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Uniform velocity occurs when an object travels equal distances in equal intervals of time without a change in speed or direction. Imagine driving a car at a constant speed on a straight highway without stopping or speeding up. That's uniform velocity! Uniform velocity is important in physics because it allows us to predict and calculate the motion of objects over time.

When dealing with uniform velocity, the formula to find the velocity is simply \[ v = \frac{d}{t} \]where

• \( v \) is the velocity
• \( d \) is the distance covered
• \( t \) is the time taken

For objects moving with uniform velocity, their path can easily be tracked using distance-time graphs, which result in a straight line. This is because equal changes in distance happen over equal intervals of time.

Understanding uniform velocity is crucial for solving various physics problems, such as the one about a person traveling on a straight path with two different velocities.

The distance-time relationship is fundamental in physics to describe how an object's position changes over time. This relationship helps us understand motion and calculate other aspects like velocity and time taken for travel.

When we say a person travels a distance of \( x \) at velocity \( v_1 \) and an equal distance at velocity \( v_2 \), we become concerned with two main aspects:

• Total distance traveled
• Total time taken for the journey

To break it down:

• Total distance is simply the sum of the distances covered at each velocity. In this case, it's \( 2x \).
• To find the total time taken, you add the time spent traveling each section: \( t = \frac{x}{v_1} + \frac{x}{v_2} \).

This is key to solving average velocity problems because it helps us determine how long each part of the journey takes compared to the whole.

JEE (Joint Entrance Examination) physics problems often test students' understanding of fundamental physics concepts, requiring practical application of theory. This problem, for example, challenges students on their comprehension of average velocity, the concept of uniform velocity, and their ability to apply the distance-time relationship.

To solve this exercise:1. **Understand the Question**: The problem asks for average velocity, given different uniform velocities over equal distances.2. **Use Known Formulas**: Utilize the formula for average velocity: \[ v = \frac{\text{Total Distance}}{\text{Total Time}} \]3. **Calculate Step-by-Step**: Break it down into calculations for total distance and time, combining them to find average velocity.4. **Match to Options**: Simplify the result using algebra to best match the multiple-choice options.
JEE problems like these are not just about getting the right answer—they aim to deepen your understanding of physics and enhance problem-solving skills through practical applications.