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Ch 02: Motion Along a Straight Line
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 15th EditionCh 02: Motion Along a Straight LineProblema 42d
Capitolo 2, Problema 42d

A hot-air balloonist, rising vertically with a constant velocity of magnitude 5.005.00 m/s, releases a sandbag at an instant when the balloon is 40.040.0 m above the ground (Fig. E2.442.44). After the sandbag is released, it is in free fall. What is the greatest height above the ground that the sandbag reaches?
Illustration of a hot-air balloonist releasing a sandbag at 40 m height.

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1
Identify the initial conditions: The sandbag is released from a height of 40.0 m above the ground with an initial upward velocity of 5.00 m/s.
Understand that after release, the sandbag is in free fall, meaning it will be subject to gravitational acceleration, which is approximately 9.81 m/s² downward.
Use the kinematic equation for vertical motion to find the maximum height: \( v^2 = u^2 + 2as \), where \( v \) is the final velocity (0 m/s at the peak), \( u \) is the initial velocity (5.00 m/s), \( a \) is the acceleration (-9.81 m/s²), and \( s \) is the displacement from the release point to the peak.
Solve for \( s \) to find the additional height gained after release: \( s = \frac{v^2 - u^2}{2a} \). Substitute \( v = 0 \), \( u = 5.00 \) m/s, and \( a = -9.81 \) m/s².
Add the displacement \( s \) to the initial height of 40.0 m to find the greatest height above the ground that the sandbag reaches.

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Free Fall

Free fall refers to the motion of an object under the influence of gravity alone, with no other forces acting on it. When the sandbag is released from the hot-air balloon, it enters free fall, meaning it will accelerate downward at a rate of approximately 9.81 m/s², the acceleration due to gravity. This concept is crucial for determining the motion of the sandbag after it is released.
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Vertical Motion & Free Fall

Initial Velocity

The initial velocity of an object is the speed and direction at which it begins its motion. In this scenario, the sandbag is released while the balloon is ascending at a constant velocity of 5.00 m/s. This initial upward velocity will affect the maximum height the sandbag reaches after being released, as it will initially continue to move upward before gravity takes over.
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Escape Velocity

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, these equations can be used to calculate the maximum height the sandbag reaches after being released, taking into account its initial upward velocity and the effects of gravity.
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Kinematics Equations
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