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