A helicopter is ascending vertically with a speed of 4.50 m/s. At a height of 65 m above the Earth, a package is dropped from a window. How much time does it take for the package to reach the ground? [Hint: The package's initial speed equals the helicopter's.]

the butterfly effect

To find the time it takes for the package to reach the ground, we can use the equations of motion.

First, let's calculate the time it takes for the package to fall from a height of 65 m using the equation:

Δy = v0t + (1/2)gt^2

Where:
Δy is the displacement or height (65 m),
v0 is the initial velocity (0 m/s, as the package is dropped),
t is the time taken, and
g is the acceleration due to gravity (-9.8 m/s^2).

Rearranging the equation, we get:

(1/2)gt^2 + v0t - Δy = 0

Plugging in the values, we have:

(1/2)(-9.8)t^2 + 0t - 65 = 0

Now, we can solve this quadratic equation using the quadratic formula:

t = (-b ± √(b^2 - 4ac)) / 2a

For this equation, we have:
a = (1/2)(-9.8) = -4.9,
b = 0, and
c = -65.

t = [ -0 ± √(0^2 - 4(-4.9)(-65)) ] / 2(-4.9)

Simplifying further:

t = [ √(0 - 1274) ] / -9.8

Since time cannot be negative in this context, we take the positive value:

t = [ √(1274) ] / 9.8

Calculating this value, we find:

t ≈ 5.1 s

Therefore, it takes approximately 5.1 seconds for the package to reach the ground.