A train travels between stations 1 and 2. The engineer of the train is instructed to start from rest at station 1 and accelerate uniformly between points A and B, then coast with a uniform velocity between points B and C, and finally accelerate uniformly between points C and D until the train stops at station 2. The distances AB, BC, and CD are all equal, and it takes 5.00 min to travel between the two stations. Assume that the uniform accelerations have the same magnitude, even when they are opposite in direction. How much of this 5.00 min period does the train spend b/w points A & B? B & C? C & D?'
To solve this problem, we need to determine the time spent between each pair of points. Let's assume that the distance between each pair of points is represented by x.
1. Time spent between points A and B:
- The train starts from rest at station 1, so its initial velocity is 0.
- The train accelerates uniformly from rest until it reaches point B.
- Let's denote the time spent between A and B as t1.
Using the equation of motion for uniformly accelerated motion:
x = 0.5 * a * t1^2, where a is the acceleration and x is the distance.
2. Time spent between points B and C:
- The train has a constant velocity between points B and C.
- Let's denote the time spent between B and C as t2.
Since the train travels with uniform velocity, we can use the formula:
x = v * t2, where v is the constant velocity and x is the distance.
3. Time spent between points C and D:
- The train decelerates uniformly from its constant velocity at point C until it stops at station 2.
- Let's denote the time spent between C and D as t3.
Using the equation of motion for uniformly accelerated motion:
x = 0.5 * (-a) * t3^2, where -a is the deceleration and x is the distance.
Given that AB = BC = CD = x, we know that the total travel time between stations 1 and 2 is 5 minutes.
Therefore, t1 + t2 + t3 = 5 minutes
Now we can solve for t1, t2, and t3.
1. Solving for t1:
x = 0.5 * a * t1^2
Since AB = BC = CD = x,
t1 = sqrt(2 * x / a)
2. Solving for t2:
x = v * t2
Since AB = BC = CD = x,
t2 = x / v
3. Solving for t3:
x = 0.5 * (-a) * t3^2
Since AB = BC = CD = x,
t3 = sqrt(2 * x / a)
To find the values of t1, t2, and t3, we need the values of x, a, and v. Unfortunately, the problem does not provide those values.
To find out how much time the train spends between points A and B, B and C, and C and D, we need to understand the concept of uniform acceleration and uniform velocity.
First, let's analyze the given information:
- The train starts from rest at station 1, so its initial velocity is 0 m/s.
- It accelerates uniformly between points A and B.
- It then travels with a uniform velocity between points B and C.
- Finally, it accelerates uniformly between points C and D until it stops at station 2.
Since the distances AB, BC, and CD are equal, we can denote each distance as "d." So, the total distance traveled by the train is 2d (from station 1 to station 2).
We are given that it takes 5.00 minutes to travel between the two stations. To use this information, we need to convert it to seconds:
5.00 minutes = 5.00 minutes × 60 seconds/minute = 300 seconds.
Now, let's find the time spent between points A and B. To do this, we need to consider the equations of motion for uniformly accelerated motion.
The equation for displacement in uniform accelerated motion is given by:
s = ut + (1/2)at²,
where:
s = displacement (in this case, the distance traveled, d),
u = initial velocity (in this case, 0 m/s),
t = time (we need to calculate this),
a = acceleration.
Between points A and B, the train is accelerating uniformly. We're told that the accelerations have the same magnitude. Let's denote the common acceleration as "a" (ignoring the fact that it changes direction).
Using the equation of motion and substituting known values, we can solve for t:
d = 0 + (1/2)at²,
2d = at²,
2d/a = t²,
√(2d/a) = t.
We have now found the time spent between points A and B, which is √(2d/a).
To find the time spent between points B and C, we need to use the fact that the train travels with uniform velocity. Since it is coasting, the acceleration is zero and therefore, there is no change in speed. Hence, the time spent between points B and C is equal to the time it takes to travel 2d, which we found to be √(2d/a) in the previous step.
Finally, for the time spent between points C and D, we can once again use the equation of motion for uniformly accelerated motion.
Using the equation of motion and substituting known values, we can solve for t:
d = 0 + (1/2)at²,
2d = at²,
2d/a = t²,
√(2d/a) = t.
The time spent between points C and D is also √(2d/a).
To summarize:
- The time spent between points A and B is √(2d/a).
- The time spent between points B and C is √(2d/a).
- The time spent between points C and D is √(2d/a).
Please note that to find the exact values for these times, we would need specific values for the distance d and the acceleration a.