Explain the terms the average velocity and instantaneous velocity. When they are equal?

When we discuss motion in physics, two key terms that come up frequently are average velocity and instantaneous velocity. In this explanation, we will define average velocity and instantaneous velocity and understand their differences and when they are equal.

Define Average Velocity

Average velocity of a particle is defined as the ratio of the total displacement to the total time taken during its motion. Mathematically, it can be represented as:

    Average velocity = (Total displacement) / (Total time taken)
                     = (r2 - r1) / (t2 - t1)
    

Here, r1 and r2 represent the initial and final positions of the particle, while t1 and t2 are the corresponding times.

The average velocity is independent of the path followed by the particle between its initial and final positions. It only depends on the initial and final positions and the time taken to travel between them.

Define Instantaneous Velocity

Instantaneous velocity refers to the velocity of a particle at a specific point in time. It is the rate of change of the particle’s position with respect to time at that instant.

Mathematically, we define instantaneous velocity as the limit of average velocity as the time interval approaches zero:

    Instantaneous velocity = Lim (Δt → 0) (Δr / Δt) = dr/dt
    

Here, dr/dt represents the derivative of the displacement with respect to time, providing the velocity at any particular instant.

When Are Average Velocity and Instantaneous Velocity Equal?

In cases where the particle is moving with uniform velocity, both the average velocity and the instantaneous velocity will be the same. This is because, in uniform motion, the velocity does not change with time, meaning the velocity at any given point in time is constant and equal to the average velocity over the time period.

In summary, we have learned how to define average velocity and instantaneous velocity. While average velocity depends on the total displacement and time, instantaneous velocity focuses on the speed and direction of motion at a specific moment in time. In uniform motion, both velocities are equal.