A train moves along a straight track towards a road crossing with a speed of 85.0 km/hr. At the moment the front car of the train arrives at the crossing the conductor applies the brakes which decelerates the train at a constant rate. If the train is 6.00 × 10^2 m long, and the end of the last
car passes the crossing with a speed of 14.0 km/hr, how long (in minutes) was the train blocking the road at the crossing? You may ignore the width of the crossing.
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To determine the time the train was blocking the road at the crossing, we need to find the time it takes for the entire length of the train to pass the crossing.
Let's break down the problem and solve it step by step:
Step 1: Convert the given speeds from km/hr to m/s.
- The initial speed of the train is 85.0 km/hr, which is converted to m/s by multiplying by (1000 m / 1 km) and dividing by (3600 s / 1 hr):
Initial speed = (85.0 km/hr) * (1000 m / 1 km) * (1 hr / 3600 s) = 23.6 m/s
- The final speed of the train is 14.0 km/hr, which is converted to m/s using the same approach:
Final speed = (14.0 km/hr) * (1000 m / 1 km) * (1 hr / 3600 s) = 3.89 m/s
Step 2: Find the deceleration (negative acceleration) of the train.
- The deceleration can be found using the equation of motion:
v^2 = u^2 + 2as
where v is the final velocity, u is the initial velocity, a is the acceleration (deceleration), and s is the distance traveled.
Plugging in the values:
(3.89 m/s)^2 = (23.6 m/s)^2 + 2a(600 m)
Solving for a:
a = (3.89 m/s)^2 - (23.6 m/s)^2 / (2 * 600 m) = -1.97 m/s^2 (notice the negative sign indicating deceleration)
Step 3: Find the time taken for the entire length of the train to pass the crossing.
- We can use the formula for the position of an object undergoing constant acceleration (deceleration) to determine the time:
s = ut + (1/2) a t^2
where s is the distance, u is the initial velocity, a is the acceleration, and t is the time.
Plugging in the known values:
600 m = (23.6 m/s)t + (1/2)(-1.97 m/s^2)t^2
Simplifying and solving the quadratic equation for t:
-0.985 t^2 + 23.6 t - 600 = 0
Using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / 2a
where a = -0.985, b = 23.6, and c = -600
Calculating the time using the positive solution:
t = (-23.6 + √((23.6)^2 - 4 * (-0.985) * (-600))) / (2 * (-0.985)) ≈ 15.59 seconds
Step 4: Convert the time to minutes.
- We have the time in seconds, so we need to convert it to minutes by dividing by 60 (60 seconds in a minute):
Time in minutes ≈ 15.59 s / 60 s/min ≈ 0.2598 min
Therefore, the train was blocking the road at the crossing for approximately 0.2598 minutes or about 15.6 seconds.