\ \text{m} \) - Angle relative to the horizontal \( 30^\circ \) --- ## **Step 1: Acceleration Components** ### **A. Airplane A** - Flying straight, so only tangential acceleration: \[ \vec{a}_A = 8\ \text{m/s}^2 \ \text{(in the direction of motion)} \] --- ### **B. Airplane B** - **Tangential acceleration** (along the path, slowing down): \[ a_{B,t} = -3\ \text{m/s}^2 \] - **Normal (centripetal) acceleration** (towards the center of the circle): \[ a_{B,n} = \frac{v_B^2}{r} = \frac{(150)^2}{300} = 75\ \text{m/s}^2 \] (Points toward the center of the circular path) --- ## **Step 2: Write Acceleration Vectors** Let’s define axes: - **x-axis**: Right, in the direction of \(A\)'s motion. - **y-axis**: Upward. ### **A. Acceleration of A** \[ \vec{a}_A = (8, 0)\ \text{m/s}^2 \] ### **B. Acceleration of B** - **Tangential acceleration**: \( -3 \) m/s² along the direction of velocity (downward and to the right at \(30^\circ\)). - **Normal acceleration**: \(75\) m/s², toward the center of the circle (perpendicular to velocity, toward the center). #### **Direction Details** - **Velocity direction**: \(30^\circ\) below the x-axis (from diagram). - **Tangential acceleration**: Same as velocity direction. - **Normal acceleration**: Perpendicular to velocity, toward the center (i.e., \(60^\circ\) to the left from the x-axis, upward). #### **Components:** - **Tangential (direction: \( -30^\circ \) from x-axis)** \[ a_{B,t,x} = -3 \cos(30^\circ) = -3 \times 0.866 = -2.598\ \text{m/s}^2 \] \[ a_{B,t,y} = -3 \sin(30^\circ) = -3 \times 0.5 = -1.5\ \text{m/s}^2 \] - **Normal (direction: \( +60^\circ \) from x-axis)** \[ a_{B,n,x} = 75 \cos(60^\circ) = 75 \times 0.5 = 37.5\ \text{m/s}^2 \] \[ a_{B,n,y} = 75 \sin(60^\circ) = 75 \times 0.866 = 64.95\ \text{m/s}^2 \] - **Total for B:** \[ a_{B,x} = a_{B,t,x} + a_{B,n,x} = -2.598 + 37.5 = 34.902\ \text{m/s}^2 \] \[ a_{B,y} = a_{B,t,y} + a_{B,n,y} = -1.5 + 64.95 = 63.45\ \text{m/s}^2 \] --- ## **Step 3: Acceleration of B Relative to A** \[ \vec{a}_{B/A} = \vec{a}_B - \vec{a}_A \] \[ \vec{a}_{B/A} = (34.902 - 8,\, 63.45 - 0)\ \text{m/s}^2 \] \[ \boxed{ \vec{a}_{B/A} = (26.902,\, 63.45)\ \text{m/s}^2 } \] - **Magnitude:** \[ |\vec{a}_{B/A}| = \sqrt{(26.902)^2 + (63.45)^2} \approx \sqrt{724.3 + 4026.6} = \sqrt{4750.9} \approx 68.9\ \text{m/s}^2 \] - **Direction (angle above x-axis):** \[ \theta = \tan^{-1}\left(\frac{63.45}{26.902}\right) \approx \tan^{-1}(2.358) \approx 67.5^\circ \] --- ## **Final Answers** ### **Relative Acceleration of B with respect to A** - **Components:** \(\boxed{(26.90,\, 63.45)\ \text{m/s}^2}\) - **Magnitude:** \(\boxed{68.9\ \text{m/s}^2}\) - **Direction:** \(\boxed{67.5^\circ}\) above the x-axis (rightward direction of A's motion). --- ### **Summary Table** | Component | Value (\(\text{m/s}^2\)) | |----------------|-------------------------| | \(x\)-axis | 26.90 | | \(y\)-axis | 63.45 | | Magnitude | 68.9 | | Direction | 67.5° above x-axis | --- If you have follow-up questions about any step, just ask!