Solving Train Speed and Length Problems

This problem involves a train moving at a certain speed and the time it takes to cross a fixed point (a pole). We need to find the length of the train based on this information. When a train crosses a pole, the distance covered by the train is equal to its own length.

We are given the following information:

• Speed of the train = 60 km/h
• Time taken to cross the pole = 1.5 minutes

We need to find the length of the train in meters. To use the formula Distance = Speed × Time, we must ensure that the units are consistent. We should convert the speed from km/h to m/s and the time from minutes to seconds.

Unit Conversion for Speed

The speed is given in kilometers per hour (km/h). We convert this to meters per second (m/s).

1 kilometer = 1000 meters

1 hour = 60 minutes = 60 × 60 seconds = 3600 seconds

So, 60 km/h can be converted as follows:

$$ \text{Speed} = 60 \text{ km/h} = \frac{60 \text{ km}}{1 \text{ hour}} $$

Converting to meters and seconds:

$$ \text{Speed} = \frac{60 \times 1000 \text{ m}}{3600 \text{ s}} $$

$$ \text{Speed} = \frac{60000 \text{ m}}{3600 \text{ s}} = \frac{600}{36} \text{ m/s} $$

Simplifying the fraction:

$$ \text{Speed} = \frac{100}{6} \text{ m/s} = \frac{50}{3} \text{ m/s} $$

The speed of the train is \(\frac{50}{3}\) m/s.

Unit Conversion for Time

The time taken to cross the pole is given in minutes. We convert this to seconds.

Time = 1.5 minutes

1 minute = 60 seconds

$$ \text{Time} = 1.5 \times 60 \text{ seconds} = 90 \text{ seconds} $$

The time taken is 90 seconds.

Calculating the Length of the Train

When a train crosses a pole, the distance covered is equal to the length of the train. We can use the formula:

Distance = Speed × Time

In this case, Distance = Length of the train.

Length of train = Speed × Time

Substituting the values we calculated:

$$ \text{Length of train} = \frac{50}{3} \text{ m/s} \times 90 \text{ s} $$

$$ \text{Length of train} = 50 \times \frac{90}{3} \text{ m} $$

$$ \text{Length of train} = 50 \times 30 \text{ m} $$

$$ \text{Length of train} = 1500 \text{ m} $$

So, the length of the train is 1500 meters.

Summary of Calculation Steps

Using the formula Distance = Speed × Time:

Length of Train = \(\frac{50}{3}\) m/s × 90 s = 1500 m

Revision Table: Train Problems Concepts

Additional Information: Speed, Distance, and Time

The relationship between speed, distance, and time is fundamental in solving problems involving motion.

• Speed: The rate at which an object moves. It is the distance covered per unit of time. Units are typically m/s or km/h.
• Distance: The total path length covered by the object. Units are typically meters or kilometers.
• Time: The duration for which the motion occurs. Units are typically seconds, minutes, or hours.

The core formula connecting these is:

$$ \text{Distance} = \text{Speed} \times \text{Time} $$

From this, we can derive:

$$ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $$

$$ \text{Time} = \frac{\text{Distance}}{\text{Speed}} $$

It is crucial to ensure that all quantities are in consistent units before performing calculations. For example, if speed is in m/s, distance should be in meters and time in seconds. If speed is in km/h, distance should be in kilometers and time in hours.

A common conversion factor is \(1 \text{ km/h} = \frac{5}{18} \text{ m/s}\). This comes from:

$$ 1 \text{ km/h} = \frac{1000 \text{ m}}{3600 \text{ s}} = \frac{10}{36} \text{ m/s} = \frac{5}{18} \text{ m/s} $$

Similarly, \(1 \text{ m/s} = \frac{18}{5} \text{ km/h}\).

Using this factor for the given speed of 60 km/h:

Speed = \(60 \times \frac{5}{18}\) m/s = \(10 \times \frac{5}{3}\) m/s = \(\frac{50}{3}\) m/s, which matches our previous calculation.