Let's analyze and solve the problem based on the provided data and instructions.

Step-by-Step Solution:

1. Analyzing the Data:

The table provided gives us:

• Low Angle (30° - 40°):

  • Average Time in Air: 2.8 seconds
  • Average Range: 14.65 meters

• High Angle (50° - 60°):

  • Average Time in Air: 3.11 seconds
  • Average Range: 11.6 meters

2. Velocity of Throw (Horizontal and Vertical Components):

The problem instructs us to find the horizontal and vertical components of the initial velocity.

Horizontal Component ($V_{x0}$):

• Formula: $V_{x0} = \frac{\text{Average Range}}{\text{Average Time in Air}}$

  For Low Angle: $V_{x0} = \frac{14.65 \, \text{m}}{2.8 \, \text{s}} \approx 5.23 \, \text{m/s}$

  For High Angle: $V_{x0} = \frac{11.6 \, \text{m}}{3.11 \, \text{s}} \approx 3.73 \, \text{m/s}$

Vertical Component ($V_{y0}$):

• Given formula in the problem (assuming upward acceleration due to gravity): $V_{y0} = g \times \left(\frac{\text{Average Time in Air}}{2}\right)$ where $g = 9.8 \, \text{m/s}^2$.

  For Low Angle: $V_{y0} = 9.8 \times \frac{2.8}{2} = 9.8 \times 1.4 = 13.72 \, \text{m/s}$

  For High Angle: $V_{y0} = 9.8 \times \frac{3.11}{2} = 9.8 \times 1.555 \approx 15.24 \, \text{m/s}$

3. Resultant Velocity Vector ($V$):

Using Pythagoras' theorem to find the resultant velocity: $V = \sqrt{V_{x0}^2 + V_{y0}^2}$

For Low Angle: $V = \sqrt{(5.23)^2 + (13.72)^2} \approx \sqrt{27.36 + 188.29} \approx \sqrt{215.65} \approx 14.68 \, \text{m/s}$

For High Angle: $V = \sqrt{(3.73)^2 + (15.24)^2} \approx \sqrt{13.91 + 232.24} \approx \sqrt{246.15} \approx 15.69 \, \text{m/s}$

4. Angle of Throw ($\theta$):

Using the formula: $\tan \theta = \frac{V_{y0}}{V_{x0}}$ we can find the angle for each throw.

For Low Angle: $\theta = \arctan\left(\frac{13.72}{5.23}\right) \approx \arctan(2.62) \approx 69.48^\circ$

For High Angle: $\theta = \arctan\left(\frac{15.24}{3.73}\right) \approx \arctan(4.09) \approx 76.09^\circ$

Summary:

• Low Angle:

  • Horizontal Velocity ($V_{x0}$): 5.23 m/s
  • Vertical Velocity ($V_{y0}$): 13.72 m/s
  • Resultant Velocity ($V$): 14.68 m/s
  • Angle of Throw ($\theta$): 69.48°

• High Angle:

  • Horizontal Velocity ($V_{x0}$): 3.73 m/s
  • Vertical Velocity ($V_{y0}$): 15.24 m/s
  • Resultant Velocity ($V$): 15.69 m/s
  • Angle of Throw ($\theta$): 76.09°

Would you like more details on any specific part of this solution?

Further Questions to Explore:

1. How would the values change if the release height were increased?
2. How does air resistance affect the calculated velocities and ranges?
3. How would we determine the maximum height reached by the ball in each throw?
4. Can we calculate the initial kinetic energy of the ball based on these values?
5. How would changing the angle of throw affect the range for a given initial speed?

Tip:

To get accurate trigonometric calculations, make sure your calculator is set to the correct mode (degrees or radians) based on the units required.