Calculating Train Length from Speed and Time

This question asks us to find the length of a train given its speed and the time it takes to cross a fixed point, specifically an electric pole. When a train crosses a pole, the distance the train covers is equal to its own length.

We are given the following information:

• Train Speed = 60 km/hour
• Time taken to cross the pole = 30 seconds

To find the length of the train, we can use the basic formula relating distance, speed, and time:

\( \text{Distance} = \text{Speed} \times \text{Time} \)

Before we can use this formula, we need to ensure that the units are consistent. The speed is given in kilometers per hour (km/h), and the time is in seconds (s). We need to convert the speed into meters per second (m/s) to get the distance (length of the train) in meters.

Speed Unit Conversion (km/h to m/s)

To convert speed from km/h to m/s, we use the conversion factors:

• 1 kilometer = 1000 meters
• 1 hour = 3600 seconds

So, to convert km/h to m/s, we multiply by \( \frac{1000}{3600} \), which simplifies to \( \frac{5}{18} \).

Let's convert the train's speed:

\( \text{Speed in m/s} = 60 \, \text{km/h} \times \frac{5}{18} \, \frac{\text{m/s}}{\text{km/h}} \)

\( \text{Speed in m/s} = \frac{60 \times 5}{18} \, \text{m/s} \)

\( \text{Speed in m/s} = \frac{300}{18} \, \text{m/s} \)

We can simplify this fraction:

\( \frac{300}{18} = \frac{150}{9} = \frac{50}{3} \, \text{m/s} \)

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

Calculating the Length of the Train

Now that we have the speed in m/s and the time in seconds, we can calculate the distance covered, which is the length of the train.

\( \text{Length of train} = \text{Speed} \times \text{Time} \)

\( \text{Length of train} = \frac{50}{3} \, \text{m/s} \times 30 \, \text{s} \)

\( \text{Length of train} = \frac{50}{3} \times 30 \, \text{meters} \)

\( \text{Length of train} = 50 \times \frac{30}{3} \, \text{meters} \)

\( \text{Length of train} = 50 \times 10 \, \text{meters} \)

\( \text{Length of train} = 500 \, \text{meters} \)

Therefore, the length of the train is 500 meters.

Let's verify this against the given options:

Our calculated length of 500 meters matches Option 3.

Revision Table: Key Concepts for Train Problems

Additional Information: Understanding Speed, Distance, and Time

Speed, distance, and time are fundamental concepts in physics and are often encountered in quantitative aptitude problems involving motion, such as those with trains, cars, or boats.

• Speed: It is the rate at which an object moves. It is defined as the distance covered per unit of time. Common units are meters per second (m/s) or kilometers per hour (km/h).
• Distance: It is the total path length covered by a moving object. Units can be meters (m), kilometers (km), etc.
• Time: It is the duration for which the motion occurs. Units can be seconds (s), minutes (min), hours (h), etc.

The relationship \( \text{Distance} = \text{Speed} \times \text{Time} \) is a cornerstone for solving motion problems. It can be rearranged to find speed or time if the other two quantities are known:

• \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)
• \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)

It is crucial to always use consistent units for speed, distance, and time when using these formulas. If distance is in meters and speed is in m/s, time will be in seconds. If distance is in kilometers and speed is in km/h, time will be in hours.

In problems involving trains or other objects with length, the concept of distance changes depending on what the train is crossing (a point, a platform, another train). Understanding these scenarios is key to correctly applying the distance-speed-time formula.