A passenger in a helicopter traveling upwards at 18 m/s accidentally drops a package out the window. If it takes 17 seconds to reach the ground, how high to the nearest meter was the helicopter when the package was dropped? To the nearest meter what was the maximum height of the package above the ground in the previous problem?

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To find the height of the helicopter when the package was dropped, we can use the equation of motion for free-falling objects. The formula is:

h = initial height + initial velocity × time + (1/2) × acceleration × time^2

In this case, since the helicopter is moving upwards, the initial velocity will be positive 18 m/s. The acceleration due to gravity is -9.8 m/s^2 because it acts in the opposite direction to the motion. The initial height is what we need to find.

Let's calculate the initial height:

Using the equation,

h = 0 + 18 × 17 + (1/2) × (-9.8) × (17)^2

h = 0 + 306 - (1/2) × 9.8 × 289

h = 306 - 1414.6

h ≈ -1108.6 meters

The negative sign indicates that the height is below the ground level. Since it's the height relative to the ground, we can consider the absolute value of the result. So, the helicopter was approximately 1109 meters below the ground when the package was dropped.

Now, let's calculate the maximum height of the package above the ground. We can start by finding the time it takes for the package to reach its peak height, where it momentarily stops before falling down. At this point, the vertical velocity becomes zero.

The final velocity (v_f) is 0 m/s, the initial velocity (v_i) is 18 m/s, and the acceleration (a) is -9.8 m/s^2. We can use the equation:

v_f = v_i + a × t

Rearranging the equation gives:

t = (v_f - v_i) / a

t = (0 - 18) / -9.8

t ≈ 1.84 seconds

Next, we can find the maximum height using the equation:

h = initial height + initial velocity × time + (1/2) × acceleration × time^2

Using the same values as before, except replacing the time with 1.84 seconds:

h = 0 + 18 × 1.84 + (1/2) × (-9.8) × (1.84)^2

h ≈ 33.6 meters

Therefore, the maximum height of the package above the ground is approximately 34 meters.