A ball is thrown upwards from a roof top, 80m above the ground. It will reach a maximum vertical height and then fall back to the ground. The height of the ball from the ground at time T is H, which is given by
H= -16t2 + 64t +80
well, you have the function. Using it, you can find the initial speed (64 ft/s), maximum height (at the vertex), and how long it takes to hit the ground (when h=0).
Incidentally, it is odd that the initial height is given in meters, but the -16t^2 is in ft/s^2.
To determine the maximum height the ball reaches, we need to find the vertex of the parabolic equation representing the height of the ball. The vertex can be found using the formula:
t = -b / (2a)
In the given equation H = -16t^2 + 64t + 80, a = -16 and b = 64. By substituting these values into the formula, we can calculate the time at which the ball reaches its maximum height.
t = -64 / (2 * (-16))
t = -64 / (-32)
t = 2
Therefore, the ball reaches its maximum height at t = 2 seconds.
To find the maximum height itself, we substitute the value of t = 2 into the equation H = -16t^2 + 64t + 80.
H = -16(2)^2 + 64(2) + 80
H = -16(4) + 128 + 80
H = -64 + 128 + 80
H = 144
Therefore, the maximum height the ball reaches is 144 meters.
After reaching its maximum height, the ball will start to fall back to the ground. The time it takes for the ball to hit the ground can be found by setting H = 0 in the equation and solving for t.
0 = -16t^2 + 64t + 80
To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, the quadratic formula is the most straightforward method.
t = (-b ± √(b^2 - 4ac)) / (2a)
Using the quadratic formula, where a = -16, b = 64, and c = 80:
t = (-64 ± √(64^2 - 4(-16)(80))) / (2(-16))
t = (-64 ± √(4096 + 5120)) / (-32)
t = (-64 ± √(9216)) / (-32)
t = (-64 ± 96) / (-32)
This gives us two possible values for t:
1. t = (-64 + 96) / (-32) = 32 / (-32) = -1
2. t = (-64 - 96) / (-32) = -160 / (-32) = 5
The negative value for t (-1) does not make physical sense in this context since time cannot be negative. Therefore, the ball hits the ground after 5 seconds.
To summarize:
- The ball reaches its maximum height of 144 meters after 2 seconds.
- The ball hits the ground after 5 seconds.
To find the time at which the ball reaches its maximum vertical height, we need to find the vertex of the parabolic equation H = -16t^2 + 64t + 80. The vertex represents the maximum or minimum point of a parabola.
The formula for finding the t-coordinate of the vertex of a parabola given in the form H = at^2 + bt + c is:
t = -b / (2a)
In this case, a = -16 and b = 64. Plugging these values into the formula, we get:
t = -64 / (2 * -16)
t = -64 / -32
t = 2
So, the time at which the ball reaches its maximum vertical height is 2 seconds.
Now, to find the maximum vertical height itself (H), we substitute this time value (t = 2) into the equation:
H = -16(2)^2 + 64(2) + 80
H = -16(4) + 128 + 80
H = -64 + 128 + 80
H = 144
Therefore, the maximum vertical height reached by the ball is 144 meters.
I'm assuming that the ht. of bldg. is 80 FT.
instead of 80m.
V = Vo + g*Tr = 0.
64 + 32Tr = 0,
Tr = 2 s. = Rise time or time to reach max height, H.
H = -16*2^2 + 64*2 + 80 = 144 Ft.
0.5g*Tf^2 = 144.
16Tf^2 = 144,
Tf = 3 s. = fall time.
T = Tr + T = 2 + 3 = 5 s. = Time in flight.