In the design of a rapid transit system, it is necessary to balance the average speed of a train against the distance between station stops. The more stops there are, the slower the train's average speed. To get an idea of this problem, calculate the time it takes a train to make a 34.0-km trip in two situations. In each case, assume that at each station the train accelerates at a rate of 1.30 m/s2 until it reaches a speed of 94.0 km/h, then stays at this speed until its brakes are applied for arrival at the next station, at which time it decelerates at -2.50 m/s2. Assume also that the train stops at each intermediate station (those not at the ends) for 19 s.
To calculate the time it takes for the train to make a 34.0-km trip in the given situations, we need to consider the acceleration, deceleration, and time spent at each station.
In the first situation, let's assume there are no intermediate stations between the starting and ending stations. In this case, the train will only accelerate and decelerate at the beginning and end of the trip.
Step 1: Convert the given values to appropriate units.
Acceleration: 1.30 m/s²
Maximum Speed: 94.0 km/h converted to m/s = 94.0 km/h × (1000 m/1 km) × (1 h/3600 s) = 26.11 m/s
Deceleration: -2.50 m/s²
Distance: 34.0 km converted to meters = 34.0 km × 1000 m/km = 34000 m
Step 2: Calculate the time taken to accelerate and decelerate.
The time taken to accelerate and decelerate can be calculated using the formula v = u + at, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is the time.
For acceleration:
v = 26.11 m/s, u = 0 m/s, a = 1.30 m/s²
Using v = u + at, we can solve for t:
t = (v - u) / a = (26.11 m/s - 0 m/s) / 1.30 m/s² = 20.08 s
For deceleration:
v = 0 m/s, u = 26.11 m/s, a = -2.50 m/s²
Using v = u + at, we can solve for t:
t = (v - u) / a = (0 m/s - 26.11 m/s) / (-2.50 m/s²) = 10.44 s
Step 3: Calculate the time spent at each station.
In this situation, there are no intermediate stations, so the train does not stop. Therefore, no time is spent at any station.
Step 4: Calculate the total time for the trip.
The total time taken for the trip can be calculated by adding the time taken to accelerate, the time taken to decelerate, and the time spent at each station (which is zero in this case).
Total time = time to accelerate + time to decelerate + time at each station
Total time = 20.08 s + 10.44 s + 0 s = 30.52 s
Therefore, in the first situation, it takes the train approximately 30.52 seconds to make a 34.0-km trip without any intermediate stations.
Now, let's consider the second situation where the train stops at each intermediate station for 19 seconds.
Step 1: Calculate the total time spent at intermediate stations.
The train stops at each intermediate station for 19 seconds. Since there are no specific details given about the number of intermediate stations, we cannot calculate the exact time spent at these stops. However, we can calculate the time spent at one station and then multiply it by the total number of intermediate stations.
Total time spent at each station = 19 s
Total number of intermediate stations = (number of stations) - 2 [subtracting the starting and ending stations]
Total time spent at intermediate stations = Total time spent at each station × Total number of intermediate stations
Step 2: Calculate the total time for the trip.
The total time taken for the trip can be calculated by adding the time taken to accelerate, the time taken to decelerate, and the total time spent at intermediate stations.
Total time = time to accelerate + time to decelerate + total time spent at intermediate stations
Total time = 20.08 s + 10.44 s + Total time spent at intermediate stations
Therefore, the calculation of the total time for the trip will depend on the number of intermediate stations.