Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.9.109b

Chapter 4, Problem 4.9.109b

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
b. Find the position of the object for all relevant times.
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

Verified step by step guidance

Start with the given acceleration equation:

a (t) = - g

, where

g = 9.8 m / s^{2}

. Integrate the acceleration equation to find the velocity function

v (t)

. The integral of

- g

with respect to

t

- g t + C

, where

C

is the constant of integration.

Determine the constant of integration

C

for the velocity function. At the moment the payload is released, the initial velocity is given as

10 m / s

(the upward velocity of the hot-air balloon). Substitute

v (0) = 10

into the velocity equation to solve for

C

Once the velocity function

v (t) = - g t + 10

is determined, integrate it to find the position function

s (t)

. The integral of

- g t + 10

with respect to

t

- \frac{g}{2} t^{2} + 10 t + C

, where

C

is the constant of integration.

Determine the constant of integration

C

for the position function. At the moment the payload is released, the initial position is given as

400 m

. Substitute

s (0) = 400

into the position equation to solve for

C

Combine all the results to write the final position function

s (t)

in terms of

t

. This function will describe the vertical motion of the payload for all relevant times.

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Key Concepts

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

Acceleration due to Gravity

Acceleration due to gravity, denoted as 'g', is the rate at which an object accelerates towards the Earth when in free fall. On Earth, this value is approximately 9.8 m/s². This constant is crucial for understanding the motion of objects under the influence of gravity, as it determines how quickly their velocity changes over time.

Velocity and Position Functions

In calculus, the velocity of an object is the derivative of its position function with respect to time. To find the position of an object at any time, one must integrate the velocity function. In this context, the initial conditions, such as the initial height and the initial velocity, play a significant role in determining the position function.

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 derive the position of the payload after it is released, taking into account its initial upward velocity and the downward acceleration due to gravity.

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