/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 87 Given the following velocity fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 87

Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions. $$v(t)=6 t^{2}+4 t-10 ; s(0)=0$$

Short Answer

Expert verified
**Question:** Based on the given velocity function $$v(t) = 6t^2 + 4t - 10$$ and the initial position $$s(0) = 0$$, find the position function $$s(t)$$ and graph both functions on the same axes. Analyze the graph for important points and describe the motion of the object. **Answer:** After integrating the velocity function and substituting the initial position, we find that the position function of the object is $$s(t) = 2t^3 + 2t^2 - 10t$$. When graphing both functions, observe the important points (intercepts, minima, and maxima) and the intersections between the two graphs. The object's motion can be described based on the behavior of the velocity and position functions, taking into account changes in direction, speed, and acceleration.

Step by step solution

01

Integrate the velocity function

To find the position function, we need to integrate the given velocity function: $$s(t) = \int (6t^2 + 4t - 10) dt$$ Now, integrate each term separately: $$\int 6t^2 dt + \int 4t dt - \int 10 dt = 6\int t^2 dt + 4\int t dt - 10\int dt$$
02

Evaluate the integration

Use the power rule for integration for each term: $$s(t) = 6 \cdot \frac{t^3}{3} + 4 \cdot \frac{t^2}{2} - 10t + C$$ Simplify the expression: $$s(t) = 2t^3 + 2t^2 - 10t + C$$
03

Find the value of the constant C

Given the initial position $$s(0) = 0$$, we substitute t=0 in the position function and solve for C: $$0 = 2(0)^3 + 2(0)^2 - 10(0) + C$$ So, the constant C is 0. Thus, the position function is: $$s(t) = 2t^3 + 2t^2 - 10t$$
04

Graph the velocity and position functions

Plot both the position function $$s(t) = 2t^3 + 2t^2 - 10t$$ and the velocity function $$v(t) = 6t^2 + 4t - 10$$ on the same graph. Remember to label the axes and indicate any intercepts, minima, and maxima. You can use a graphing calculator or software like Desmos or GeoGebra to plot the functions.
05

Analyze the graph

Observe the graph for any important features or points. Note any intersections between the position and velocity functions or critical points such as local extrema and inflection points. Describe the motion of the object moving along the line based on the behavior of the velocity and position functions.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use analytical methods to evaluate the following limits. $$\lim _{n \rightarrow \infty} \frac{1+2+\cdots+n}{n^{2}}( \text {Hint}:$$ $$\left.1+2+\cdots+n=\frac{n(n+1)}{2}.\right)$$

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime \prime}(x)=672 x^{5}+24 x, F^{\prime \prime}(0)=0, F^{\prime}(0)=2, F(0)=1$$

Show that \(f(x)=\log _{a} x\) and \(g(x)=\) \(\log _{b} x,\) where \(a>1\) and \(b>1,\) grow at a comparable rate as \(x \rightarrow \infty\).

Differentials Consider the following functions and express the relationship between a small change in \(x\) and the corresponding change in \(y\) in the form \(d y=f^{\prime}(x) d x\) $$f(x)=(4+x) /(4-x)$$

Given the following velocity functions of an object moving along a line, find the position function with the given initial position. Then graph both the velocity and position functions. $$v(t)=2 \cos t ; s(0)=0$$
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.