Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.7.102
Capitolo 4, Problema 4.7.102

Applications


Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?

Guida verificata passo dopo passo
1
Identify the known quantities: the acceleration \(a = 20 \ \text{m/s}^2\) and the time \(t = 1 \ \text{minute}\). Convert the time to seconds since acceleration is in meters per second squared. So, \(t = 60 \ \text{seconds}\).
Recall the kinematic equation that relates velocity, acceleration, and time when starting from rest: \(v = v_0 + a t\). Since the rocket starts from rest, the initial velocity \(v_0 = 0\).
Substitute the known values into the equation: \(v = 0 + (20)(60)\).
Simplify the expression to find the velocity after 60 seconds, which will give the speed of the rocket 1 minute after liftoff.
Interpret the result as the rocket's speed in meters per second at 1 minute after liftoff.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Constant Acceleration

Constant acceleration means the velocity of an object changes at a steady rate over time. In this problem, the rocket's acceleration is 20 m/s², which means its speed increases by 20 meters per second every second.
Video consigliato:
Percorso guidato
06:15
Derivatives Applied To Acceleration

Velocity-Time Relationship

When acceleration is constant, velocity changes linearly with time. The velocity after a certain time can be found using the formula v = at, where 'a' is acceleration and 't' is time elapsed.
Video consigliato:
Percorso guidato
06:29
Derivatives Applied To Velocity

Unit Conversion and Time Calculation

Time must be expressed in consistent units when applying formulas. Since acceleration is in meters per second squared and time is given in minutes, converting 1 minute to 60 seconds is essential for accurate calculation.
Video consigliato:
5:10
Evaluate Composite Functions - Values on Unit Circle
Pratica correlata
Domanda del libro di testo

Theory and Examples


Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).

212
views
Domanda del libro di testo

Theory and Examples

67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then

[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd ≥ 16

253
views
Domanda del libro di testo

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = (x² − 3) / (x − 2), x ≠ 2

187
views
Domanda del libro di testo

Checking the Mean Value Theorem


Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.


f(x) = √(x(1 − x)), [0, 1]

188
views
Domanda del libro di testo

93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \(\y\)'' as positive, negative, or zero.

" style="" width="254">

236
views
Domanda del libro di testo

32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.

126
views