a helicopter is ascending vertically with a speed of 5.20 m/s. at a height of 110 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: v0 for the package equals the speed of the helicopter).

To calculate the time it takes for the package to reach the ground, we can use the formula for free fall motion:

h = (1/2)gt^2

Where:
h = height (in this case, 110 m)
g = acceleration due to gravity (approximately 9.8 m/s^2)
t = time

Since the package is dropped from a window, it starts from rest and its initial velocity (v0) is 0.

We can rearrange the equation to solve for time:

t = √(2h / g)

Substituting the given values:
t = √(2 * 110 / 9.8)

Simplifying:
t = √(220 / 9.8)
t = √22.45
t ≈ 4.74 seconds

Therefore, it takes approximately 4.74 seconds for the package to reach the ground.

To determine the time it takes for the package to reach the ground, we can use the equations of motion. Specifically, we will use the equation for displacement:

s = v0 * t + (1/2) * a * t^2

where:
s = displacement (in this case, the height of the package, which is 110 m)
v0 = initial velocity (equal to the ascending speed of the helicopter, which is 5.20 m/s)
t = time taken for the package to fall
a = acceleration (which is equal to the acceleration due to gravity, approximately 9.8 m/s^2)

Rearranging the equation, we get:
(1/2) * a * t^2 + v0 * t - s = 0

Since the package is dropping, we consider the acceleration due to gravity to be negative. Thus, we can write:
(-1/2) * g * t^2 + v0 * t - s = 0

Substituting the known values, we have:
(-1/2) * 9.8 * t^2 + 5.20 * t - 110 = 0

This is a quadratic equation that we can solve using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)

In our case, the equation becomes:
t = (-(5.20) ± √((5.20)^2 - 4 * (-1/2) * (-110))) / (2 * (-1/2) * (-9.8))

Simplifying further, we have:
t = (-5.20 ± √(27.04 + 2156)) / (-9.8)

Now, we can calculate the values within the square root:
t = (-5.20 ± √(2183.04)) / (-9.8)

t = (-5.20 ± 46.7) / (-9.8)

Considering both positive and negative solutions, we have two possible values for t:
t1 = (-5.20 + 46.7) / (-9.8) ≈ 4.75 s
t2 = (-5.20 - 46.7) / (-9.8) ≈ -5.64 s

Since time cannot be negative, the package takes approximately 4.75 seconds to reach the ground.