To analyze the motion of the ball, we decompose it into horizontal and vertical components and use the principles of projectile motion. The parameters are:

• Initial speed ($v_0$) = 10 m/s
• Angle of projection ($\theta$) = $37^\circ$
• Acceleration due to gravity ($g$) = $9.8 \, \text{m/s}^2$

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1. Horizontal and Vertical Velocities

The horizontal ($v_{x0}$) and vertical ($v_{y0}$) components of the initial velocity are given by: $v_{x0} = v_0 \cos \theta$ $v_{y0} = v_0 \sin \theta$

$v_{x0} = 10 \cos(37^\circ) \approx 10 \cdot 0.7986 = 7.99 \, \text{m/s}$ $v_{y0} = 10 \sin(37^\circ) \approx 10 \cdot 0.6018 = 6.02 \, \text{m/s}$

2. Time in Air

The time to reach the peak is: $t_{\text{up}} = \frac{v_{y0}}{g} = \frac{6.02}{9.8} \approx 0.614 \, \text{s}$ The total time of flight ($T$) is twice the time to reach the peak: $T = 2 \cdot t_{\text{up}} = 2 \cdot 0.614 = 1.228 \, \text{s}$

3. Horizontal Range

The horizontal range ($R$) is given by: $R = v_{x0} \cdot T$ $R = 7.99 \cdot 1.228 \approx 9.81 \, \text{m}$

4. Maximum Height

The maximum height ($H$) is given by: $H = \frac{v_{y0}^2}{2g}$ $H = \frac{(6.02)^2}{2 \cdot 9.8} = \frac{36.24}{19.6} \approx 1.85 \, \text{m}$

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Final Results

1. Horizontal velocity ($v_{x0}$): $7.99 \, \text{m/s}$
2. Vertical velocity ($v_{y0}$): $6.02 \, \text{m/s}$
3. Time in air ($T$): $1.228 \, \text{s}$
4. Horizontal range ($R$): $9.81 \, \text{m}$
5. Maximum height ($H$): $1.85 \, \text{m}$

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Would you like further clarification or visualization? Here are some related questions to consider:

1. How would the results change if the angle of projection were $45^\circ$?
2. What happens to the time of flight if the initial speed is doubled?
3. How does air resistance affect the range and time of flight in real-life scenarios?
4. What equations describe the ball's motion at any point in time?
5. Can this motion be represented graphically in both horizontal and vertical directions?

Tip: For quick calculations, remember that the maximum range for a projectile occurs at $45^\circ$ assuming no air resistance.