A train 400 m long is moving on a straight track with a speed of 80.1 km/h. The engineer applies the brakes at a crossing, and later the last car passes the crossing with a speed of 16.9 km/h. Assuming constant acceleration, determine how long the train blocked the crossing. Disregard the width of the crossing
To determine how long the train blocked the crossing, we can use the equations of motion.
Let's convert the speeds from km/h to m/s:
Initial speed (v0) = 80.1 km/h = (80.1 * 1000) / 3600 m/s = 22.25 m/s
Final speed (v) = 16.9 km/h = (16.9 * 1000) / 3600 m/s = 4.69 m/s
The length of the train is given as 400 m.
We need to find the time (t) it takes for the change in speed to occur.
We can use the second equation of motion:
v^2 = v0^2 + 2aΔd
Where:
v = final velocity (4.69 m/s)
v0 = initial velocity (22.25 m/s)
a = acceleration
Δd = change in distance (400 m)
Rearranging the equation to solve for acceleration (a):
a = (v^2 - v0^2) / (2Δd)
Substituting the given values:
a = (4.69^2 - 22.25^2) / (2 * 400)
Calculating the acceleration:
a = (-453.4526) / 800
a ≈ -0.567 m/s^2
Now, we can find the time it takes for the train to come to a stop:
v = v0 + at
Substituting the known values:
0 = 22.25 + (-0.567)t
Rearranging the equation to solve for time (t):
t = (0 - 22.25) / -0.567
t ≈ 39.19 seconds
Therefore, the train blocked the crossing for approximately 39.19 seconds.