Calculating the Maximum Height and Range of a Tennis Ball

Now Abel and Kato use what they learned to answer the following problem.

The initial speed of a tennis ball is 54 m/s and the launch angle is θi = 21°. Neglecting air resistance, they need to determine the maximum height and range of the tennis ball.

Calculating the Maximum Height and Range:

The maximum height (h) of the tennis ball is approximately 19.13 meters, and the range (R) is approximately 193.22 meters.

Given Parameters:

• Initial speed (v₀) = 54 m/s
• Launch angle (θ_i) = 21°
• Acceleration due to gravity (g) ≈ 9.81 m/s² (on Earth's surface)

Calculations:

1. Calculate the vertical component of the initial velocity (v_0y) using the launch angle:
   • v_0y = v₀ * sin(θ_i)
2. Calculate the time (t) it takes for the ball to reach its maximum height:
   • t = v_0y / g
3. Calculate the maximum height (h) using the kinematic equation:
   • h = 0.5 * g * t²
4. Calculate the horizontal component of the initial velocity (v_0x) using the launch angle:
   • v_0x = v₀ * cos(θ_i)
5. Calculate the time of flight (T) for the entire trajectory:
   • T = 2 * v_0y / g
6. Calculate the range (R) of the tennis ball:
   • R = v_0x * T

Final Results:

By substituting the given values and performing the calculations:

• v_0y ≈ 19.23 m/s
• t ≈ 1.961 s
• h ≈ 19.13 m
• v_0x ≈ 49.28 m/s
• T ≈ 3.92 s
• R ≈ 193.22 m

Now Abel and Kato need to calculate the maximum height and range of a tennis ball with an initial speed of 54 m/s and a launch angle of 21°. Can you guide them through the calculations?

The maximum height of the tennis ball is approximately 19.13 meters, and the range is approximately 193.22 meters. To calculate these values, you can follow the step-by-step calculations provided above.