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 4.9.109a

Chapter 4, Problem 4.9.109a

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 .
a. Find the velocity 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

is the acceleration due to gravity, 9.8 m/s². This equation represents the rate of change of velocity with respect to time.

Integrate the acceleration equation to find the velocity function. The integral of

- g

with respect to time

t

- g t + C

, where

C

is the constant of integration.

Determine the constant of integration

C

using the initial velocity of the object. The payload is released from a hot-air balloon that is rising at a rate of 10 m/s, so the initial velocity

v (0)

is 10 m/s. Substitute

t = 0

and

v (0) = 10

into the velocity equation to solve for

C

Substitute the value of

C

back into the velocity equation. The velocity function will now be expressed as

- g t + 10

, where

g

is 9.8 m/s².

Interpret the velocity function. This equation describes the velocity of the payload at any time

t

after it is released. The negative term

- g t

accounts for the deceleration due to gravity, while the constant term

10

represents the initial upward velocity of the payload.

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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 Function

The velocity function describes the speed and direction of an object's motion at any given time. It is derived from the acceleration function by integrating the acceleration with respect to time. In this context, finding the velocity involves integrating the constant acceleration due to gravity, which will yield a linear function representing the object's velocity over time.

Initial Conditions

Initial conditions are the starting values that define the state of a system at the beginning of an observation. In this problem, the initial conditions include the initial height of the payload (400 m) and the initial velocity (10 m/s upward). These conditions are essential for solving the motion equations, as they allow for the accurate determination of the object's position and velocity at any time during its descent.

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