In a diving siblings showcase, 11yr old Hannah takes off from a 24.61 foot platform with an initial velocity of 9.7 ft/sec at the same time that her brother Tyler, who is older takes off from a 32.81 foot platform with an initial velocity of 4.51ft/sec. Who enters the water first? By how much? Round to the nearest hundredth. Please show all work thanks
Once u have the vertical motion model would the best way to solve for t be to use the quadratic formula??
Assuming the initial velocity is upward, the height of the divers is
H(t) = 24.61 + 9.7t - 16t^2
T(t) = 32.81 + 4.51t - 16t^2
Just solve each for t when the height is zero to see who takes longer. The quadratic formula would be the way to go.
I was mildly surprised at the answer.
Yeah they both hit the water at the same time yet the book asks who hits it first.
well, the answers are identical to a few decimal places. Just check the 3rd or 4th place to see which is actually a bit smaller.
Wow jeez so Tyler enters the water 0.00002 of a second I didn't catch that because I just rounded it. Thanks Steve!!! :)
To determine who enters the water first, we need to calculate the time it takes for each of them to reach the water.
We can use the formula:
h = ut + (1/2)gt^2
where:
- h is the distance travelled (height of the platform)
- u is the initial velocity
- g is the acceleration due to gravity (approximately 32.2 ft/sec^2)
- t is the time taken
For Hannah:
- h = 24.61 ft
- u = 9.7 ft/sec
- g = 32.2 ft/sec^2
24.61 = (9.7)t + (1/2)(32.2)t^2
Simplifying the equation:
12.1t^2 + 9.7t - 24.61 = 0
We can solve this quadratic equation to find the value of t. Using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
where:
a = 12.1
b = 9.7
c = -24.61
t = (-9.7 ± √(9.7^2 - 4*12.1*(-24.61))) / (2*12.1)
Simplifying further:
t = (-9.7 ± √(94.09 + 1186.884)) / 24.2
t = (-9.7 ± √(1280.974)) / 24.2
Taking the positive value:
t = (-9.7 + √(1280.974)) / 24.2
Calculating this value:
t ≈ 0.715 seconds
Now let's do the same calculation for Tyler:
- h = 32.81 ft
- u = 4.51 ft/sec
- g = 32.2 ft/sec^2
32.81 = (4.51)t + (1/2)(32.2)t^2
Simplifying the equation:
16.1t^2 + 4.51t - 32.81 = 0
Using the quadratic formula, we find:
t ≈ 1.049 seconds
Therefore, Tyler enters the water first, and the time difference is approximately 1.049 - 0.715 = 0.334 seconds.
Rounding the time difference to the nearest hundredth, we find that Tyler enters the water approximately 0.33 seconds before Hannah.