Certainly! Let's solve this step by step.
To determine the maximum height reached by the golf ball, we start with the given relationship:
[tex]\[ h \text{ is proportional to } v^2 \][/tex]
This can be written mathematically as:
[tex]\[ h = k \cdot v^2 \][/tex]
where [tex]\( k \)[/tex] is the proportionality constant.
First, we need to determine the value of [tex]\( k \)[/tex] using the initial information provided:
[tex]\[ v = 20 \, \text{m/s}, \, h = 8 \, \text{m} \][/tex]
Substituting these values into the equation:
[tex]\[ 8 = k \cdot (20)^2 \][/tex]
[tex]\[ 8 = k \cdot 400 \][/tex]
Solving for [tex]\( k \)[/tex]:
[tex]\[ k = \frac{8}{400} \][/tex]
[tex]\[ k = 0.02 \][/tex]
Now that we have found the proportionality constant [tex]\( k \)[/tex], we can use it to find the maximum height [tex]\( h \)[/tex] when the speed [tex]\( v \)[/tex] is 35 m/s:
[tex]\[ v = 35 \, \text{m/s} \][/tex]
Using our equation again:
[tex]\[ h = k \cdot v^2 \][/tex]
Substituting [tex]\( k \)[/tex] and [tex]\( v \)[/tex]:
[tex]\[ h = 0.02 \cdot (35)^2 \][/tex]
[tex]\[ h = 0.02 \cdot 1225 \][/tex]
[tex]\[ h = 24.5 \, \text{m} \][/tex]
Therefore, the maximum height reached by the golf ball when its speed is 35 m/s is 24.5 meters.
