# What is the definition of velocity?

We know that displacement is change in an object's position (here position means 'position vector'). Then velocity will be change in position of the object with respect to time, simply displacement/time taken. But I was taught that velocity is change in displacement with respect to time, then it means velocity will be equal to change of (change in object position =displacement) with respect to time. So, which one definition is correct?

• And we all taught that the slope of displacement v/s time graph gives velocity.so , will the slope of displacement v/s time graph gives velocity OR slope of position v/s time graph gives velocity? Commented Jun 14, 2023 at 22:40
• "Displacement" is ambiguous. I think it pretty much always refers to a difference between two positions, but it could be the difference between positions of the same object at different moments in time, or it could be the difference between the position of an object and some "reference position." Sounds like your instructor was using the word in the second sense—displacement from some reference position. Commented Jun 15, 2023 at 1:30

The change in an objects position is equal to the displacement.

$$\vec r_{\rm displacement} = \vec r_{\text {final position}}-\vec r_{\text {initial position}}$$.

i was taught that velocity is change in displacement with respect to time

should be, velocity is displacement (ie change in position) with respect to time

I might also add that given a position vector is relative to some origin,

$$\vec r_{\text {final position}}-\vec r_{\text {initial position}}= \vec r_{\text {final displacement from origin}}-\vec r_{\text {initial displacement from origin}}$$.

Your mention of the gradient of a displacement against time graph does highlight the use of change of displacement per unit time as a definition of velocity.
So look at this way.
Somewhere on the displacement axis the displacement is zero and all the other displacements are measured relative to displacement equal zero.
(How close is that to the definition of a position vector?)

Thus the change in displacement is,
$$\text{final displacement relative to position where displacement is zero}$$
$$\text {minus}$$
$$\text{initial displacement relative to position where displacement is zero}$$

As the reference to zero displacement is generally understood an abbreviated form is used, $$\text{change in displacement = final displacement - initial displacement}$$

• then what will be the definition of velocity ? Commented Jun 14, 2023 at 22:38
• are you saying that position = displacement? but you mentioned above displacement as change in object position Commented Jun 14, 2023 at 23:02
• will the slope of displacement v/s time graph gives velocity OR slope of position v/s time graph gives velocity? whats your opinion on it? Commented Jun 14, 2023 at 23:03
• Do you think that the two slopes are different?
– nasu
Commented Jun 15, 2023 at 0:56

I think your confusion lies in the difference between change in position, and displacement. They are both different only when the object you are considering did not start from the origin. Otherwise they are both the same.
For example, at $$t=0$$ let your body start from the origin and reach $$x=1$$ in one second.

Let $$s(t)$$ be your displacement at time $$t$$ and let $$p(t)$$ be your position at time $$t$$.
Now the displacement of a body is indeed change in position. But when the initial position is zero, it just ends up being position anyway. $$s(t)=p(t)-p(0)=p(t)-0=p(t)\\ \therefore s(t)=p(t)$$ So your velocity after one second would be $$1ms^{-1}$$ in this case.

Now suppose the body does not start from the origin, but from somewhere else, say $$x=1$$; and reach $$x=2$$ in one second.

In this case, $$s(t)=p(t)-1$$ And velocity would be defined as $$v=\frac{s(t)}{t}$$ but $$v\neq \frac{p(t)}{t}$$ Velocity would still remain $$1ms^{-1}$$.

The slope of both position-time graph and the displacement-time graph would be the same. And it'll give you the instantaneous velocity.
You can now generalise this to n-dimensions using vectors.
Hope this helps!

Displacement means the same the same thing as position. They are exact synonyms. Displacement means the difference between the objects position and an arbitrary but fixed origin position. It does NOT mean the difference between an objects current position and its previous position.

The difference between an objects current and previous positions is the change in position or change in displacement.