A subway accelerate gradually (a= 1.2 m/s/s) from rest at one station until it is halfway to the next station and then decelerates gradually (a=-1.2 m/s/s) for the final half of the trip until it comes to rest. The two stations 1100 m apart.
A) what is the maximum speed the train reaches?
B) how long does it take to travel between the stations?
1100.2 = 550
550 = (1/2) a t^2
550 = .5 * 1.2 * t^2
t = 30.3 seconds to go halfway (same for the second half)
v = a t = 1.2 * 30.3 = 36.3 m/s
2 t = 60.6 seconds
To find the answers to these questions, we can use the equations of motion. Let's break down the problem into two parts: the acceleration phase and the deceleration phase.
Acceleration Phase:
1) We need to find the time it takes for the train to reach halfway between the two stations. The formula to find time is given by: t = (v - u) / a, where v is the final velocity, u is the initial velocity, t is the time, and a is the acceleration.
Given:
Initial velocity (u) = 0 (as the train starts from rest)
Acceleration (a) = 1.2 m/s^2 (positive because it is accelerating)
Distance (s) = 1100 m / 2 = 550 m (since the train travels halfway between the two stations)
Plugging in these values into the equation, we get:
t = (v - 0) / 1.2
550 = (v - 0) / 1.2
Simplifying:
550 * 1.2 = v
v = 660 m/s
So, the maximum speed the train reaches during the acceleration phase is 660 m/s.
Deceleration Phase:
2) Now, we need to find the time it takes for the train to come to rest after reaching halfway between the two stations. Here, the acceleration is negative (-1.2 m/s^2) because the train is decelerating.
Given:
Initial velocity (u) = 660 m/s (since the train is at maximum speed now)
Acceleration (a) = -1.2 m/s^2 (negative because it is decelerating)
Distance (s) = 550 m (since the remaining distance is the same as the earlier half)
Using the same formula as before, we get:
t = (v - u) / a
0 = (v - 660) / -1.2
0 = -1.2v + 792
Simplifying:
1.2v = 792
v = 660 m/s
The negative sign is ignored because we are interested in the magnitude of the velocity.
Now, we know that the train's maximum speed during the deceleration phase is also 660 m/s.
To find the total time taken to travel between the stations, we sum up the times taken during the acceleration and deceleration phases.
Total time taken = time during acceleration phase + time during deceleration phase
Total time taken = (550 / 1.2) + (550 / 1.2)
Total time taken = 458.33 + 458.33
Total time taken = 916.66 seconds
Therefore, it takes approximately 916.66 seconds to travel between the two stations.