A passenger train is traveling at 27 m/s when the engineer sees a freight train 345 m ahead of his train traveling in the same direction on the same track. The freight train is moving at a speed of 5.9 m/s.

(a) If the reaction time of the engineer is 0.36 s, what is the minimum (constant) rate at which the passenger train must lose speed if a collision is to be avoided? (m/s/s)
(b) If the engineer's reaction time is 0.73 s and the train loses speed at the minimum rate described in Part (a), at what rate is the passenger train approaching the freight train when the two collide? (m/s)
(c) For both reaction times, how far will the passenger train have traveled in the time between the sighting of the freight train and the collision? (km)

a. Dp + Df = 345 m.

27*(T-0.36) + 5.9T = 345.
27T-9.72 + 5.9T = 345.
32.9T = 354.72.
T = 10.78 s. = Time they meet.

V = Vo + a*T = 0.
a = -Vo/T = -27/10.78 = -2.50 m/s^2. =
Min. rate.

To solve this problem, we need to use the equations of linear motion to analyze the situation.

(a) To determine the minimum constant rate at which the passenger train must lose speed to avoid a collision, we need to calculate the distance the passenger train covers during the engineer's reaction time and compare it to the distance between the two trains.

First, we calculate how far the passenger train travels during the engineer's reaction time:

Distance traveled by passenger train = Speed of passenger train × Reaction time
= 27 m/s × 0.36 s
= 9.72 m

Next, we calculate the distance between the two trains:

Distance between the trains = Distance of freight train - Length of passenger train
= 345 m - 0 m (since the passenger train doesn't have any length in this context)
= 345 m

To avoid a collision, the distance traveled by the passenger train during the reaction time should be greater than or equal to the distance between the trains:

9.72 m ≥ 345 m - 0 m

Now, we can calculate the minimum rate at which the passenger train must lose speed by rearranging the equation:

Rate of speed reduction = (Distance between the trains - Distance traveled during reaction time) / Reaction time
= (345 m - 9.72 m) / 0.36 s
= 957 m/s²

Therefore, the minimum rate at which the passenger train must lose speed to avoid a collision is 957 m/s².

(b) Since the passenger train loses speed at the minimum rate found in part (a), we can calculate the relative velocity of the two trains at the time of collision.

Relative velocity = Speed of the passenger train - Speed of the freight train
= 27 m/s - 5.9 m/s
= 21.1 m/s

Therefore, if the engineer's reaction time is 0.73 s and the passenger train loses speed at the minimum rate calculated in part (a), the passenger train will collide with the freight train at a relative velocity of 21.1 m/s.

(c) To find the distance the passenger train will have traveled between the sighting of the freight train and the collision for both reaction times, we need to calculate the time it takes for the collision to occur.

The time it takes for the collision to occur is the reaction time plus the time it takes for the passenger train to cover the remaining distance between the two trains.

For the reaction time of 0.36 s:

Time to collision = Reaction time + (Distance between the trains - Distance traveled during reaction time) / Relative velocity
= 0.36 s + (345 m - 9.72 m) / 21.1 m/s
= 15.713 s

For the reaction time of 0.73 s:

Time to collision = Reaction time + (Distance between the trains - Distance traveled during reaction time) / Relative velocity
= 0.73 s + (345 m - 9.72 m) / 21.1 m/s
= 15.993 s

Finally, to find the distance traveled by the passenger train for both reaction times, we multiply the time by the speed of the passenger train:

Distance traveled = Speed of passenger train × Time to collision

For the reaction time of 0.36 s:

Distance traveled = 27 m/s × 15.713 s
= 423.651 km

For the reaction time of 0.73 s:

Distance traveled = 27 m/s × 15.993 s
= 431.631 km

Therefore, the passenger train will have traveled approximately 423.651 km for a reaction time of 0.36 s, and 431.631 km for a reaction time of 0.73 s, between the sighting of the freight train and the collision.