This problem asks us to calculate the time taken by a bus to cover a certain distance, given its speed. We are given the distance in kilometers and the speed in meters per second. To solve this, we need to ensure the units are consistent, typically converting the speed to kilometers per hour or the distance to meters and time to seconds. Converting speed to km/hr is a common approach in these types of problems.

Understanding the Problem: Distance, Speed, and Time

The relationship between distance, speed, and time is fundamental in physics and everyday motion problems. The formula connecting them is:

\begin{equation*} \text{Distance} = \text{Speed} \times \text{Time} \end{equation*}

From this formula, we can derive the formula for time:

\begin{equation*} \text{Time} = \frac{\text{Distance}}{\text{Speed}} \end{equation*}

In this question, we are given:

• Distance = 162 km
• Speed = 15 m/s

Before we can use the formula to find the time, we must convert the speed from meters per second (m/s) to kilometers per hour (km/hr) so that the distance and speed units are compatible (kilometers and kilometers per hour).

Converting Speed Units: Meters per Second to Kilometers per Hour

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

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

So, to convert meters to kilometers, we divide by 1000, and to convert seconds to hours, we divide by 3600. When converting m/s to km/hr, we effectively multiply by \(\frac{3600}{1000}\) or 3.6.

Speed in km/hr = Speed in m/s \(\times \frac{3600}{1000}\)

Let's apply this to the given speed:

Speed = 15 m/s

Speed in km/hr = \(15 \times 3.6\) km/hr

Calculating the speed in km/hr:

\(15 \times 3.6 = 54\)

So, the speed of the bus is 54 km/hr.

Calculating the Time Taken for the Journey

Now that we have the distance in kilometers and the speed in kilometers per hour, we can calculate the time taken using the formula:

\begin{equation*} \text{Time} = \frac{\text{Distance}}{\text{Speed}} \end{equation*}

Substitute the values:

Time = \(\frac{162 \text{ km}}{54 \text{ km/hr}}\)

Let's perform the division:

\(\frac{162}{54}\)

We can simplify this fraction or perform the division directly. \(54 \times 3 = 162\). So, \(\frac{162}{54} = 3\).

The unit for time will be hours, as we used distance in km and speed in km/hr.

Time = 3 hours.

Therefore, the time taken by the bus to cover a distance of 162 km at a speed of 15 m/s is 3 hours.

Revision Table: Key Concepts in Distance, Speed, Time

Additional Information: Importance of Consistent Units

It is crucial in any physics or mathematical calculation involving different quantities that their units are consistent. For example, when using the formula Distance = Speed \(\times\) Time, if speed is in km/hr, then time must be in hours to get the distance in kilometers. Using inconsistent units, like speed in m/s and time in hours, without conversion would lead to an incorrect result. Always check and convert units before performing calculations.

Understanding unit conversion factors, such as the conversion between meters and kilometers or seconds and hours, is a fundamental skill required for solving problems involving speed, distance, and time. These conversion factors are derived from basic definitions of the units themselves.