A passenger train is traveling at 30 m/s when the engineer sees a freight train 357 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 0.43 s, what is the minimum (constant) rate at which the passenger train must lose speed if a collision is to be avoided?
m/s2
(b) If the engineer's reaction time is 0.87 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
23322
To solve this problem, we need to consider the distance between the two trains, the relative speed between them, and the reaction time of the engineer.
Let's tackle the problem step by step:
(a) First, let's calculate the relative speed between the passenger train and the freight train. Since they are traveling in the same direction, we subtract their velocities:
Relative speed = Passenger train's speed - Freight train's speed
Relative speed = 30 m/s - 5.9 m/s
Relative speed = 24.1 m/s
Now, we need to calculate how long it takes for the engineer to react. Given that the reaction time is 0.43 s, we can use the formula:
Reaction distance = Relative speed * Reaction time
Reaction distance = 24.1 m/s * 0.43 s
Reaction distance = 10.363 m
To avoid a collision, the passenger train must reduce its speed by at least the same amount as the reaction distance within the available distance.
Distance available = Distance between the trains - Reaction distance
Distance available = 357 m - 10.363 m
Distance available = 346.637 m
The passenger train must lose speed over this available distance, so we can calculate the required acceleration:
Minimum acceleration = (0 m/s - 30 m/s) / (2 * Distance available)
Minimum acceleration = -30 m/s / (2 * 346.637 m)
Minimum acceleration = -0.04332 m/s^2 (rounded to five decimal places)
Therefore, the minimum constant rate at which the passenger train must lose speed to avoid a collision is approximately -0.04332 m/s^2.
(b) Now, let's calculate the approaching rate when the two trains collide. Given that the reaction time is 0.87 s, we can use the same formula as before to calculate the reaction distance:
Reaction distance = Relative speed * Reaction time
Reaction distance = 24.1 m/s * 0.87 s
Reaction distance = 20.967 m
To find out the rate at which the passenger train is approaching the freight train, we need to subtract the distance available (when the passenger train starts reducing speed) from the distance the freight train travels during the reaction time:
Distance traveled by the freight train = Freight train's speed * Reaction time
Distance traveled by the freight train = 5.9 m/s * 0.87 s
Distance traveled by the freight train = 5.133 m
Rate of approach = Distance traveled by the freight train - Distance available
Rate of approach = 5.133 m - 346.637 m
Rate of approach = -341.504 m
Therefore, if the passenger train loses speed at the minimum rate to avoid collision, the two trains will collide at an approaching rate of approximately -341.504 m/s.
(c) To find how far the passenger train will have traveled between the sighting of the freight train and the collision, we need to calculate the total time between these two events.
Total time = Reaction time + Time to collide
Given that the reaction time is 0.43 s and that we want to find the time to collide, we can use the equation:
Distance traveled by the passenger train = Passenger train's speed * Total time
We can rearrange this equation to find the time to collide:
Time to collide = Distance traveled by the passenger train / Passenger train's speed
Distance traveled by the passenger train = Distance between the trains - Distance available
Distance traveled by the passenger train = 357 m - 346.637 m
Distance traveled by the passenger train = 10.363 m
Time to collide = 10.363 m / 30 m/s
Time to collide = 0.345 s
For both reaction times, the time between the sighting of the freight train and the collision is 0.345 seconds.
Now, to find how far the passenger train will have traveled, we multiply its speed by the total time:
Distance traveled by the passenger train = Passenger train's speed * Total time
Distance traveled by the passenger train = 30 m/s * 0.345 s
Distance traveled by the passenger train = 10.35 m
Therefore, for both reaction times, the passenger train will have traveled approximately 10.35 meters between the sighting of the freight train and the collision.
Note: The given distance traveled by the passenger train in part (c) seems very short, so please double-check the problem statement and ensure accurate calculations.