Problem 82 - Master Resource Book in JEE Main Physics

Question

A passenger arriving in a new town wishes to go from the station to a hotel located \(10 \mathrm{~km}\) away on a straight road from the station. A dishonest cabman takes him along a circuitoius path \(23 \mathrm{~km}\) long and reaches the hotel in \(28 \mathrm{~min}\). What are the average speed of the taxi and the magnitude of average velocity respectively (in \(\mathrm{km} / \mathrm{h}) ?\) (a) \(49.3\) and \(21.43\) (b) \(48.3\) and \(24.43\) (c) 21 and 20 (d) \(21.3\) and \(49.3\)

Step-by-Step Solution

Answer

(a) 49.3 and 21.43

Step 1: Understand the Problem

We are given a problem involving distance, time, and velocity. A passenger travels 23 km with a taxi taking 28 minutes, but the shortest path is 10 km. We need to find both the average speed and the average velocity.

Step 2: Convert Time to Hours

Since average speed and average velocity need to be in \(\text{km/h}\), convert the travel time from minutes into hours: \[ 28 \text{ minutes} = \frac{28}{60} \text{ hours}. \]

Step 3: Calculate the Average Speed

Average speed is the total distance traveled divided by the total time. The distance traveled is 23 km and the time taken is \(\frac{28}{60}\) hours. \[ \text{Average speed} = \frac{23}{\frac{28}{60}} = \frac{23 \times 60}{28} \approx 49.29 \text{ km/h}. \]

Step 4: Calculate the Magnitude of Average Velocity

The magnitude of average velocity is the displacement (the shortest straight line distance) divided by the total time taken. The displacement is 10 km.\[ \text{Average velocity} = \frac{10}{\frac{28}{60}} = \frac{10 \times 60}{28} \approx 21.43 \text{ km/h}. \]

Step 5: Choose the Correct Answer

Based on our calculations: the average speed is approximately 49.3 km/h, and the average velocity is approximately 21.43 km/h. Thus, the correct answer is option (a) \(49.3\) and \(21.43\).

Key Concepts

Distance and Displacement

In everyday language, we often use the terms "distance" and "displacement" interchangeably, but they have distinct meanings in physics. **Distance** refers to the total length of the path traveled, no matter how twisted or curved the route may be. It's a scalar quantity, which means it only has magnitude and no direction. In the context of the taxi problem, the distance traveled by the passenger was 23 km because that is the length of the path the taxi took.

On the other hand, **displacement** is concerned with the change in position. It's a vector quantity, which means it includes both magnitude and direction. Displacement is essentially the shortest straight line from the starting point to the endpoint. For the passenger, the displacement is just 10 km, as it's the direct distance from the station to the hotel, irrespective of the actual path traveled.

Recognizing the differences between distance and displacement is crucial for solving problems related to motion as it guides what information to use for calculating average speed and average velocity.

Time Conversion

Time conversion is a useful skill in many areas, especially in physics where time must often be converted from one unit to another for calculations. In this exercise, the travel time was originally given in minutes. To align with standard units for speed and velocity, typically kilometers per hour (km/h), we need to convert that time to hours.

To convert minutes to hours, you divide the minutes by 60 because there are 60 minutes in an hour. In our example, 28 minutes is converted as follows:

• \( rac{28}{60} = 0.4667 \) hours.
  

This conversion allows us to calculate speed and velocity in km/h, ensuring consistency in the units used throughout the calculations.

Average Speed Formula

Average speed is a straightforward concept but often misunderstood. It calculates how fast something is moving along a path, regardless of its direction. To find average speed, you take the total distance traveled and divide it by the total time taken.

Using the formula:
\[ \text{Average speed} = \frac{\text{Total distance}}{\text{Time}} \]

For the taxi problem, the total distance is the 23 km actually traveled, while the time is the converted 0.4667 hours. Plugging in these values, we get:

• \( \text{Average speed} = \frac{23}{0.4667} \approx 49.29 \) km/h.
  

This calculation shows how fast the taxi was moving on average over the entire path it took.

Average Velocity Formula

Average velocity might sound similar to average speed, but it has a key difference: velocity considers direction. It's calculated using the displacement rather than the total path traveled.

The formula for average velocity is:
\[ \text{Average velocity} = \frac{\text{Displacement}}{\text{Time}} \]

Here, the displacement is the straight-line distance between the start and end points, which is 10 km in this scenario, and the time remains as 0.4667 hours.

• \( \text{Average velocity} = \frac{10}{0.4667} \approx 21.43 \) km/h.
  

Average velocity gives us a measure of how quickly the position changes over time, emphasizing the directness and efficiency of the travelling route.