Skip to main content
Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 3 - DerivativesProblem 3.4.10e
Chapter 3, Problem 3.4.10e

Lunar projectile motion A rock thrown vertically upward from the surface of the moon at a velocity of 24 m/sec (about 86 km/h) reaches a height of s = 24t − 0.8t² m in t sec.
e. How long is the rock aloft?

Verified step by step guidance
1
To determine how long the rock is aloft, we need to find the time 't' when the rock returns to the surface of the moon. This occurs when the height 's' is zero.
Set the height equation to zero: \( s = 24t - 0.8t^2 = 0 \).
This is a quadratic equation in the form \( at^2 + bt + c = 0 \), where \( a = -0.8 \), \( b = 24 \), and \( c = 0 \).
Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for 't'.
Calculate the discriminant \( b^2 - 4ac \) and then find the two possible values for 't'. The positive value will be the time the rock is aloft.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
Was this helpful?

Key Concepts

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

Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. In this problem, the rock's motion is described by a quadratic equation, which models its vertical displacement over time on the moon, where gravity is weaker than on Earth.
Recommended video:
06:51
Derivatives Applied To Acceleration Example 2

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0. The height function s = 24t − 0.8t² is a quadratic equation representing the rock's height over time. Solving this equation for when s = 0 will determine the time when the rock returns to the moon's surface, indicating how long it is aloft.
Recommended video:
5:02
Solving Logarithmic Equations

Roots of Equations

Finding the roots of an equation involves determining the values of the variable that make the equation true, typically where the function equals zero. For the height equation s = 24t − 0.8t², the roots represent the times when the rock is at ground level. Solving for these roots will reveal the duration the rock remains in the air.
Recommended video:
5:02
Solving Logarithmic Equations
Related Practice
Textbook Question

Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


e. When is it moving fastest (highest speed)? Slowest?


s = 4 - 7t + 6t² - t³, 0 ≤ t ≤ 4

Textbook Question

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


d. ƒ(g(x)), x = 0

Textbook Question

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.


" style="" width="250">


Find the derivatives with respect to x of the following combinations at the given value of x.


f. √f(x), x = 2

Textbook Question

Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


f. When is it farthest from the axis origin?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

Textbook Question

Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


d. When does it speed up and slow down?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

Textbook Question

Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.



d. During what year did home prices drop most rapidly and what is an estimate of this rate?