To find the minimum time of travel, we need to determine the values of t and t1 that minimize the total time for the journey.
Let's break down the problem into two parts:
1. Acceleration phase:
During the acceleration phase, the train is accelerating with an acceleration of a1 = 0.100 m/s^2. We need to find the time t1 for this phase.
Using the formula for distance traveled during constant acceleration:
d = v0*t + (1/2)*a*t^2
Where:
d = distance covered (1.36 km or 1360 m)
v0 = initial velocity (0 m/s, as the train starts from rest)
By substituting these values, we can rearrange the formula to find t1:
1360 = 0*t1 + (1/2)*0.100*t1^2
2720 = 0.050*t1^2
t1^2 = 2720/0.050
t1^2 = 54400
t1 = sqrt(54400)
t1 ≈ 233.24 seconds
So, the time spent during the acceleration phase is approximately 233.24 seconds.
2. Deceleration phase:
During the deceleration phase, the train is braking with an acceleration of a2 = -0.410 m/s^2. We need to find the time t2 for this phase.
Using the same formula as above, but this time with the negative acceleration, we get:
1360 = 0*t2 + (1/2)*(-0.410)*t2^2
1360 = (-0.205)*t2^2
t2^2 = 1360/(-0.205)
t2^2 ≈ -6649.76
But since time cannot be negative, this means that the deceleration phase is not needed. The train will reach the destination without applying the brakes.
Therefore, the minimum time of travel, t, is equal to t1, which is approximately 233.24 seconds.