a train starts from rest and accelerates unifotmly, until it has travled 3.7 km and acquired a velocity of 30 m/s. the train then moves at a constant velocity of 30 m/s for 410 s. the train then slows down uniformly at 0.065 m/s^2, until it reaches a halt. what distance does the train travel while it is slowing down
V^2 = Vo^2 + 2ad
a=(V^2-Vo^2)/2d=(900-0)/7400=0.122 m/s^2
d2 = 30m/s * 410s = 12300 m.
d3 = (V^2-Vo^2)/2a
d3 = (0-900)/-0.130 = 6923 m.
To find the distance traveled while the train is slowing down, we need to calculate the time it takes for the train to come to a halt first.
We know that the train initially starts from rest and accelerates uniformly until it reaches a velocity of 30 m/s. We can use the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (30 m/s)
u = initial velocity (0 m/s)
a = acceleration
Since the train starts from rest, the initial velocity (u) is 0 m/s. Rearranging the equation, we get:
a = (v^2 - u^2) / (2s)
Substituting the values, we find:
a = (30^2 - 0^2) / (2 * 3.7 km)
Note that we need to convert the distance from kilometers to meters:
a = (900 - 0) / (2 * 3.7 * 1000)
Next, we calculate the acceleration (a):
a = 900 / 7400
a ≈ 0.122 m/s^2
So, the train accelerates at a rate of approximately 0.122 m/s^2.
Now, we can calculate the time it takes for the train to reach a constant velocity. We can use the formula:
v = u + at
Since the train initially starts from rest, the initial velocity (u) is 0 m/s. Rearranging the equation, we get:
t = (v - u) / a
Substituting the values, we find:
t = (30 - 0) / 0.122
t ≈ 245.9 s
Therefore, the train takes approximately 245.9 seconds to reach a constant velocity of 30 m/s.
Finally, we can calculate the distance traveled while the train is slowing down. We can use the equation of motion:
v^2 = u^2 + 2as
Since the train comes to a halt, the final velocity (v) is 0 m/s. Rearranging the equation, we get:
s = (v^2 - u^2) / (2a)
Substituting the values, we find:
s = (0^2 - 30^2) / (2 * -0.065)
s = (-900) / (-0.13)
s ≈ 6923.1 m
Therefore, the train travels approximately 6923.1 meters while it is slowing down.