Calculating Train Distance Covered

The question asks us to find the distance a train covers in a specific amount of time, given its speed. This is a classic problem involving speed, distance, and time.

Understanding Speed, Distance, and Time

The relationship between speed, distance, and time is fundamental in physics and everyday life. The formula connecting these three quantities is:

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

In this problem, we are given the speed of the train and the time it travels. We need to calculate the distance.

Units Consistency

It is very important to ensure that the units of speed and time are consistent. The speed is given in kilometers per hour (km/hr), and the time is given in minutes. To use the formula correctly, we need to convert the time into hours.

• Given Speed = 72 km/hr
• Given Time = 15 minutes

To convert minutes to hours, we divide the number of minutes by 60 (since there are 60 minutes in an hour).

\(\text{Time in hours} = \frac{\text{Time in minutes}}{60}\)

\(\text{Time in hours} = \frac{15}{60} \text{ hours}\)

\(\text{Time in hours} = \frac{1}{4} \text{ hours}\) or \(0.25 \text{ hours}\)

Calculating the Distance

Now that we have the speed in km/hr and the time in hours, we can use the distance formula:

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

Substitute the values:

\(\text{Distance} = 72 \text{ km/hr} \times \frac{1}{4} \text{ hr}\)

\(\text{Distance} = \frac{72}{4} \text{ km}\)

\(\text{Distance} = 18 \text{ km}\)

So, the train covers a distance of 18 kilometers in 15 minutes.

Verifying the Answer

The calculated distance is 18 km, which matches one of the given options.

Let's quickly summarize the steps:

• Identify the given speed and time.
• Ensure units are consistent (convert time from minutes to hours).
• Use the formula: Distance = Speed × Time.
• Perform the calculation.

Detailed Calculation Steps

Step 1: Convert Time to Hours

Time given is 15 minutes. There are 60 minutes in 1 hour.

\(\text{Time in hours} = 15 \text{ minutes} \times \frac{1 \text{ hour}}{60 \text{ minutes}} = \frac{15}{60} \text{ hours} = \frac{1}{4} \text{ hours}\)

Step 2: Use the Distance Formula

Speed = 72 km/hr

Time = \(\frac{1}{4}\) hours

Distance = Speed \(\times\) Time

Distance = \(72 \text{ km/hr} \times \frac{1}{4} \text{ hr}\)

Distance = \(\frac{72}{4}\) km

Distance = \(18\) km

The distance covered by the train in 15 minutes is 18 km.

Revision Table: Speed, Distance, Time

Remember to always check and convert units so they are consistent before performing calculations.

Additional Information: Unit Conversions for Speed Problems

Handling units correctly is crucial in speed, distance, and time problems. Here are some common conversions:

• Time Conversion:
  • 1 hour = 60 minutes
  • 1 minute = 60 seconds
  • 1 hour = 3600 seconds
• Speed Conversion (Example: km/hr to m/s):

  To convert km/hr to m/s, you can use the conversion factor:

  \(1 \text{ km/hr} = 1 \times \frac{1000 \text{ meters}}{3600 \text{ seconds}} = \frac{1000}{3600} \text{ m/s} = \frac{5}{18} \text{ m/s}\)

  So, to convert speed from km/hr to m/s, multiply by \(\frac{5}{18}\).

• Speed Conversion (Example: m/s to km/hr):

  To convert m/s to km/hr, you multiply by the reciprocal of \(\frac{5}{18}\), which is \(\frac{18}{5}\).

  \(1 \text{ m/s} = 1 \times \frac{18}{5} \text{ km/hr}\)

These conversions are very useful when dealing with problems where units are mixed, such as speed in km/hr and time in seconds, or distance in meters and time in hours.