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 4.9.109d

Chapter 4, Problem 4.9.109d

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 .
d. Find the time when the object strikes the ground.
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 by identifying the key components of the problem: the initial height of the payload is 400 m, the initial velocity is 10 m/s (upward), and the acceleration due to gravity is -9.8 m/s² (downward).

Write the position function s(t) for the payload. The general formula for position is s(t) = s₀ + v₀t + (1/2)at², where s₀ is the initial height, v₀ is the initial velocity, and a is the acceleration due to gravity. Substitute s₀ = 400 m, v₀ = 10 m/s, and a = -9.8 m/s².

Set the position function s(t) equal to 0 to find the time when the payload strikes the ground. This represents the moment when the height of the payload is zero.

Solve the quadratic equation obtained in the previous step for t. Use the quadratic formula t = (-b ± √(b² - 4ac)) / 2a, where a, b, and c are the coefficients from the quadratic equation derived from the position function.

Select the positive root of the quadratic equation, as time cannot be negative. This positive root represents the time when the payload strikes the ground.

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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.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. They relate displacement, initial velocity, final velocity, acceleration, and time. In this context, these equations can be used to calculate the time it takes for the payload to hit the ground after being released from the balloon, considering its initial height and the effects of gravity.

Initial Conditions

Initial conditions refer to the starting parameters of a motion problem, such as initial height, initial velocity, and time. In this scenario, the payload is released from a height of 400 m with an initial upward velocity of 10 m/s. These conditions are essential for accurately applying kinematic equations to determine the time of impact with the ground.

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if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.

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if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.

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