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Ch. 02 - Describing Motion: Kinematics in One Dimension
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 02 - Describing Motion: Kinematics in One DimensionProblem 66
Chapter 2, Problem 66

A helicopter is ascending vertically with a constant speed of 6.40 m/s. At a height of 105 m above the Earth, a package is dropped from the helicopter. How much time does it take for the package to reach the ground? [Hint: What is v₀ for the package?]

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Identify the known values: The initial velocity of the package, \(v_0\), is the same as the helicopter's vertical speed, which is \(6.40\, \text{m/s}\) upward. The initial height of the package above the ground is \(h_0 = 105\, \text{m}\). The acceleration due to gravity, \(g\), is \(9.80\, \text{m/s}^2\) downward. The final position of the package when it reaches the ground is \(h = 0\).
Write the kinematic equation for vertical motion: \(h = h_0 + v_0 t - \frac{1}{2} g t^2\). Substitute \(h = 0\), \(h_0 = 105\, \text{m}\), \(v_0 = 6.40\, \text{m/s}\), and \(g = 9.80\, \text{m/s}^2\) into the equation.
Rearrange the equation to form a standard quadratic equation: \(0 = 105 + 6.40t - \frac{1}{2}(9.80)t^2\). Simplify further to \(0 = -4.90t^2 + 6.40t + 105\).
Solve the quadratic equation \(-4.90t^2 + 6.40t + 105 = 0\) using the quadratic formula: \(t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = -4.90\), \(b = 6.40\), and \(c = 105\).
Calculate the discriminant \(\Delta = b^2 - 4ac\) and determine the two possible solutions for \(t\). Discard the negative solution since time cannot be negative. The positive solution will give the time it takes for the package to reach the ground.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Free Fall

Free fall refers to the motion of an object under the influence of gravity alone, without any air resistance. When the package is dropped from the helicopter, it begins to accelerate downward due to Earth's gravitational pull, which is approximately 9.81 m/s². Understanding free fall is crucial for calculating the time it takes for the package to reach the ground.
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Initial Velocity

The initial velocity (v₀) of an object is the speed at which it starts its motion. In this scenario, the package is dropped from a helicopter ascending at 6.40 m/s, meaning its initial velocity when released is also 6.40 m/s upward. This initial velocity affects the time it takes for the package to reach the ground, as it will initially move upward before gravity pulls it down.
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Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this problem, we can use the kinematic equation that incorporates initial velocity, acceleration due to gravity, and displacement to determine the time it takes for the package to hit the ground.
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