# Is it possible to have zero velocity and zero acceleration?

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1. Feb 17, 2015

### persia77

is it possible to have zero velocity and zero acceleration for a moving object
for example with trajectory x= (t-2)^3
at t=2
a=v=0

2. Feb 17, 2015

### A.T.

How do you define "moving"?

3. Feb 17, 2015

### persia77

move=change of location

4. Feb 17, 2015

### PeroK

I guess you mean instantaneously 0. For example, an object could stop for a second and then start again. It would have 0 velocity and acceleration for 1 second as part of its motion.

If you imagine an object at rest subject to a force proportional to t. F = t, say. Then at t = 0 it would have 0 velocity and acceleration, yet it would start moving at t = 0.

5. Feb 17, 2015

### Khashishi

It's clear that you understand velocity and acceleration from your given example. The language of math is clear on this one. I wouldn't worry about "moving" or not.

6. Feb 17, 2015

### Staff: Mentor

I'm sitting on my couch right now, with zero velocity and zero acceleration.

7. Feb 17, 2015

### nasu

But are you "a moving object"? :)

8. Feb 17, 2015

### phinds

There are an infinite number of Frames of Reference in which he is a moving object. Only in his own Frame of Reference is he not moving (and that's not counting the arm motions required to drink his beer and click the TV remote).

9. Feb 17, 2015

### Staff: Mentor

It's not my frame, it's my couch's -- I'm just in (on) it temporarily.

10. Feb 17, 2015

### phinds

Fair enough

11. Feb 18, 2015

### persia77

you are not moving object in your coachs frame
you should say a frame that you are a moving object in it and have instantaneous zero velocity and acceleration

12. Feb 18, 2015

### phinds

Do you understand that "moving" and "having velocity" mean exactly the same thing? You are asking if something can be moving and not moving at the same time. Well ... no. At the time when something has zero velocity it is not moving. If you want something that is moving irregularly (fast sometime, slow sometimes, not at all sometimes) then you can certainly have something that is moving most of the time and not moving some of the time. BUT ... if you are transitioning from moving to not moving, that is called "acceleration" so you can't have something that is moving some of the time and then not moving and having zero acceleration as it come to a stop. Once it has been AT at a stop for a small amount of time, then it will have zero acceleration.

I'm not sure you understand what velocity and acceleration imply and/or I'm not sure what you are asking. It sounds very confused. In your example, yes v=0 at t=0 but the acceleration at that point is not zero because it is transitioning from v=0 to v not = 0 and that is called acceleration.

Do you understand how, mathematically, to get the acceleration of your example? If x = (t-2)^3 then what operation do you perform to get the acceleration? If you perform that operation, you will see clearly that the acceleration is not zero.

13. Feb 18, 2015

### Staff: Mentor

Why? Doesn't that make the original question sort of circular and contradictory? Basically when you combine the two, you get: "Is it possible to be stationary in a frame where you are moving?" Well, no -- if you are stationary, you aren't moving and if you are moving, you aren't stationary.

14. Feb 18, 2015

### PeroK

There is something of a paradox here, since at t = 2, we have:

x = 0, v = 0 and a = 0

Yet, for any t > 2, the particle has a non-zero displacement. It's instantaneously at rest and, also, instantaneously has 0 acceleration.

It's just the same puzzle as how the function $y=x^3$ manages to get back up off the x-axis after x=0.

15. Feb 18, 2015

### phinds

It's constrained motion. It's moving that way by definition, whatever the force(s) required to make it do so.

I do thing, however, that you have pointed out a flaw in my own thinking. In this example, which is as good as any, i see (ignoring the math) that the object is going from zero motion to non-zero motion so I see it as accelerating, BUT ... the math says a=0.

16. Feb 18, 2015

### persia77

you dont understand what velocity and acceleration is
acceleration is derivation of velocity
do you understand meaning of derivation ?

17. Feb 18, 2015

### persia77

my first example shows your statement is wrong
a moving object
with trajectory x= (t-2)^3
at t=2
instantaneous a=v=0

18. Feb 18, 2015

### PeroK

But, it's not "moving" at t = 2. It's moving at all other times, but it's instantaneously at rest at t = 2.

19. Feb 18, 2015

### persia77

a moving object can have instantaneous zero velocity at some moment

20. Feb 18, 2015

### PeroK

It can have 0 velocity for as long as you like. I already pointed that out to you. It can move, stop for a second, then move again. It's still a "moving object", it's just not moving all the time.

21. Feb 18, 2015

### persia77

if you drop a ball in the air (to up) it has instantaneous zero velocity at top of its trajectory but it is a moving object all the time

22. Feb 18, 2015

### PeroK

The mathematics says otherwise.

23. Feb 18, 2015

### jbriggs444

This seems to amount to nothing more than a word game. We need to pin down a definition of "moving".

Definition:

An object whose position is given by x(t) is "moving" at time t if and only if for all epsilon > 0 there is a strictly positive delta such that |delta| < epsilon and x(t+delta) != x(t).

Definition:

An object is "moving" over an open interval if it is moving at all times that fall within that interval.

Claim:

An object can be "moving" over the interval from t=0 through t=4 and have zero velocity and zero acceleration at t=2.

24. Feb 18, 2015

### persia77

no
The mathematics says exactly same

25. Feb 18, 2015

### PeroK

At the top of its trajectory, in which direction is it moving?