A train 500 m long is moving on a straight track with a speed of 84.2 km/h. The engineer applies the brakes at a crossing, and later the last car passes the crossing with a speed of 16.4 km/h. Assuming constant acceleration, determine how long the train blocked the crossing.Disregard the width of the crossing.

To determine the time it takes for the train to pass the crossing, we can calculate the time taken for the train to slow down from 84.2 km/h to 16.4 km/h.

First, let's convert the speeds from km/h to m/s for consistent units:
Speed of the train initially (V1) = 84.2 km/h = 84.2 * (1000/3600) = 23.4 m/s
Speed of the train finally (V2) = 16.4 km/h = 16.4 * (1000/3600) = 4.6 m/s

Now, let's use the formula for acceleration:

a = (V2 - V1) / t

Where:
a = acceleration
V1 = initial velocity
V2 = final velocity
t = time taken

Rearrange the formula to solve for t:

t = (V2 - V1) / a

Since the acceleration is constant, we can calculate it using the formula:

a = (V2 - V1) / t

Substituting the values:

t = (4.6 - 23.4) / a

To find the acceleration (a), we can use the formula:

a = (V2 - V1) / t1,

where t1 is the time taken for the train to initially slow down.

The initial velocity of the train (V1) is 23.4 m/s,
the final velocity (V2) is 0 m/s since the train comes to a stop,
and we need to find the time taken (t1).

Using the formula:

a = (V2 - V1) / t1

0 = (0 - 23.4) / t1

Now solving for t1:

t1 = (0 - 23.4) / 0 = undefined (division by zero)

This calculation shows that the train would theoretically take infinite time to come to a complete stop.

Therefore, the train never stops completely. It just slows down and continues moving, indicating that the train never blocked the crossing.

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.

Given:
- Length of the train (L): 500 meters
- Initial speed of the train (u): 84.2 km/h
- Final speed of the last car (v): 16.4 km/h

To find the time it takes to stop the train, we first need to convert the speeds from km/h to m/s:
- Initial speed (u) = 84.2 km/h = (84.2 * 1000) / 3600 = 23.4 m/s
- Final speed (v) = 16.4 km/h = (16.4 * 1000) / 3600 = 4.56 m/s

Now, let's calculate the acceleration:
Using the equation v^2 = u^2 + 2aS, where a is the acceleration and S is the distance:
0 = (23.4)^2 + 2a * 500
Simplifying the equation gives:
0 = 547.56 + 1000a

Now, let's solve for acceleration (a):
1000a = -547.56
a = -0.54756 m/s^2

Since the acceleration is negative, it means the train is decelerating (braking).

Next, we can use the equation v = u + at to find the time (t) it takes for the train to stop:
0 = 23.4 + (-0.54756)t
-23.4 = -0.54756t
t = 42.69 seconds (rounded to two decimal places)

Therefore, the train blocked the crossing for approximately 42.69 seconds.

distance engineer moved = 500 m = .5 km

average speed = (84.2 +16.4)/2 = 50.3
km/hr
(during constant acceleration you can use the average of first and last for the distance and time)

time = distance/speed = .5km / 50.3 km/hr

= .00994 hours
* 3600 = 35.8 seconds