Calculating Average Velocity for a Round Trip

The question asks for the magnitude of the average velocity of a car that starts from Bengaluru, travels 50 km towards south, and immediately returns to Bengaluru. The total time taken for this entire round trip is given as 2 hours.

To solve this problem, we need to understand the definition of average velocity and the concept of displacement.

Understanding Average Velocity and Displacement

• Average Velocity: Average velocity is defined as the total displacement divided by the total time taken for the motion. Mathematically, it is given by:

  Concept	Formula
  Average Velocity	\( \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \)

  
  
• Displacement: Displacement is a vector quantity that represents the shortest distance between the initial position and the final position of an object. It also includes the direction of motion.
• Distance: Distance is a scalar quantity that represents the total length of the path traveled by an object.

In this problem, the car starts from Bengaluru (initial position) and returns to Bengaluru (final position).

Step-by-Step Calculation

Step 1: Determine the Initial and Final Positions

The car starts at Bengaluru. Let's consider Bengaluru as the starting point.

The car travels 50 km south and then returns to Bengaluru. So, the final position of the car is also Bengaluru.

Step 2: Calculate the Total Displacement

Displacement is the change in position. Since the initial position and the final position are the same (both are Bengaluru), the total displacement of the car for the round trip is zero.

\( \text{Total Displacement} = \text{Final Position} - \text{Initial Position} \)

\( \text{Total Displacement} = \text{Bengaluru} - \text{Bengaluru} = 0 \text{ km} \)

Step 3: Note the Total Time Taken

The question states that the total time taken for the round trip is 2 hours.

\( \text{Total Time} = 2 \text{ hours} \)

Step 4: Calculate the Average Velocity

Now, we use the formula for average velocity:

\( \text{Average Velocity} = \frac{\text{Total Displacement}}{\text{Total Time}} \)

\( \text{Average Velocity} = \frac{0 \text{ km}}{2 \text{ hours}} \)

\( \text{Average Velocity} = 0 \text{ km/hr} \)

The average velocity of the car for the round trip is 0 km/hr.

Step 5: Determine the Magnitude of Average Velocity

The magnitude of a velocity is its speed in that direction. Since the velocity is 0 km/hr, its magnitude is also 0 km/hr.

\( \text{Magnitude of Average Velocity} = |0 \text{ km/hr}| = 0 \text{ km/hr} \)

Analyzing the Options

• Option 1: is 0 - This matches our calculated magnitude of average velocity.
• Option 2: is 50 km/hr - This would be related to average speed. The total distance traveled is 50 km (south) + 50 km (north) = 100 km. Average speed = Total Distance / Total Time = 100 km / 2 hours = 50 km/hr. Average speed is different from average velocity.
• Option 3: is 25 km/hr - This is the average speed for one leg of the journey (50 km / 2 hours for the round trip would imply 1 hour for each leg, so 50 km / 1 hour = 50 km/hr speed, or 50 km / 2 hours for the entire trip would be 25 km/hr as an average speed, which is incorrect). This is not the average velocity.
• Option 4: cannot be calculated without knowing acceleration - Average velocity only depends on total displacement and total time, not on the acceleration during the journey.

Therefore, the correct option is that the magnitude of the average velocity of the car for this round trip is 0.

Revision Table: Key Concepts

Additional Information: Velocity and Speed

It is crucial to distinguish between velocity and speed, especially when dealing with motion where direction changes or the object returns to its starting point. Velocity is a vector quantity and considers direction, while speed is a scalar quantity and only considers the magnitude of motion. For a round trip where the object returns to its origin, the total displacement is always zero, resulting in an average velocity of zero, regardless of the distance covered or the time taken. However, the average speed will be non-zero as long as the object has traveled some distance.