To find the height of the water tower and the time it takes for the tennis ball to reach the ground, we will walk through the calculations step-by-step.
### Step-by-Step Solution:
#### Given:
- Initial horizontal velocity of the tennis ball, [tex]\( v_x = 10.0 \)[/tex] m/s.
- Horizontal distance covered by the tennis ball, [tex]\( d_x = 20.0 \)[/tex] m.
- Acceleration due to gravity, [tex]\( g = 9.81 \)[/tex] m/s² (acting downward).
#### To Find:
1. The height of the water tower, [tex]\( h \)[/tex]
2. The time it takes for the tennis ball to reach the ground, [tex]\( t \)[/tex].
### Step 1: Calculate the Time to Reach the Ground
The tennis ball is thrown horizontally, so the initial vertical velocity is [tex]\( 0 \)[/tex]. It travels a horizontal distance of [tex]\( 20.0 \)[/tex] meters at a constant horizontal speed of [tex]\( 10.0 \)[/tex] m/s. The time it takes to cover this horizontal distance can be determined using the formula:
[tex]\[ t = \frac{d_x}{v_x} \][/tex]
Plugging in the given values:
[tex]\[ t = \frac{20.0 \text{ m}}{10.0 \text{ m/s}} = 2.0 \text{ seconds} \][/tex]
So, the tennis ball takes [tex]\( 2.0 \)[/tex] seconds to reach the ground.
### Step 2: Calculate the Height of the Water Tower
The height of the water tower can be calculated using the vertical motion equation for free fall. The formula for the distance fallen under gravity (starting from rest) is:
[tex]\[ h = \frac{1}{2} g t^2 \][/tex]
Using the time calculated in Step 1, and the given acceleration due to gravity:
[tex]\[ h = \frac{1}{2} \times 9.81 \text{ m/s}^2 \times (2.0 \text{ s})^2 \][/tex]
Calculating this:
[tex]\[ h = \frac{1}{2} \times 9.81 \times 4 \][/tex]
[tex]\[ h = 19.62 \text{ meters} \][/tex]
### Results:
- The tennis ball takes [tex]\( 2.0 \)[/tex] seconds to reach the ground.
- The height of the water tower is [tex]\( 19.62 \)[/tex] meters.
These results provide the required information about the time and height for the tennis ball thrown horizontally from the top of the water tower.
