a person jumps off of a 6 foot high diving board with an initial velocity of 13 feet per second. how many seconds does it take the person to hit the water?
the equation would be
h = -16t^2 + 13t + 6 , where h is the height above the water in ft, and t is in seconds
so we want h=0
0 = -16t^2 + 13t +6
16t^2 - 13t - 6 = 0
t = (13 ± √553)/32
= appr 1.14 or a negative, which we must reject
it would take about 1.14 seconds
To find out how many seconds it takes for the person to hit the water, we can use the kinematic equation for displacement:
s = ut + (1/2)at^2
where:
- s is the displacement (in this case, the height of the diving board, which is 6 feet),
- u is the initial velocity (13 feet per second),
- a is the acceleration (acceleration due to gravity, which is approximately 32.2 feet per second squared),
- t is the time taken.
Using the equation, we can rearrange it to solve for t:
(1/2)at^2 + ut - s = 0
Substituting the given values:
(1/2)(32.2)t^2 + 13t - 6 = 0
Now, we can solve this quadratic equation to find the value of t. We can use factoring, completing the square, or the quadratic formula.
Factoring is not practical in this case, so let's use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / 2a
For our quadratic equation, the values are:
a = (1/2)(32.2) = 16.1
b = 13
c = -6
Substituting these values into the quadratic formula:
t = (-13 ± sqrt(13^2 - 4(16.1)(-6))) / (2(16.1))
Simplifying:
t = (-13 ± sqrt(169 + 387.84)) / 32.2
t = (-13 ± sqrt(556.84)) / 32.2
t = (-13 ± 23.61) / 32.2
We have two t-values since we have a plus and minus sign. However, since the time cannot be negative, we discard the negative solution.
t = (-13 + 23.61) / 32.2
t = 10.61 / 32.2
t ≈ 0.33 seconds
Therefore, it takes approximately 0.33 seconds for the person to hit the water.