there are two train stations 4.2 km apart on a train track. the train must stop at both stations. the acceleration of the train while speeding up or slowing down is 0.35 m/2^2 in magnitude and the maximum speed of the train is 28 m/s. calculate the time for the train to travel between stations.
To calculate the time for the train to travel between the two stations, we need to break down the motion into three phases: speeding up, traveling at a constant speed, and slowing down.
Phase 1: Speeding up
Let's denote the time taken to accelerate from rest to the maximum speed as t1.
The distance covered during this phase can be calculated using the equation:
distance = (speed)^2 / (2 * acceleration)
Here, the speed is initially zero and the acceleration is 0.35 m/s^2 (as given).
distance = (28 m/s)^2 / (2 * 0.35 m/s^2) = 1120 m
Phase 2: Traveling at a constant speed
The distance remaining to be covered between the two stations is 4.2 km (or 4200 m). Since the train is traveling at a constant speed of 28 m/s, the time taken for this phase can be calculated using:
time = distance / speed
time = 4200 m / 28 m/s = 150 seconds
Phase 3: Slowing down
Similar to Phase 1, the time taken to decelerate from the maximum speed to rest can be denoted as t3.
Using the same equation as before:
distance = (speed)^2 / (2 * acceleration)
distance = (28 m/s)^2 / (2 * 0.35 m/s^2) = 1120 m
Total time:
Now, we can calculate the total time taken as the sum of the times taken in each of the three phases:
total time = t1 + 150 + t3
Since the train must stop at both stations, it must accelerate and decelerate the same amount of time. Therefore, t1 = t3.
total time = 2 * t1 + 150
To calculate the value of t1 (or t3), we can use the equation:
distance = (initial speed * time) + (0.5 * acceleration * t^2)
For phase 1:
1120 m = (0 * t1) + (0.5 * 0.35 m/s^2 * t1^2)
1120 = 0.175 * t1^2
t1^2 = 6400
t1 = 80 seconds
Substituting this value back into the total time equation:
total time = 2 * 80 sec + 150 sec
total time = 160 sec + 150 sec
total time = 310 seconds
Therefore, the time for the train to travel between the two stations is 310 seconds.