To determine the time it takes for the two trains to pass each other, we need to consider their initial speeds, lengths, and the effect of their accelerations. Let's break this down step by step.
Understanding the Problem
We have two trains:
- Train A: Length = 90 m, Speed = 60 km/h, Acceleration = 0.3 m/s²
- Train B: Length = 90 m, Speed = 40 km/h, Acceleration = 0.15 m/s²
Since they are moving in opposite directions, we can add their speeds to find the relative speed when they pass each other.
Converting Speeds
First, let's convert the speeds from km/h to m/s:
- Speed of Train A: 60 km/h = (60 * 1000 m) / (3600 s) = 16.67 m/s
- Speed of Train B: 40 km/h = (40 * 1000 m) / (3600 s) = 11.11 m/s
Calculating Initial Relative Speed
The initial relative speed of the two trains is:
Relative Speed = Speed of Train A + Speed of Train B
Relative Speed = 16.67 m/s + 11.11 m/s = 27.78 m/s
Considering Acceleration
Next, we need to account for the accelerations of both trains. The distance they need to cover to completely pass each other is the sum of their lengths:
Total Distance = Length of Train A + Length of Train B
Total Distance = 90 m + 90 m = 180 m
Using Kinematic Equations
Since both trains are accelerating, we can use the kinematic equation for distance:
d = ut + (1/2)at²
Where:
- d = distance
- u = initial speed
- a = acceleration
- t = time
Setting Up the Equation
For Train A:
d_A = 16.67t + (1/2)(0.3)t²
For Train B:
d_B = 11.11t + (1/2)(0.15)t²
The total distance covered when they pass each other is:
180 = d_A + d_B
Substituting the distances:
180 = (16.67t + 11.11t) + (0.15t² + 0.15t²)
Combining terms gives:
180 = 27.78t + 0.225t²
Rearranging the Equation
This can be rearranged into a standard quadratic form:
0.225t² + 27.78t - 180 = 0
Solving the Quadratic Equation
Using the quadratic formula, t = [ -b ± √(b² - 4ac) ] / 2a, where:
- a = 0.225
- b = 27.78
- c = -180
Calculating the discriminant:
b² - 4ac = (27.78)² - 4(0.225)(-180)
Calculating gives:
Discriminant = 771.0884 + 162 = 933.0884
Now, applying the quadratic formula:
t = [ -27.78 ± √933.0884 ] / (2 * 0.225)
Calculating the square root and solving for t gives us two potential solutions, but we only consider the positive value since time cannot be negative.
Final Calculation
After performing the calculations, we find:
t ≈ 6.17 seconds
Conclusion
Thus, the time taken for the two trains to completely pass each other is approximately 6.17 seconds. Therefore, the correct answer is option d) 6.17s.
