# Can we say that the instantaneous velocity of an object is the displacement in zero time?

Can we say that the instantaneous velocity of an object is the displacement in zero time?

In the image above the instantaneous velocity of the object as change in time gets closer and closer to zero is zero. What does it mean?

• Well, the units don't match. – lionelbrits Jan 15 '15 at 12:53
• Well velocity and displacement don't have the same units. You could say that instantaneous velocity is the displacement divided by time, taking the limit of smaller and smaller time intervals. – lionelbrits Jan 15 '15 at 12:57
• Who said that they possess same units? – user66452 Jan 15 '15 at 13:00
• I know that it is for an instant of time. – user66452 Jan 15 '15 at 13:03

No. Velocity is diplacement per unit time - not per zero time.

The unit is: $\mathrm{[m/s]}$ (or other "displacement per unit of time"). For any type of velocity - instantaneous, average or other - the value is displacement per unit of time.

Instantaneous velocity $v=ds/dt$ tells you your velocity at this very moment. If you continued from now on without changing that speed, then the velocity would be the displacement during the next unit of time.

You must be thinking this because $v=\Delta s/\Delta t \to ds/dt$ as $\Delta t$ gets very, very small. But $dt$ is not zero time duration. It is an negligibly small time duration.

Nothing happens during zero time duration - no displacement. But during the very first small, negligible time duration, you also have a very small, negligible displacement. Those two divided by each other give you the instantaneous velocity.

Instantaneous velocity $v_i$ at a time $t$ is the velocity of a body at that time $t$. It is more like this: Instantaneous position is answer to the question "where are you now?". Similarly instantaneous velocity is the answer to the question what is your velocity now?

It is limiting case of time of measurement of displacement tending to zero (but not zero).
Displacement: $\delta s$
Time of measurement of that displacement: $\delta t$

Instantaneous velocity is $$v_i= \lim_{\delta t\rightarrow 0} \frac{\delta s}{\delta t}$$

Experimentally, the smaller the time interval ($\delta t$) of displacement, the closer the velocity measured ($v=\frac{\delta s}{\delta t}$) is to the instantaneous velocity

So, Can we say that the instantaneous velocity of an object is the displacement in zero time? NO
We can say the instantaneous velocity of an object is the displacement per time interval, the time interval forever getting closer to zero