Question

A train \(200 \mathrm{~m}\) long travels at the speed of \(72 \mathrm{~km} / \mathrm{h}\). A man is running at \(3.6 \mathrm{~km} / \mathrm{h}\) in the same direction in which the train is going. The train will pass the man in : (a) \(10 \mathrm{~s}\) (b) \(12 \frac{3}{19} s\) (c) \(10 \frac{10}{10} \mathrm{~s}\) (d) none of these

Short answer

Answer: (d) none of these

Step-by-step solution

Convert the speeds to the same unit

The train's speed is given in km/h and the man's speed is given in km/h as well. To make it easier to calculate, let's convert these speeds to meters per second (m/s). We multiply the speed by (1000 meters/1 km) and (1 hour/3600 seconds) to get the conversion. Train's speed = \(72 \frac{km}{h} \times \frac{1000}{1} \frac{m}{km} \times \frac{1}{3600} \frac{h}{s} = \frac{72000}{3600} \frac{m}{s} = 20 \frac{m}{s}\) Man's speed = \(3.6 \frac{km}{h} \times \frac{1000}{1} \frac{m}{km} \times \frac{1}{3600} \frac{h}{s} = \frac{3600}{3600} \frac{m}{s} = 1 \frac{m}{s}\)

Find the relative speed

Since the train and the man are moving in the same direction, we'll subtract the man's speed from the train's speed to find their relative speed. Relative speed = (Train's speed) - (Man's speed) = \(20 \frac{m}{s} - 1 \frac{m}{s} = 19 \frac{m}{s}\)

Calculate the time taken for the train to pass the man

The train needs to cover its own length to pass the man fully. So, we'll divide the length of the train (200 meters) by the relative speed (19 m/s) to find the time required. Time taken = \(\frac{Length\, of\, the\, train}{Relative\, speed} = \frac{200m}{19\frac{m}{s}} = \frac{200}{19} s\)

Comparing the calculated time to the given options

We found that the time taken for the train to pass the man is \(\frac{200}{19} s\). Now, let's compare this to the given options: (a) \(10s \nRightarrow\) incorrect (b) \(12\frac{3}{19}s \nRightarrow\) incorrect (c) \(10\frac{10}{10}s \nRightarrow\) incorrect Since none of these options match our calculated time \(\frac{200}{19} s\), the correct answer is: (d) none of these

Key concepts

Relative Speed

When two objects are moving along the same path, understanding how fast one is moving in relation to the other is essential in solving problems involving moving vehicles, such as cars, trains, or even runners. This concept is known as relative speed.

To calculate the relative speed of two objects moving in the same direction, you subtract the speed of the slower object from the speed of the faster object. In the case of a train passing a man, if both are moving in the same direction, the relative speed would be the train's speed minus the man's speed. It is a crucial concept not only in textbook exercises but also in real-life situations like determining the time it'll take one vehicle to overtake another one on the highway.

In the example provided, the train and the man are moving in the same direction. Thus, the relative speed is found by subtracting the man's speed from the train's speed. With this approach, you calculate how quickly the train is approaching the man, which is instrumental in finding out how long it will take for the train to completely pass by the man.

Unit Conversion

One of the key skills in solving physics and maths problems, like the train passing problem, is the ability to convert units. Unit conversion is vital because it allows you to bring different measurements into the same unit system, facilitating calculations that lead to the correct solution.

The train speed is initially given in kilometers per hour (km/h), which needs to be converted into meters per second (m/s) to work effectively with the length of the train in meters. To convert km/h to m/s, you multiply the speed by the conversion factor \(\frac{1000\,m}{1\,km}\) and \(\frac{1\,h}{3600\,s}\) because there are 1000 meters in a kilometer and 3600 seconds in an hour. Remembering and accurately applying these conversion factors is essential for quantitative problems across various scientific and engineering disciplines.

Time Distance Calculation

In many quantitative problems, you will need to calculate the time it takes for an object to travel a certain distance. This is commonly known as time distance calculation. It relies on the simple formula: Time = Distance / Speed.

In our scenario, to figure out how long it will take the train to pass the man, you'll need to divide the length of the train (the distance the train needs to cover to pass the man) by the relative speed of the train with respect to the man. It’s important to ensure that the units of distance and speed match to avoid any errors in your calculation. This skill is not only fundamental for solving problems in textbooks but is also applicable in planning trips, estimating delivery times, and many other practical situations.

The train needs to cover a distance equal to its length to pass the man completely. Therefore, by dividing the train's length by their relative speed, the calculation yields the time required for the train to pass the man.

Quantitative Aptitude

The problem presented falls under the broader category of quantitative aptitude, which is a measure of a person's ability to handle numerical data efficiently and accurately. This involves various mathematical skills, such as number theory, algebra, geometry, and data interpretation. In this particular problem, skills like unit conversion and time distance calculation come into play.

Having a strong quantitative aptitude is important not just for solving academic tasks but also for making informed decisions in professional and daily life. Whether you are comparing prices, budgeting, or interpreting statistics, the ability to process numerical data with precision is invaluable. Practice and familiarity with these concepts enhance one's ability to approach complex problems with a systematic and analytical mindset which is a key asset in many career paths including finance, engineering, and technology.