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

The average speed of the train, from the time deceleration begins until the last car passes, is

(80.1 + 16.9)/2 = 48.5 m/s.
The train moves 400 m during that time, so it took 0.4 km/48.5 km = 8.247*10^-3 h to leave the crossing. That is 29.7 seconds

To determine how long the train blocked the crossing, we need to calculate the time it took for the train to slow down from 80.1 km/h to 16.9 km/h.

Step 1: Convert the speeds from km/h to m/s.
Speed of the train initially = 80.1 km/h
Speed of the train initially = 80.1 * (1000/3600) m/s
Speed of the train initially = 22.25 m/s

Speed of the train at the crossing = 16.9 km/h
Speed of the train at the crossing = 16.9 * (1000/3600) m/s
Speed of the train at the crossing = 4.6944 m/s

Step 2: Calculate the acceleration.
Using the equation v = u + at where v is final velocity, u is initial velocity, a is acceleration, and t is time, we can rearrange to solve for acceleration.
Final velocity (v) = 4.6944 m/s
Initial velocity (u) = 22.25 m/s
Time taken (t) = unknown
Acceleration (a) = unknown

v = u + at
4.6944 = 22.25 + a * t

Step 3: Calculate the distance covered during the deceleration.
Using the equation s = ut + (1/2)at^2 where s is distance, u is initial velocity, t is time, and a is acceleration, we can rearrange to solve for distance.
Distance (s) = unknown
Initial velocity (u) = 22.25 m/s
Time taken (t) = unknown
Acceleration (a) = unknown

s = ut + (1/2)at^2
400 = 22.25 * t + (1/2) * a * t^2

Step 4: Substitute the equations from Step 2 and Step 3.
4.6944 = 22.25 + a * t (eq. 1)
400 = 22.25 * t + (1/2) * a * t^2 (eq. 2)

Step 5: Solve the system of equations.
Rearrange eq. 1 to express a in terms of t:
a = (4.6944 - 22.25) / t
a = -17.5556 / t

Substitute this value of a into eq. 2:
400 = 22.25 * t + (1/2) * (-17.5556 / t) * t^2
400 = 22.25 * t - 8.7778 * t
400 = 13.4722 * t

t = 400 / 13.4722
t ≈ 29.67 seconds

Therefore, the train blocked the crossing for approximately 29.67 seconds.

To determine how long the train blocked the crossing, we need to calculate the time it takes for the train to come to a complete stop from its initial speed of 80.1 km/h.

Step 1: Convert the speeds from km/h to m/s.
The initial speed of the train is 80.1 km/h, which is (80.1 * 1000) / 3600 = 22.25 m/s.
The final speed of the last car is 16.9 km/h, which is (16.9 * 1000) / 3600 = 4.69 m/s.

Step 2: Calculate the deceleration (negative acceleration) of the train.
Since the train is slowing down, the acceleration is negative. We can use the following formula to calculate the deceleration:

acceleration = (final speed - initial speed) / time

Rearranging the formula gives us:

time = (final speed - initial speed) / acceleration

In this case, the final speed is 0 m/s (since the train comes to a stop) and the initial speed is 22.25 m/s. Therefore, the deceleration can be calculated as:

deceleration = (final speed - initial speed) / time
0 - 22.25 = -22.25 m/s

Step 3: Calculate the time it takes for the train to come to a complete stop.
Using the formula from step 2, we can find the time:

time = (final speed - initial speed) / acceleration
time = (0 - 22.25) / -22.25
time = 1 second

Step 4: Calculate the distance traveled during the deceleration.
To find the distance traveled during the deceleration, we can use the following formula:

distance = initial speed * time + (1/2) * acceleration * (time^2)

In this case, the initial speed is 22.25 m/s, the time is 1 second, and the acceleration is -22.25 m/s^2. Plugging these values into the formula, we get:

distance = 22.25 * 1 + (1/2) * (-22.25) * (1^2)
distance = 22.25 - 11.125
distance = 11.125 meters

Step 5: Calculate the time the train blocked the crossing.
The train's length is given as 400 meters, and during the deceleration, it traveled a distance of 11.125 meters. Therefore, the time the train blocked the crossing can be calculated using the formula:

time_blocked = (total distance - distance traveled during deceleration) / initial speed

Plugging in the values, we get:

time_blocked = (400 - 11.125) / 22.25
time_blocked = 18.022 seconds

Therefore, the train blocked the crossing for approximately 18.022 seconds.