A commuter train can minimize the time t between two stations by accelerating (a1 = 0.1 m/s^2) for a time t1 then undergoing a negative acceleration (a2 = -0.5 m/s^2) by using his brakes for a time t2. Since the stations are only 1 km apart, the train never reaches its maximum velocity. Find the minimum time of travel t, and the time t1.
The Attempt at a Solution
basically I listed
a1 = 0.1 m/s^2
a2 = - 0.5 m/s^2
Then I integrated the equations into a piecewise equation for velocity and position.
from 0 < t <= t1 v(t) = t/10
from t1 <= t < t2 v(t) = -t/2 + v(t1)
from 0 < t <= t1 x(t) = t^2 / 20
from t1 <= t < t2 x(t) = -t^2 / 4 + (t1)(t) / 10 + t1^2 / 20
Then I set v(t2) = 0
getting t1 = 5t2.
And I set x(t2) = 1000m using the second equation to solve for t1 and t2.
I eventaully got t1 = 129 s. and t = 154.8 s.
I'm not sure about this answer because from the piecewise function for x(t).
The first piece of the function doesn't equal the second piece at t = t1 so I just want to know what I did wrong. I'm pretty sure I integrated correctly.