During halftime of a football game, a sling shot launches T-shirts at the crowd. A T-shirt is launched from a height of 6 feet with an initial upward velocity of 80 feet per second. The T-shirt is caught 33 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?
16 = (1/2) g = 32/2
v = 80 - 32 t
v = 0 at top
so
at top t = 80/32 = 2.5 seconds upward to top
h = 6 + 80 t - (16) t^2
at top
h = 6 + 80 (2.5) - 16 * 6.25 = 6 + 200 - 100 = 106 feet at top
range is 6 feet to 106 feet
I think speedycake may have been doing a similar problem starting at 5 feet and 64 ft/s.
To solve this problem, we can use the kinematic equations of motion.
1. Finding the time taken to reach the maximum height:
The formula to find the time taken to reach the maximum height is given by: t = (v - u) / g
where v is the final velocity, u is the initial velocity, and g is the acceleration due to gravity.
In this case, the initial velocity (u) is 80 feet per second, and the final velocity (v) is 0 feet per second (at the maximum height). The acceleration due to gravity (g) is approximately -32 feet per second squared.
t = (0 - 80) / -32
t = 2.5 seconds
Therefore, the T-shirt will take 2.5 seconds to reach its maximum height.
2. Finding the maximum height:
The formula to find the maximum height is given by: h = u * t + (1/2) * g * t^2
where h is the height, u is the initial velocity, g is the acceleration due to gravity, and t is the time.
In this case, u is 80 feet per second, g is -32 feet per second squared, and t is 2.5 seconds.
h = 80 * 2.5 + (1/2) * (-32) * (2.5)^2
h = 200 - 100
h = 100 feet
Therefore, the maximum height reached by the T-shirt is 100 feet.
3. Finding the range of the function:
The range of the function that models the height of the T-shirt over time is the set of all possible heights reached by the T-shirt.
In this case, the T-shirt starts at a height of 6 feet and reaches a maximum height of 100 feet. Therefore, the range of the function is [6, 100] feet.
So, the time taken to reach the maximum height is 2.5 seconds, the maximum height is 100 feet, and the range of the function is [6, 100] feet.