Use the position equation given below, where s represents the height of the object (in feet),
v0
represents the initial velocity of the object (in feet per second),
s0
represents the initial height of the object (in feet), and t represents the time (in seconds), as the model for the problem.
s = −16t2 + v0t + s0
You drop a coin from the top of a building. The building has a height of 1070 feet.
(a) Use the position equation to write a mathematical model for the height of the coin.
s =
(b) Find the height of the coin after 4 seconds.
s =
ft
(c) How long does it take the coin to strike the ground? (Round your answer to two decimal places.)
t =
sec
"drop a coin" means v0 = 0
"The building has a height of 1070 feet" means s0 = 1070
a)
so you have
s = -16t^2 + 1070
b) set t = 4 and find s
c)
"strike the ground" means s = 0
0 = -16t^2 + 1070
16t^2 = 1070
t^2 = 66.875
t = ....
(a) The mathematical model for the height of the coin is s = -16t^2 + v0t + s0, where v0 represents the initial velocity of the coin and s0 represents the initial height of the coin.
s = -16t^2 + v0t + s0
(b) To find the height of the coin after 4 seconds, we substitute t = 4 into the equation:
s = -16(4)^2 + v0(4) + 1070
s = -256 + 4v0 + 1070
s = 4v0 + 814
The height of the coin after 4 seconds is 4v0 + 814 feet.
(c) To find how long it takes the coin to strike the ground, we need to find the value of t when s = 0 (because the height of the ground is 0):
0 = -16t^2 + v0t + 1070
This is a quadratic equation. We can either solve it using factoring, the quadratic formula, or completing the square. Let's use the quadratic formula:
t = (-v0 +/- sqrt(v0^2 - 4(-16)(1070))) / (2(-16))
Simplifying further, we have:
t = (-v0 +/- sqrt(v0^2 + 10880)) / (-32)
Since the coin is being dropped from rest, the initial velocity v0 is 0. Plugging this into the equation:
t = (0 +/- sqrt(0^2 + 10880)) / (-32)
t = +/- sqrt(10880) / (-32)
Taking the positive square root, we have:
t = sqrt(10880) / (-32)
t = -sqrt(680) / 8
Rounded to two decimal places, the time it takes the coin to strike the ground is approximately t = 2.59 seconds.
(a) The mathematical model for the height of the coin is s = -16t^2 + v0t + s0
(b) To find the height of the coin after 4 seconds, we substitute t = 4 into the equation:
s = -16(4)^2 + v0(4) + 1070
s = -256 + 4v0 + 1070
s = 4v0 + 814
Therefore, the height of the coin after 4 seconds is s = 4v0 + 814 ft.
(c) To find how long it takes for the coin to strike the ground, we set the height (s) equal to 0 and solve for t:
0 = -16t^2 + v0t + 1070
Now, we can use the quadratic formula to solve for t:
t = (-v0 ± sqrt(v0^2 - 4(-16)(1070))) / (2(-16))
t = (-v0 ± sqrt(v0^2 + 10880)) / (-32)
Since the coin is dropped, the initial velocity (v0) is 0. Substituting this into the equation:
t = (-0 ± sqrt(0^2 + 10880)) / (-32)
t = ± sqrt(10880) / -32
Taking the positive square root (as the negative one wouldn't make sense in this context):
t = sqrt(10880) / -32
Rounding to two decimal places, the time it takes for the coin to strike the ground is t ≈ -9.51 seconds.
(a) The mathematical model for the height of the coin can be written as:
s = -16t^2 + v0t + s0
Since the coin is being dropped from the top of a building, the initial velocity (v0) is 0. Also, the initial height (s0) of the coin is the height of the building, which is 1070 feet. Therefore, we can substitute these values into the equation:
s = -16t^2 + 0t + 1070
s = -16t^2 + 1070
(b) To find the height of the coin after 4 seconds, we can substitute t = 4 into the equation:
s = -16(4)^2 + 1070
s = -16(16) + 1070
s = -256 + 1070
s = 814 ft
Therefore, the height of the coin after 4 seconds is 814 feet.
(c) To find the time it takes for the coin to strike the ground, we need to find when the height (s) becomes 0. We can set the equation to 0 and solve for t:
0 = -16t^2 + 1070
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 0, and c = 1070. Substituting these values into the formula:
t = (0 ± √(0^2 - 4(-16)(1070))) / (2(-16))
t = (0 ± √(0 - (-68640))) / (-32)
t = (0 ± √(68640)) / (-32)
t = (0 ± 262) / (-32)
Using the positive square root because negative time is not applicable in this context:
t = (262) / (-32)
t = -8.19
Rounding to two decimal places, the time it takes for the coin to strike the ground is approximately 8.19 seconds.