can a particle have zero displacement and non-zero average velocity? if yes or no...why so?

Yes, a particle can indeed have zero displacement while having a non-zero average velocity. This concept can seem a bit counterintuitive at first, but let’s break it down to clarify how this is possible.

Understanding Displacement and Average Velocity

To grasp this idea, we first need to define two key terms: displacement and average velocity.

• Displacement is the shortest distance from the initial position to the final position of an object, along with the direction. If an object returns to its starting point, its displacement is zero.
• Average velocity is defined as the total displacement divided by the total time taken. It is a vector quantity, meaning it has both magnitude and direction.

Scenario Illustration

Imagine a particle moving in a circular path. Let’s say it starts at point A, moves around the circle, and returns to point A after a certain period. In this case:

• The initial position and final position are the same, so the displacement is zero.
• However, during its motion, the particle has traveled a distance equal to the circumference of the circle. If we consider the time taken for this journey, the average velocity can be calculated as:

Average Velocity = Total Displacement / Total Time

Since the total displacement is zero (as it returns to the starting point), the average velocity is:

Average Velocity = 0 / Total Time = 0

In this specific scenario, the average velocity is zero. However, if we consider a different example where the particle moves back and forth along a straight line, it can have a non-zero average velocity.

Another Example

Consider a particle that moves from point A to point B and then back to point A, but it does so in a way that takes longer on the return trip. For instance:

• The particle moves from A to B (5 meters) in 2 seconds.
• Then, it returns from B to A (5 meters) in 3 seconds.

In this case:

• The total displacement is zero because the particle ends up where it started.
• The total time taken is 5 seconds (2 seconds + 3 seconds).

Now, if we calculate the average velocity:

Average Velocity = Total Displacement / Total Time = 0 / 5 = 0

So, in this case, the average velocity is still zero. However, if we modify the scenario slightly, say the particle moves from A to B and then to C (where C is not the same as A), it can have a non-zero average velocity.

Key Takeaways

In summary, while a particle can have zero displacement, it cannot have a non-zero average velocity at the same time. The average velocity is directly tied to the displacement over time. If the particle returns to its starting point, the displacement is zero, leading to an average velocity of zero. However, if the particle moves to a different point, it can have a non-zero average velocity even if it travels back and forth along a path.