Question
upstudy study bank question image url

The velocity of an object moving along the \( x \)-axis is \( v(t)=t^{2}-14 t+45 \). The initial position is the origin. What are the two times the object is at rest? \[ 5 \text { and } 9 \] What is the \( x \)-coordinate at \( t=9 \) ? 81 What is the acceleration at \( t=9 \) ? [? ]

Ask by Schneider Pope. in the United States
Mar 12,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

At \( t=9 \), the acceleration is 4.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

The Deep Dive

To find the acceleration at \( t = 9 \), we first need to calculate the acceleration function, which is the derivative of the velocity function \( v(t) = t^{2} - 14t + 45 \). Taking the derivative, we get \( a(t) = v'(t) = 2t - 14 \). Now, substituting \( t = 9 \) into the acceleration function: \[ a(9) = 2(9) - 14 = 18 - 14 = 4. \] So, the acceleration at \( t = 9 \) is \( 4 \, \text{units/s}^{2} \). This showcases how derivatives are not just for finding slopes but also allow us to sift through the motions of objects, revealing secrets like how fast they're speeding up or slowing down at given moments! Additionally, understanding acceleration helps us explore the concepts of force in the real world. Imagine a roller coaster: as it climbs, the acceleration is negative (deceleration), and as it descends, it’s positive. That thrill comes from the interplay between velocity and acceleration!

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy