text: Chapter 12.4, Problem 50E
  
  ### Calculus
  
  10th Edition
  Ron Larson + 1 other
  ISBN: 9781285057095
  
  Chapter
  Section
  
  ### Calculus
  
  10th Edition
  Ron Larson + 1 other
  ISBN: 9781285057095
  Textbook Problem
  1 views
  
  # Motion Along an Involute of a Circle The figure shows a particle moving along a path modeled by r ( t ) = âŒ© cos Ï€ t + Ï€ t sin Ï€ t , sin Ï€ t âˆ’ Ï€ t cos Ï€ t âŒª .The figure also shows the vectors v(t) and a(t) for t = 1 and t = 2 .(a) Find a T and a N at t = 1 and t = 2 .(b) Determine whether the speed of the particle is increasing or decreasing at each of the indicated values of t. Give reasons for your answers.
  
  (a)
  
  To determine
  
  To Calculate: The value of the tangential and normal components of acceleration as, aT and aN at t=1 and t=2 if the position vector is,
  
  r(t)= cosπt+πtsinπt, sinπtπtcosπt.
  
  Explanation
  
  Given:
  
  The provided position vector is, r(t)= âŒ©cosÏ€t+Ï€tsinÏ€t, sinÏ€tâˆ’Ï€tcosÏ€tâŒª.
  
  Formula Used:
  
  The tangential velocity and tangential acceleration are:
  
  v(t)=dr(t)dt and a(t)=dv(t)dt.
  
  Calculation:
  
  The velocity is calculated as shown below,
  
  Now, find the magnitude of the velocity vector to find speed of the particle as shown below,
  
  â€–v(t)â€–= Ï€2t â€¦â€¦ (1)
  
  Now, find the acceleration vector of the particle as calculated below,
  
  Now, find the tangential component of acceleration as shown below,
  
  Since, cos2Ï€+sin2Ï€t=1
  
  The value is found as shown below,
  
  aT=Ï€2
  
  At t=1 and t=2 the tangent vector is aT=Ï€2
  
  (b)
  
  To determine
  Whether the speed of particle is increasing or decreasing at each of the indicated value of t from t=1 to t=2 if the position vector is,
  
  r(t)= cosπt+πtsinπt, sinπtπtcosπt.
  
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