/*! 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 85 Given the following velocity fun... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 85

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. $$v(t)=2 \sqrt{t} ; s(0)=1$$

Short Answer

Expert verified
Short Answer: To find the position function, we integrated the given velocity function, \(v(t)=2\sqrt{t}\), resulting in the position function \(s(t) = \frac{4}{3}t^{\frac{3}{2}} + 1\). To graph both functions, create a graph with time on the x-axis, and position and velocity on the y-axis. Then, plot the functions \(v(t) = 2\sqrt{t}\) and \(s(t) = \frac{4}{3}t^{\frac{3}{2}} + 1\).

Step by step solution

01

Integrate the velocity function

To find the position function, we need to integrate the given velocity function, \(v(t)=2\sqrt{t}\). We can write the square root of t as \(t^{\frac{1}{2}}\). Integrate: $$s(t) = \int v(t) dt = \int 2t^{\frac{1}{2}} dt$$
02

Apply the power rule for integration

Using the power rule for integration, we will add 1 to the exponent and divide by the new exponent: $$s(t) = 2 \cdot \frac{t^{\frac{3}{2}}}{\frac{3}{2}} + C$$ $$s(t) = \frac{4}{3}t^{\frac{3}{2}} + C$$
03

Determine the constant of integration

To find the constant of integration, we need to use the given initial position value, \(s(0) = 1\). Plug \(t=0\) into the position function: $$1 = \frac{4}{3}(0)^{\frac{3}{2}} + C$$ Since \((0)^{\frac{3}{2}} = 0\), this simplifies to: $$1 = C$$
04

Write the position function

Now that we have the constant of integration, we can write the complete position function: $$s(t) = \frac{4}{3}t^{\frac{3}{2}} + 1$$
05

Graph the velocity and position functions

To graph both the velocity and position functions, you can create a graphical representation using various available software like Desmos, GeoGebra or even by hand. Create a graph with time on the x-axis, and position and velocity on the y-axis, then plot the functions \(v(t) = 2 \sqrt{t}\) and \(s(t) = \frac{4}{3}t^{\frac{3}{2}} + 1\). Note: Since we cannot provide an actual graph as output, here are the steps for graphing both 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

Find the function \(F\) that satisfies the following differential equations and initial conditions. $$F^{\prime \prime}(x)=\cos x, F^{\prime}(0)=3, F(\pi)=4$$

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. $$v(t)=e^{-2 t}+4 ; s(0)=2$$

Determine the following indefinite integrals. Check your work by differentiation. $$\int\left(\csc ^{2} \theta+1\right) d \theta$$

Determine the following indefinite integrals. Check your work by differentiation. $$\int(4 \cos 4 w-3 \sin 3 w) d w$$

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

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Probability and Statistics

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

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.