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 + s0A C-141 Starlifter flying at 20,000 feet over level terrain drops a 700-pound supply package.
(a) How long will it take until the supply package strikes the ground? (Round your answer to one decimal place.)
1 seconds
(b) The plane is flying at 500 miles per hour. How far will the supply package travel horizontally during its descent? (Round your answer to one decimal place.)
2 miles
a. d = ho + Vo*t -0.5g*t^2.
20000 + 0 - 16t^2 = 0
16t^2 = 20000
t^2 = 1250
Tf = 35.4 s. = Fall time.
b. Dx = 500mi/h * (35.4/3600)h = 4.9 Mi.
To find the answer to both parts of this problem, we will use the position equation provided. We will substitute the given values into the equation and solve for the unknowns.
(a) How long will it take until the supply package strikes the ground?
In the given equation, s represents the height of the object, which in this case is the supply package above the ground. When the package strikes the ground, the height (s) will be zero. We can set up the equation as follows:
0 = -16t^2 + v0t + s0
We are given that the initial velocity (v0) is not given, but the package is dropped from a stationary plane, so the initial velocity is zero (v0 = 0). The initial height (s0) is given as 20,000 feet.
0 = -16t^2 + 0t + 20,000
Simplifying the equation:
-16t^2 + 20,000 = 0
Now we can solve this quadratic equation for t. We'll use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = -16, b = 0, and c = 20,000. Substituting these values into the quadratic formula:
t = (-0 ± √(0 - 4(-16)(20,000))) / 2(-16)
t = (√(0 + 1,280,000)) / -32
t = (√1,280,000) / -32
Calculating the square root:
t ≈ ± 35.8 / -32
Since time cannot be negative in this context, we take the positive value:
t ≈ 35.8 / -32
t ≈ -1.1 seconds
However, time cannot be negative since it represents a duration. Therefore, we discard the negative value.
Hence, it will take about 1.1 seconds until the supply package strikes the ground.
(b) How far will the supply package travel horizontally during its descent?
To find the horizontal distance traveled, we need to determine the time it takes for the package to strike the ground, which we found to be approximately 1.1 seconds. The horizontal distance traveled can be calculated by multiplying the horizontal velocity by the time.
We are given that the plane is flying at 500 miles per hour, but we need to convert it to feet per second. There are 5280 feet in a mile and 3600 seconds in an hour:
Horizontal velocity = 500 miles/hour * 5280 feet/mile * 1 hour/3600 seconds
Converting units:
Horizontal velocity ≈ (500 * 5280) / 3600 feet/second
Horizontal velocity ≈ 733.33 feet/second
Now we can calculate the horizontal distance:
Horizontal distance = Horizontal velocity * time
Plugging in the values:
Horizontal distance ≈ 733.33 feet/second * 1.1 seconds
Calculating:
Horizontal distance ≈ 806.67 feet
Therefore, the supply package will travel approximately 806.67 feet horizontally during its descent (approximately 0.15 miles).