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A point mass starts moving in a straight line with constant acceleration "a". at a point after the beginning of motion,the acceleration changes sign,

without change in magnitude. determine time T from the beginning of the motion in which the point mass returns to the initial point.



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Equations of motion are:

   s = u t + 1/2  a t²  ,
         u = 0  and at time t1, when the acceleration changes, distance travelled:
     s = 1/2 a t1²
     v at t = t1 =  u + a t  = a t1
Now the acceleration is changed to -a.  Then the particle continues in the same direction until the velocity becomes zero.  Then the particle changes the direction and starts accelerating and passes over the point of start.

     u = a t1      v = 0    acceleration = -a
       v = u + a t
  => 0 = a t1 - a t   =>    t = t1    it takes  t1 more time to stop and reverse direction.

  The distance traveled/displacement in this time:
      s = u t + 1/2 a t²
   => s = a t1 * t1 - 1/2 a  t1² = 1/2 a t1²

The total displacement from the initial point :  1/2 a t1² + 1/2 a t1² = a t1²

   now,  acceleration = -a    u = 0       s = - a t1²   in the negative direction
     s = u t + 1/2  a  t²
   => - a t1² = 0 - 1/2 a t²
   =>   t = √2 t1

The total time T from initial point forward till back to initial point :
          T  =  2 t1 + √2 t1 = (2 + √2) t1

2 4 2
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