Final answer:
The minimum time needed to avoid a collision between two trains undergoing the same deceleration and coming to rest is determined by the train with the higher initial velocity; the time to stop is calculated as the initial velocity of that train divided by the deceleration.
Step-by-step explanation:
To calculate the minimum time needed to avert a collision when both trains apply equal deceleration and must come to rest, we use the kinematic equation for uniformly accelerated motion:
v = u + at
Where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
Since both trains must come to rest (v = 0), and if we consider that they apply brakes simultaneously with the same deceleration:
0 = v1 - at
and
0 = v2 - at
The time taken by each train to stop is:
t1 = v1 / a
and
t2 = v2 / a
Since the accelerations are equal for both trains:
t = t1 = t2
The relative velocity between the two trains is vrel = v1 + v2 since they are moving in opposite directions. The total stopping time is determined by the train with the higher velocity, so:
t = max(v1, v2) / a
Therefore, the minimum time to prevent a collision is the highest value of t obtained from the above expression for each train.