A hot-air balloonist, rising vertically with a constant velocity of magnitude 5.005.005.00 m/s, releases a sandbag at an instant when the balloon is 40.040.0 40.0 m above the ground (Fig. E2.442.442.44). 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.

Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 15th Edition

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Young & Freedman Calc 15th Edition

Ch 02: Motion Along a Straight Line

Problem 42a

Chapter 2, Problem 42a

A hot-air balloonist, rising vertically with a constant velocity of magnitude $5.00$ m/s, releases a sandbag at an instant when the balloon is $40.0$ m above the ground (Fig. E $2.44$ ). After the sandbag is released, it is in free fall. Compute the position and velocity of the sandbag at $0.250$ s and $1.00$ s after its release.

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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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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, 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.

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.

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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An egg is thrown nearly vertically upward from a point near the cornice of a tall building. The egg just misses the cornice on the way down and passes a point $30.0$ m below its starting point $5.00$ s after it leaves the thrower's hand. Ignore air resistance. What is the initial speed of the egg?

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Textbook Question

A hot-air balloonist, rising vertically with a constant velocity of magnitude $5.00$ m/s, releases a sandbag at an instant when the balloon is $40.0$ m above the ground (Fig. E $2.44$ ). After the sandbag is released, it is in free fall. What is the greatest height above the ground that the sandbag reaches?

Textbook Question

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