from 4 feet above a swimming pool, susan throws a ball upward witha velocity of 32 feet per second. how long does it take the ball to reach the water?
(Vf)^2 = Vo^2 + 2gd = 0,
(32)^2 - 2*32d = 0,
1024 - 64d = 0,
-64d = -1024,
d(up) = 16ft.
Vf = Vo + gt = 0,
32 - 32t = 0,
-32t = -32,
t(up) = 1s.
d(down) = 4 + 16 = 20ft.
d = Vo + 0.5gt^2 = 20ft.
d = 0 + 0.5*32t^2 = 20,
16t^2 = 20,
t^2 = 1.25,
t(down) = 1.12s.
t(up) + t(down) = 1 + 1.12 = 2.12s =
time for ball to reach water.
To find the time it takes for the ball to reach the water, we can use the equations of motion. Specifically, we will use the equation:
h = h0 + v0t - 0.5gt^2
where:
h is the height of the ball at time t
h0 is the initial height (4 feet above the water)
v0 is the initial velocity (32 feet per second)
g is the acceleration due to gravity (32 feet per second squared, assuming we are neglecting air resistance)
t is the time we want to find
Since the ball reaches the water when h = 0, we can rewrite the equation as:
0 = h0 + v0t - 0.5gt^2
Substituting the known values, we have:
0 = 4 + 32t - 0.5 * 32 * t^2
Simplifying the equation further, we can rewrite it as a quadratic equation:
0 = 4 + 32t - 16t^2
Now we can solve this quadratic equation to find the value of t.
To find the time it takes for the ball to reach the water, we can use the equation of motion for free fall:
s = ut + (1/2)at^2
Where:
s = displacement (change in height)
u = initial velocity
t = time
a = acceleration due to gravity
In this case, the initial height above the water is 4 feet, the initial velocity is 32 feet per second, the acceleration due to gravity is 32 feet per second squared (assuming we neglect air resistance).
Using these values, we can set up the equation:
0 = 4 + (32)t + (1/2)(-32)t^2
Simplifying, we get:
0 = 4 + 32t - 16t^2
Rearranging the equation, we have a quadratic equation:
16t^2 - 32t - 4 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 16, b = -32, and c = -4.
Substituting these values into the quadratic formula:
t = (-(-32) ± √((-32)^2 - 4(16)(-4))) / (2(16))
Simplifying further:
t = (32 ± √(1024 + 256)) / 32
t = (32 ± √(1280)) / 32
t = (32 ± 35.78) / 32
To solve for t, we have two possible solutions:
1. t = (32 + 35.78) / 32 = 67.78 / 32 = 2.12 seconds
2. t = (32 - 35.78) / 32 = -3.78 / 32 = -0.12 seconds (ignore this negative value)
Therefore, it takes approximately 2.12 seconds for the ball to reach the water.