SOLUTION: You are flying in an open plane at an altitude of 7056 feet and you drop a Coca Cola bottle out of the window. The bottle will hit the ground after how many seconds.

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: You are flying in an open plane at an altitude of 7056 feet and you drop a Coca Cola bottle out of the window. The bottle will hit the ground after how many seconds.      Log On


   

Question 278229: You are flying in an open plane at an altitude of 7056 feet and you drop a Coca Cola bottle out of the window. The bottle will hit the ground after how many seconds.
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
The general form of the equation for this situation is:
s%28t%29+=+%281%2F2%29at%5E2+%2B+v%5B0%5Dt+%2B+s%5B0%5D
where
s(t) = the distance from the ground at time t
a = the acceleration due to gravity
v%5B0%5D = the initial velocity (IOW velocity at t = 0)
s%5B0%5D = the initial distance from the ground (IOW distance from teh ground at t = 0)
Also:
  • since distance above ground is usually considered positive and since gravity pulls back toward the ground, acceleration due to gravity is a negative number
  • the units of distance for s(t), a, v%5B0%5D and s%5B0%5D are the same. If it is feet, a = -32 (on earth). If it is meters, a = -9.8 (on earth). Convert any other unit of distance to one of these two.
  • the units of time should be seconds. Convert any other unit of time to seconds.

Using this on your problem:
Since units of distance are feet, a = -32
Since the bottle is dropped, v%5B0%5D+=+0
s%5B0%5D+=+7056
This gives us the equation:
s%28t%29+=+%281%2F2%29%28-32%29t%5E2+%2B+%280%29t+%2B+7056
which simplifies to:
s%28t%29+=+%28-16%29t%5E2+%2B+7056

The problem is to find the time when the bottle hits the ground. In Mathematics terms, find t when s(t) = 0:
0+=+%28-16%29t%5E2+%2B+7056
This is a quadratic equation. To solve it we want one side equal to zero. The left side is already zero. Then we factor (or use the Quadratic Formula). We can factor out 16 (or -16). I'll factor out -16 because I want the coefficient of the squared term to be positive:
0+=+%28-16%29%28t%5E2+-+441%29
Since 441 is a well-known perfect square ;), the second factor is a difference of squares which can be factored with the pattern: a%5E2+-+b%5E2+=+%28a%2Bb%29%28a-b%29
0+=+-16%28t%2B21%29%28t-21%29
From the Zero Product Property we know that this product is zero only if one of the factors is zero. -16 cannot be zero but the other two factors can:
t+21 = 0 or t-21 = 0
Solving these we get
t = -21 or t = 21

Mathematically both of these numbers fit the equation. But realistically we must reject t = -21 because negative time is not relevant to the problem. So the only realistic solution is t = 21.