Expert: Alon Mandes - 3/9/2010

Question
A subatomic particle is traveling in a linear accelerator between time 0 nanoseconds and time pi\pi Ù  nanoseconds, i.e., 0 is less than t and t is less than or equal to pi. Its position is given by s(t)=2sint+cos(2t) , where s is in meters.

(a) At what values of t is the particle at rest? Give your answer as a set of values in braces, e.g., {12,45,34,122}. Use exact numbers.
b) When is the particle moving forward? Give your answer in interval notation.
c) When is the particle moving backward? Give your answer in interval notation.
d) Find the total distance traveled between t=0 and t=pi/2

Answer
Hello Lynn,
In this section we will use the same drill as we did before .
a) s'(t)=2cos(t)-2sin(2t). Solving cos(t)=sin(2t) is a little bit problematic, therefore
  we will use the trigonometrical identity : sin(2t)=2sin(t)cos(t). thus, we gain :
  cos(t)=2sin(t)cos(t) --> sin(t)=0.5 --> t1=pi/6 and t2=5*pi/6 . We must not forget also
  that t=pi/2 also gives us s'(t)=0, and here's why :
  s'(pi/2)=2cos(pi/2)-2sin(2*pi/2)=2*0-2*sin(pi)=0-0=0.
  Thus t = { pi/6 , pi/2 , 5*pi/6 } ("IF the interval of time is 0<t<pi or 0<t<2*pi. Its not
  clear what you wrote !").
b) The particle moving forward when : t=(0,pi/6) or t=(pi/2,5*pi/6)
c) The particle moving backward when : t=(pi/6,pi/2) or t=(5*pi/6,pi)
d) to find total distance is D=s(pi/2)-s(0)=2sin(pi/2)+cos(pi)-2sin(0)-cos(0)=2-1-0-1=0 .

Alon.