During halftime of a basketball 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 64 feet per second. The T-shirt is caught 41 feet above the court. 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?
the height h = 4 + 64t - 16t^2
find the vertex of that parabola for maximum height
also, you can now determine the range
To find the time it takes for the T-shirt to reach its maximum height, you can use the formula for the time it takes to reach the maximum point in projectile motion, which is given by:
t = (Vf - Vo) / g
Where:
- t is the time taken
- Vf is the final velocity (which is 0 at the maximum height)
- Vo is the initial velocity
- g is the acceleration due to gravity (approximately 32.2 feet per second squared)
In this case, the initial velocity (Vo) is 64 feet per second, and the acceleration due to gravity (g) is 32.2 feet per second squared.
Plugging in these values into the formula, we get:
t = (0 - 64) / 32.2
t = -64 / 32.2
t ≈ -1.988 seconds
It's important to note here that time cannot be negative, so we discard the negative value. Hence, it will take approximately 1.988 seconds for the T-shirt to reach its maximum height.
To find the maximum height, we can use the formula for the height of an object in projectile motion, given by:
h = h0 + V0*t - (1/2)*g*t^2
Where:
- h is the height
- h0 is the initial height (4 feet)
- V0 is the initial velocity (64 feet per second)
- t is the time
Plugging in the values, we get:
h = 4 + 64*1.988 - (1/2)*32.2*(1.988)^2
h ≈ 69.47 feet
Therefore, the maximum height the T-shirt reaches is approximately 69.47 feet.
The range of the function that models the height of the T-shirt over time will be the horizontal distance traveled by the T-shirt. In projectile motion, the range can be calculated using the formula:
R = V0*t
Where:
- R is the range
- V0 is the initial velocity (64 feet per second)
- t is the time taken to reach the maximum height (1.988 seconds)
Plugging in the values, we get:
R = 64 * 1.988
R ≈ 127.23 feet
Hence, the range of the function that models the height of the T-shirt over time is approximately 127.23 feet.