Answer :
To find the direction of the acceleration of the ball, let’s break down the steps:
1. Convert angles to radians:
[tex]\[ \text{initial\_angle\_rad} = \text{initial\_angle} \times \frac{\pi}{180} = -37.0^{\circ} \times \frac{\pi}{180} \][/tex]
[tex]\[ \text{final\_angle\_rad} = \text{final\_angle} \times \frac{\pi}{180} = 150.0^{\circ} \times \frac{\pi}{180} \][/tex]
2. Calculate initial and final velocity components:
The initial velocity components [tex]\( v_{x_i} \)[/tex] and [tex]\( v_{y_i} \)[/tex] can be calculated using:
[tex]\[ v_{x_i} = \text{initial\_velocity} \cdot \cos(\text{initial\_angle\_rad}) \][/tex]
[tex]\[ v_{y_i} = \text{initial\_velocity} \cdot \sin(\text{initial\_angle\_rad}) \][/tex]
The final velocity components [tex]\( v_{x_f} \)[/tex] and [tex]\( v_{y_f} \)[/tex] can be calculated using:
[tex]\[ v_{x_f} = \text{final\_velocity} \cdot \cos(\text{final\_angle\_rad}) \][/tex]
[tex]\[ v_{y_f} = \text{final\_velocity} \cdot \sin(\text{final\_angle\_rad}) \][/tex]
Substituting the values provided:
[tex]\[ v_{x_i} = 1.6851209261997877 \, \text{m/s} \][/tex]
[tex]\[ v_{y_i} = -1.2698296988508218 \, \text{m/s} \][/tex]
[tex]\[ v_{x_f} = -3.290896534380867 \, \text{m/s} \][/tex]
[tex]\[ v_{y_f} = 1.9000000000000000 \, \text{m/s} \][/tex]
3. Calculate changes in velocity components:
[tex]\[ \Delta v_x = v_{x_f} - v_{x_i} = -3.290896534380867 - 1.6851209261997877 = -4.976017460580655 \, \text{m/s} \][/tex]
[tex]\[ \Delta v_y = v_{y_f} - v_{y_i} = 1.9000000000000000 + 1.2698296988508218 = 3.1698296988508217 \, \text{m/s} \][/tex]
4. Calculate acceleration components:
[tex]\[ a_x = \frac{\Delta v_x}{\text{contact\_time}} = \frac{-4.976017460580655}{0.19} = -26.189565582003446 \, \text{m/s}^2 \][/tex]
[tex]\[ a_y = \frac{\Delta v_y}{\text{contact\_time}} = \frac{3.1698296988508217}{0.19} = 16.68331420447801 \, \text{m/s}^2 \][/tex]
5. Calculate the direction of acceleration:
The angle of the acceleration vector can be calculated using:
[tex]\[ \theta = \tan^{-2}\left(\frac{a_y}{a_x}\right) \][/tex]
Substituting the values:
[tex]\[ \theta = \tan^{-1}\left(\frac{16.68331420447801}{-26.189565582003446}\right) \approx 147.50199063224693^{\circ} \][/tex]
6. Adjusting the angle to be between 0 and 360 degrees:
Since the angle is already within the range after adjusting for the arctangent function's range, there's no need for further adjustments.
The direction of the acceleration of the ball is:
[tex]\[ \boxed{147.50199063224693^{\circ}} \][/tex]
1. Convert angles to radians:
[tex]\[ \text{initial\_angle\_rad} = \text{initial\_angle} \times \frac{\pi}{180} = -37.0^{\circ} \times \frac{\pi}{180} \][/tex]
[tex]\[ \text{final\_angle\_rad} = \text{final\_angle} \times \frac{\pi}{180} = 150.0^{\circ} \times \frac{\pi}{180} \][/tex]
2. Calculate initial and final velocity components:
The initial velocity components [tex]\( v_{x_i} \)[/tex] and [tex]\( v_{y_i} \)[/tex] can be calculated using:
[tex]\[ v_{x_i} = \text{initial\_velocity} \cdot \cos(\text{initial\_angle\_rad}) \][/tex]
[tex]\[ v_{y_i} = \text{initial\_velocity} \cdot \sin(\text{initial\_angle\_rad}) \][/tex]
The final velocity components [tex]\( v_{x_f} \)[/tex] and [tex]\( v_{y_f} \)[/tex] can be calculated using:
[tex]\[ v_{x_f} = \text{final\_velocity} \cdot \cos(\text{final\_angle\_rad}) \][/tex]
[tex]\[ v_{y_f} = \text{final\_velocity} \cdot \sin(\text{final\_angle\_rad}) \][/tex]
Substituting the values provided:
[tex]\[ v_{x_i} = 1.6851209261997877 \, \text{m/s} \][/tex]
[tex]\[ v_{y_i} = -1.2698296988508218 \, \text{m/s} \][/tex]
[tex]\[ v_{x_f} = -3.290896534380867 \, \text{m/s} \][/tex]
[tex]\[ v_{y_f} = 1.9000000000000000 \, \text{m/s} \][/tex]
3. Calculate changes in velocity components:
[tex]\[ \Delta v_x = v_{x_f} - v_{x_i} = -3.290896534380867 - 1.6851209261997877 = -4.976017460580655 \, \text{m/s} \][/tex]
[tex]\[ \Delta v_y = v_{y_f} - v_{y_i} = 1.9000000000000000 + 1.2698296988508218 = 3.1698296988508217 \, \text{m/s} \][/tex]
4. Calculate acceleration components:
[tex]\[ a_x = \frac{\Delta v_x}{\text{contact\_time}} = \frac{-4.976017460580655}{0.19} = -26.189565582003446 \, \text{m/s}^2 \][/tex]
[tex]\[ a_y = \frac{\Delta v_y}{\text{contact\_time}} = \frac{3.1698296988508217}{0.19} = 16.68331420447801 \, \text{m/s}^2 \][/tex]
5. Calculate the direction of acceleration:
The angle of the acceleration vector can be calculated using:
[tex]\[ \theta = \tan^{-2}\left(\frac{a_y}{a_x}\right) \][/tex]
Substituting the values:
[tex]\[ \theta = \tan^{-1}\left(\frac{16.68331420447801}{-26.189565582003446}\right) \approx 147.50199063224693^{\circ} \][/tex]
6. Adjusting the angle to be between 0 and 360 degrees:
Since the angle is already within the range after adjusting for the arctangent function's range, there's no need for further adjustments.
The direction of the acceleration of the ball is:
[tex]\[ \boxed{147.50199063224693^{\circ}} \][/tex]
