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I was studying about the fundamentals in physics and I read a defination of average velocity. It was like that average velocity is a physical quantity which gives the measure of overall rapidity of a moving particle in a given time interval. And it is useful for the comparison of overall rapidity among two particle within a given time interval. Then the defined the instantaneous velocity as the limiting value of the average velocity when the time interval tends to zero. What does it physically signify? According to the defination it is the rapidity of the particle at a given instant of time but I am not getting that. Suppose a situation where we are comparing two particles and both of them are moving randomly , with a different instantaneous speed at different instants. If we consider any instant where suppose the first particle has a larger speed than the second . What does it physically mean? Does it have any physical significance? As both the particles are changing speeds with respect to the time it may happen that the next instant the second is having a larger speed the first one.

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The velocity of a particle can change with time, so it is a function $\mathbf{v}(t)$. The “instantaneous velocity” at time $t_1$ is simply the value of this function when $t=t_1$. Its has physical significance as “how fast, and in what direction, the particle is moving at $t_1$”.

Instantaneous velocity should be reasonably intuitive, because instantaneous speed is what the speedometer in every car displays! (Speed is just the magnitude of the velocity vector.)

There is nothing odd about object $A$ moving faster than object $B$ at $t_1$ and vice versa at $t_2$. One has a velocity function $\mathbf{v}_A(t)$ and one has velocity function $\mathbf{v}_B(t)$. For example, car $A$ might be speeding up while car $B$ is slowing down.

In physics, instantaneous velocity is much more important than average velocity.

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this video explains brilliantly where you are getting confused, and that's good.

https://youtu.be/WUvTyaaNkzM

and to keep the stack exchange rules, i'll also explain it in my own words. suppose a particle is moving, and you take the snapshots of it while while it is moving, if you look at only one snapshot of it, you can't certainly tell whether it's moving or not because it seems to be stationary in the snapshot. Now consider the problem of telling how fast and in which direction the particle will tend to move if you are given a snapshot of it at a particular time and the initial conditions. That velocity which will tell you in which direction and how fast will the particle will tend to move just after the time instant of which we are given a snapshot. The idea is to make the difference between the time interval at which the snapshot was taken and the next time interval at which we intend to find the particle motion details infinitesimally small, which is just another way to express the fact we are just finding instantaneous velocity. Hope it helps

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