A train accelerates from rest at a rate of 4m/s^2 for 5 minutes. Then it continues with constant velocity for 10 minutes. Then it slows down to a complete stop while traveling 200 meters.
a. What was the total displacement of the train?
b. What was the average velocity for the entire trip?
c. What was the average speed for the entire trip?
To find the answers to these questions, we need to break down the motion of the train into different parts and use the appropriate equations.
a. To find the total displacement, we need to find the displacements in each part of the journey and then add them up.
1. First, let's find the displacement during the acceleration phase using the equation:
displacement = (initial velocity * time) + (0.5 * acceleration * time^2)
Here, the initial velocity is 0 m/s, the acceleration is 4 m/s^2, and the time is 5 minutes.
Note that we need to convert the time from minutes to seconds by multiplying it by 60.
So, the displacement during the acceleration phase is:
displacement = (0 * 300) + (0.5 * 4 * (300)^2)
2. Next, let's find the displacement during the constant velocity phase.
Since the train is moving with constant velocity, there is no acceleration and the displacement is simply the product of the velocity and time.
The velocity is constant, so the displacement is:
displacement = velocity * time
Here, the velocity is constant and given as the final velocity of the acceleration phase. We have to find it.
3. Finally, let's find the displacement during the deceleration phase.
We are given that the train comes to a complete stop while traveling 200 meters, so the displacement during the deceleration phase is -200 meters.
Note that the negative sign indicates the direction opposite to the initial direction of motion.
After calculating the above displacements, we can find the total displacement by adding them up.
b. To find the average velocity for the entire trip, we need to calculate the total displacement and divide it by the total time taken for the trip.
c. To find the average speed for the entire trip, we need to divide the total distance (which is the sum of the distances traveled in each part) by the total time taken for the trip. Note that speed is the magnitude of velocity and does not have a direction.
Now, let's calculate the answers using the given information.
To find the answers to these questions, we will break down the train's motion into individual segments and calculate the required values for each segment.
Segment 1: Acceleration
Given:
Initial velocity (u) = 0 (since it starts from rest)
Acceleration (a) = 4 m/s^2
Time (t) = 5 minutes = 5 * 60 seconds = 300 seconds
Using the equation v = u + at, we can find the final velocity (v):
v = 0 + (4 * 300)
v = 1200 m/s
To find the displacement (s) during this segment, we can use the equation s = ut + (1/2)at^2:
s = 0 * 300 + (1/2) * 4 * (300^2)
s = 0 + 2 * 90000
s = 180000 meters
Segment 2: Constant velocity
Given:
Time (t) = 10 minutes = 10 * 60 seconds = 600 seconds
Since the train maintains a constant velocity, the displacement (s) during this segment will be zero.
Segment 3: Deceleration
Given:
Initial velocity (u) = 1200 m/s (since the train was traveling at this velocity after the acceleration)
Final velocity (v) = 0 m/s
Displacement (s) = 200 meters
Using the equation v^2 = u^2 + 2as, we can calculate the deceleration (a):
0 = (1200^2) + 2 * a * 200
-1440000 = 400a
a = -3600 m/s^2
To find the time (t) taken to decelerate, we can use the equation v = u + at:
0 = 1200 + (-3600) * t
3600t = 1200
t = 1200 / 3600
t = 1/3 seconds
Now that we have the individual values, we can find the total values.
a. Total Displacement:
The total displacement is the sum of displacements during each segment.
Total displacement = displacement of Segment 1 + displacement of Segment 2 + displacement of Segment 3
Total displacement = 180000 + 0 + 200
Total displacement = 180200 meters
b. Average Velocity:
Average velocity is the total displacement divided by the total time taken.
Total time taken = time taken for Segment 1 + time taken for Segment 2 + time taken for Segment 3
Total time taken = 300 + 600 + (1/3)
Total time taken = 900 + (1/3) seconds
Average velocity = Total displacement / Total time taken
Average velocity = 180200 / (900 + (1/3))
Average velocity = 180200 / 900.333
Average velocity ≈ 200.19 m/s
c. Average Speed:
Average speed is the total distance covered divided by the total time taken.
Total distance covered = displacement of Segment 1 + displacement of Segment 2 + displacement of Segment 3
Total distance covered = 180000 + 0 + 200
Total distance covered = 180200 meters
Average speed = Total distance covered / Total time taken
Average speed = 180200 / (900 + (1/3))
Average speed = 180200 / 900.333
Average speed ≈ 200.19 m/s
Therefore,
a. The total displacement of the train is 180200 meters.
b. The average velocity for the entire trip is approximately 200.19 m/s.
c. The average speed for the entire trip is approximately 200.19 m/s.