Answer:
Comparing the heights, the first ball reaches higher, reaching 25 feet compared to the second ball's 16.25 feet.
Step-by-step explanation:
To find out how high each ball goes, we can use the equation for the height off the ground as a function of time:
[tex]\[ f(t) = -\frac{1}{2}at^2 + v_0t + c \][/tex]
Where:
[tex]- \( a \) is the acceleration due to gravity (32 feet per second\(^2\))[/tex]
[tex]- \( v_0 \) is the initial velocity[/tex]
[tex]- \( c \) is the initial height off the ground (which is 10 feet for one ball and 0 for the other)[/tex]
Let's calculate the height each ball reaches:
For the first ball (launched at 40 feet per second):
[tex]- \( v_0 = 40 \) feet per second[/tex]
[tex]- \( c = 0 \) (since it starts from the ground)[/tex]
[tex]\[ f(t) = -\frac{1}{2}(32)t^2 + (40)t + 0 \][/tex]
[tex]\[ f(t) = -16t^2 + 40t \][/tex]
To find the maximum height, we can use the vertex formula: [tex]\( t = -\frac{b}{2a} \)[/tex]
[tex]\[ t = -\frac{40}{2(-16)} = \frac{40}{32} = 1.25 \text{ seconds} \][/tex]
Now, plug [tex]\( t = 1.25 \)[/tex] into the equation to find the maximum height:
[tex]\[ f(1.25) = -16(1.25)^2 + 40(1.25) \][/tex]
[tex]\[ f(1.25) = -16(1.5625) + 50 \][/tex]
[tex]\[ f(1.25) = -25 + 50 = 25 \text{ feet} \][/tex]
So, the first ball reaches a height of 25 feet.
For the second ball (launched at 20 feet per second and starting 10 feet off the ground):
[tex]- \( v_0 = 20 \) feet per second[/tex]
[tex]- \( c = 10 \) feet (it starts 10 feet off the ground)[/tex]
[tex]\[ f(t) = -\frac{1}{2}(32)t^2 + (20)t + 10 \][/tex]
[tex]\[ f(t) = -16t^2 + 20t + 10 \][/tex]
Using the vertex formula again:
[tex]\[ t = -\frac{b}{2a} = -\frac{20}{2(-16)} = \frac{20}{32} = 0.625 \text{ seconds} \][/tex]
Now, plug \( t = 0.625 \) into the equation to find the maximum height:
[tex]\[ f(0.625) = -16(0.625)^2 + 20(0.625) + 10 \][/tex]
[tex]\[ f(0.625) = -16(0.390625) + 12.5 + 10 \][/tex]
[tex]\[ f(0.625) = -6.25 + 22.5 = 16.25 \text{ feet} \][/tex]
So, the second ball reaches a height of 16.25 feet.
Comparing the heights, the first ball reaches higher, reaching 25 feet compared to the second ball's 16.25 feet.
