During halftime of a football game, a sling shot launches T-shirts at the crowd. A T-shirt is launched from a height of 4 feet with an initial upward velocity of 88 feet per second. The T-shirt is caught 32 feet above the field. How long will it take the T-shirt to reach its maximum height? What is the maximum height? What is the range of the function that models the height of the T-shirt over time?
I have the amount of time it will take the t-shirt to reach its maximum height.
You have T for max height
h(t) = 4 + 88t - 16t^2
so now evaluate H = h(T)
The range is naturally [0,H]
a. V = Vo + g*T = 0
88 + (-32)T = 0
T =
b. V^2 = Vo^2 + 2g*h = 0
88^2 + (-64)h = 0
h = 121 Ft. above launching point.
ho+h = 4 + 121 = 125 Ft. above gnd.
To find the time it takes for the T-shirt to reach its maximum height, we can use the kinematic equation for vertical motion:
𝑦 = 𝑦₀ + 𝑣₀𝑡 - 1/2𝑔𝑡²
Where:
- 𝑦 is the final height (maximum height)
- 𝑦₀ is the initial height (4 feet)
- 𝑣₀ is the initial upward velocity (88 feet per second)
- 𝑔 is the acceleration due to gravity (-32 feet per second squared, considering it's upward motion)
We want to find the time it takes for the T-shirt to reach its maximum height, so we set 𝑦 to the maximum height and solve for 𝑡.
𝑦 = 𝑦₀ + 𝑣₀𝑡 - 1/2𝑔𝑡²
𝑦 = 4 + 88𝑡 - 1/2(32)𝑡²
Since we already have the expression for 𝑦 in terms of 𝑡, we can differentiate 𝑦 with respect to 𝑡 and set it equal to zero to find the time at its maximum.
𝑑𝑦/𝑑𝑡 = 0 + 88 - (32)𝑡 = 0
88 - 32𝑡 = 0
-32𝑡 = -88
𝑡 = 88/32
𝑡 ≈ 2.75 seconds
Therefore, it will take approximately 2.75 seconds for the T-shirt to reach its maximum height.
To find the maximum height, substitute this value of 𝑡 back into the equation for 𝑦:
𝑦 = 4 + 88(2.75) - 1/2(32)(2.75)²
Simplifying the equation will give you the maximum height.
Lastly, the range of the function that models the height of the T-shirt over time will depend on how long the T-shirts are launched. If they are launched continuously until they reach the field, the range would be from the initial height of 4 feet to the final height of 32 feet.