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2010 FREE RESPONSE QUESTION 6 Velocity and Acceleration

2010 FREE RESPONSE QUESTION 6 Velocity and Acceleration. Prepared by: Etjen Vincani. Two particles move along the x-axis, for 0<t<6, the position of Particle P at time t is given by P(t)=2cos(Pi/4t) while the position of Particle R at time t is given by R(t)=t^3-6t^2+9t+3.

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2010 FREE RESPONSE QUESTION 6 Velocity and Acceleration

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  1. 2010 FREE RESPONSE QUESTION 6Velocity and Acceleration Prepared by: Etjen Vincani

  2. Two particles move along the x-axis, for 0<t<6, the position of Particle P at time t is given by P(t)=2cos(Pi/4t) while the position of Particle R at time t is given by R(t)=t^3-6t^2+9t+3 A) For 0<t<6, find all times t to which particle R is moving to the Right B) For 0<t<6, find all times t during which the particles travel in opposite directions C) Find the acceleration of particle P at time t = 3. Is particle P speeding up, slowing down, or doing neither at time t = 3 ? Explain your reasoning. D) Write, but do not evaluate, an expression for the average distance between the two particles on the interval 1<t<3

  3. How do you determine Velocity Velocity = the first derivative of the position function HOW DO YOU DETERMINE ACCELERATION? Acceleration is the derivative of Velocity or the second derivative of Acceleration

  4. A) For 0<t<6, find all times t to which particle R is moving to the Right Particle R will move to the right when r´(t) is greater then 0. Thus the particle will be moving to the right when 3t2 -12t+9 >0. If you factor this you get (t-3)(t-1) >0 Thus the Particle will be moving to the right in the interval 0≤t<1 and 3<t≤6 WARNING PLEASE MAKE SURE NEVER TO---

  5. B) For 0<t<6, find all times t during which the particles travel in opposite directions • The particles are traveling in opposite directions when the derivatives of p(t) and r(t) have opposite signs or when the product of the two derivatives is less then 0. • P’(t) R’(t)= -π/2· 3(t-1)(t-3) sin(πt/4)<0 • (t-1)(t-3) sin(πt/4)<0 • T=1 and T=3 • Thus the particles move in opposite directions when 0<t<1 and when 3<t<4 • Or you can simply graph the derivatives as Shown and analyze the graph

  6. C) Find the acceleration of particle P at time t = 3. Is particle P speeding up, slowing down, or doing neither at time t = 3 ? Explain your reasoning. Since P“(t)= -π2/4cos •(π/4t) the acceleration at t=3 will be P“(3) = -π2/4 •(-√2/2) >0 Thus because the acceleration is positive the velocity is increasing when t=3 In addition using your calculator you can use the derivative feature as shown to get your result which is 1.37, a positive number which proves the velocity is increasing.

  7. D) Write, but do not evaluate, an expression for the average distance between the two particles on the interval 1<t<3 • The distance between the particles is |p(t)-r(t)| so the average distance on the interval 1<t<3 is 1/2∫ |p(t)-r(t)| dt

  8. THE END CITATIONS http://home.roadrunner.com/~askmrcalculus/index.html http://www.slideshare.net/canadagirl28/meghans-calculus-powerpoint

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