How to Determine if a Particle Has Returned to the Origin?

[The_Engineer]

Homework Statement

A point begins at rest at x = 0 and accelerates at 1.09 m/s^2 to the right for 10 s. It then continues at constant velocity of 10.9 m/s for 8 more seconds. In the third phase of its motion, it decelerates at 5 m/s^2 and is observed to be passing again through the origin when the total time of travel equals 28 s. Determine the whether or not the particle has passed returned to the origin.

Homework Equations

The Attempt at a Solution

I am splitting up the motion into the 3 phases mentioned.
a1 = 1.09 ∴ v1 = 1.09t
v2 = 10.9
a3 = -5 ∴ v3 = -5t + c → v3(0) = v2(8) = 10.9 ∴ v3 = -5t + 10.9

Now I will get the displacements by integrating all of the velocity equations over their respective time intervals.
r1 = ∫v1 from 0 to 10 = 54.5
r2 = ∫v2 from 0 to 8 = 87.2
r3 = ∫v3 from 0 to 10 = -141

Adding all of the displacements,
r1 + r2 + r3 = 0.7 therefore the particle is almost at the origin but hasn't passed it again...



I don't know the correct answer but I know I am wrong... Am I going about this correctly?

 

[voko]

You are doing it correctly.

The catch is in determining when the particle passes the origin, and then rounding that to the accuracy of the time measurements given in the problem.

Put simple, how much more time is required to reach the origin?