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Ch 02: Motion Along a Straight Line
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoYoung & Freedman Calc 14th EditionCh 02: Motion Along a Straight LineProblema 44a
Capitolo 2, Problema 44a

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. Compute the position and velocity of the sandbag at 0.2500.250 s and 1.001.00 s after its release.
Diagram showing a balloonist releasing a sandbag at 40.0 m height with velocity 5.00 m/s.

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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.
Use the kinematic equation for position: \( y = y_0 + v_0 t + \frac{1}{2} a t^2 \), where \( y_0 = 40.0 \) m, \( v_0 = 5.00 \) m/s, \( a = -9.81 \) m/s² (acceleration due to gravity), and \( t \) is the time after release.
Calculate the position of the sandbag at \( t = 0.250 \) s using the kinematic equation for position.
Calculate the position of the sandbag at \( t = 1.00 \) s using the same kinematic equation for position.
Use the kinematic equation for velocity: \( v = v_0 + a t \) to find the velocity of the sandbag at \( t = 0.250 \) s and \( t = 1.00 \) s.

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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. In this scenario, once the sandbag is released from the hot-air balloon, it begins to accelerate downward at a rate of approximately 9.81 m/s², the acceleration due to gravity. This means that its velocity will increase as it falls, and its position can be calculated using kinematic equations.
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Vertical Motion & Free Fall

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. For the sandbag, we can use these equations to determine its position and velocity at specific times after release, taking into account its initial upward velocity of 5.00 m/s and the downward acceleration due to gravity.
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Kinematics Equations

Initial Conditions

Initial conditions are the starting parameters of a motion problem, including initial position and velocity. In this case, the sandbag is released from a height of 40.0 m with an initial upward velocity of 5.00 m/s. These conditions are crucial for accurately applying kinematic equations to predict the sandbag's subsequent motion and determine its position and velocity at given time intervals.
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