Question

Train \(A\) is \(56 \mathrm{~m}\) long and train \(\mathrm{B} 54 \mathrm{~m}\) long. They are travelling in opposite direction with velocity \(15(\mathrm{~m} / \mathrm{s})\) and \(5(\mathrm{~m} / \mathrm{s})\) respectively. The time of crossing is. (A) \(12 \mathrm{~s}\) (B) \(6 \mathrm{~s}\) (C) \(3 \mathrm{~s}\) (D) \(18 \mathrm{~s}\)

Short answer

(B) \(6\, s\)

Step-by-step solution

Calculate relative velocities

To determine the relative velocities of the two trains, we sum their individual velocities because they are traveling in opposite directions: \[\text{Relative velocity} = \text{Train A's velocity} + \text{Train B's velocity}\] \[\text{Relative velocity} = 15\, m/s + 5\, m/s = 20\, m/s\]

Calculate the combined length of the trains

Next, we find the sum of the lengths of both trains: \[\text{Combined length} = \text{Train A's length} + \text{Train B's length}\] \[\text{Combined length} = 56\, m + 54\, m = 110\, m\]

Calculate the time needed to cross each other

We now have the relative velocity and combined length of both trains. We can calculate the time it takes for the trains to cross entirely by dividing the combined length by the relative velocity: \[\text{Time of crossing} = \frac{\text{Combined length}}{\text{Relative velocity}}\] \[\text{Time of crossing} = \frac{110\, m}{20\, m/s} = 5.5\, s\] Since the given options are (A) 12 s, (B) 6 s, (C) 3 s and (D) 18 s, the closest option to our answer of 5.5 s is:

Answer

(B) \(6\, s\)

Key concepts

Problems on Trains

When working through train problems, a common scenario you'll encounter involves calculating how long it takes for two trains to cross each other. These problems test your understanding of relative velocity, train length, and other kinematics concepts.
Trains usually travel on designated tracks, and their lengths can range greatly. In our example, we have Train A, which is 56 meters long, and Train B, which is 54 meters long. These problems often provide you with the speeds of the trains, requiring you to determine the time it takes for them to completely pass each other.
This kind of problem is particularly interesting because you need to account for the direction each train is traveling. If the trains are moving towards each other, their velocities add up, making the crossing time shorter.

Kinematics Problems

Kinematics is a branch of physics that deals with the motion of objects. In these types of exercises, key concepts include displacement, speed, velocity, and time. When solving kinematics problems involving trains, you'll use these elements to solve for unknowns.

• Velocity: The speed of something in a given direction.
• Displacement: The object’s overall change in position.
• Time: The duration over which the object is in motion.

In our train example, you calculate the relative velocity, which is the sum of both train's speeds because they move in opposite directions. This relative velocity helps you determine how quickly the trains pass each other when combined with their total lengths.
Kinematics problems are foundational in physics, offering real-world applications in areas like transportation design, safety, and engineering.

Time of Crossing

The 'time of crossing' refers to the time it takes for two objects traveling towards each other to completely pass by one another. To calculate this, you need to know both the relative velocity and the combined length of the objects involved.
In our specific train problem, the time of crossing is calculated using the formula:
\[\text{Time of crossing} = \frac{\text{Combined length of trains}}{\text{Relative velocity}}\]
This equation tells us how long it takes for the entire length of one train to pass the entire length of another. In practice, this requires careful handling of units (e.g., making sure distances are in meters and speeds are in meters per second) to ensure the result is in seconds.
Understanding this concept is critical for analyzing many real-world scenarios, from scheduling train departure times to ensuring safety measures are in place at railway crossings.