Can 0 acceleration be termed as constant acceleration?

Today I started having a discussion about how acceleration should be considered constant if its numerical value is zero because '0' is also a numerical constant. There was a contradiction stating that acceleration=0 should not be considered constant because the rate of change in velocity doesn't exist in that case (basically there is no acceleration) hence 0 will not be considered constant in this case. I think as the numerical value of acceleration is not changing it should be constant.

I am providing the question that started this whole discussion and what I believe to be the answer.

The displacement-time graph of a moving object is a straight line. Then,

(a) its acceleration may be uniform
(b) its velocity may be uniform
(c) its acceleration may be variable
(d) both its velocity and acceleration may be uniform

In the attached question I think the last option will be correct because yes, a graph showing constant acceleration can also have zero acceleration. It should be totally possible as 0 is a constant.

• I think this is a matter of opinion and perhaps semantics. I would not mind if someone told me to solve a problem or described a situation with a constant acceleration of 0. I think mathematicians would agree Aug 6 '21 at 8:36
• @Alwin, I would not confuse problems of terminology and matter of opinion. The difference between using a term in physics and in common language is a fact, not an opinion. Furthermore, many conceptual problems originate from linguistic misunderstanding. I acknowledge that once more on PSE, these obvious and quite well-known facts in Physics Education Research are completely ignored once more on PSE. That's a pity. Aug 6 '21 at 17:18
• Hi Sourav. If you have a question about why this answer was closed or how you might improve it, you may ask on the meta site. Aug 6 '21 at 18:53
• Why is this question closed as opinion based? This question is perfectly valid. Aug 6 '21 at 19:10
• This question is the subject of a discussion on Meta.
– rob
Aug 7 '21 at 7:32

Technical language often uses the same words as everyday language, but with a different meaning. Just to remain close to the subject matter of the question, in everyday language, we distinguish among acceleration and deceleration, while in Physics one deals with positive or negative acceleration.

The case of zero acceleration is along the same line. In everyday language it would be weird to speak about a constan acceleration equal to zero. Still, in Physics that's possible and common, and it becomes important when it is useful to avoid to list special cases. The closest example I can think of is the case of velocity. Again, in informal language constant velocity would imply movement. In physics it is more useful to include also the zero-velocity among the constant velocity cases. For example, it turns out useful when stating the First Principle of Newtonian dynamics. Another example could be a periodic motion with zero frequency. A strange object for everyday language, but a very useful concept in mathematics and physics.

• It's common in physics to speak of constant acceleration of 0? I am surprised, I don't think I have ever heard constant velocity described as thus, because it's confusing. Aug 6 '21 at 10:43
• @Allure As I wrote, it is common not to treat the case of zero value differently from other constant values. When people say that, given one inertial system, all the systems with any change of origin and all possible relative values of the velocity are inertial, are including the case of relative zero velocity. Aug 6 '21 at 12:17

There may be mathematical and philosophical discussions over the ambiguity of the number zero. But physically, there is no confusion.

• If a physical property doesn't change (over time in this case), then it is constant. Regardless of its value.

This has got nothing to do with how the acceleration happens to influence an underlying velocity. If is doesn't change, then it is constant. An acceleration at, say, $$10 \,\mathrm{m/s^2}$$ that never changes is thus constant. An acceleration at, say, $$1 \,\mathrm{m/s^2}$$ that never changes is also constant. And an acceleration at $$0 \,\mathrm{m/s^2}$$ that never changes is likewise also constant.

In the first case, the velocity changes a lot, in the second it changes less and in the latter it doesn't change. This is a gradual difference in how the value of acceleration happens to influence the velocity - nevertheless, all three examples of accelerations are constant.

For instance the kinematic motion equations, such as $$s=s_0+v_0t+\frac12 at^2,$$ which only apply in cases of constant acceleration, are perfectly fine to use with an acceleration value of $$0$$ precisely because it does not change - it is a constant value.

Zero acceleration can be a constant acceleration. It is, of course, also possible to have a non-constant zero acceleration.

For example, a particle with equation of motion $$x=a+bt +ct^2$$ has a constant acceleration of $$2c$$. If $$c$$ happens to be zero then this is a constant zero acceleration. On the other hand, a particle with equation of motion $$x=dt^3$$ with $$d\ne 0$$ has zero acceleration at $$t=0$$ but this acceleration is not constant.

A constant zero acceleration is certainly not a varying (or non-constant) acceleration. So if you don’t want to call zero a constant acceleration then you have three categories of acceleration - constant, varying and zero. Presumably you have to treat position, velocity, momentum, energy etc. in a similar way for consistency. Is a temperature of $$0^\circ\mathrm C$$ not a constant temperature ?? This would be very confusing.

I don’t see any contradiction between “no acceleration” and “constant acceleration”. If I have no money then I have a constant amount of money - that constant amount just happens to be zero.

If the case of constant zero acceleration is excluded then the motion can be unambiguously described as “constant non-zero acceleration” or “constant positive acceleration”.

• How is a non-constant zero acceleration possible? Can you provide an example? Aug 7 '21 at 10:46
• @PeterMortensen Acceleration can be zero at an instant in time, but need not be constant. I gave an example in the 2nd paragraph of my answer. Aug 7 '21 at 23:30
• @PeterMortensen Here, the acceleration is instantaneously zero at time $t_0.$ Aug 13 '21 at 16:11

0 (like 1 or $$\pi$$ ) is a number and not a constant. The term constant (like in “constant acceleration”) is only relevant when you discuss if the acceleration happens to be 0 for some moment $$t_0$$ in time $$a(t_0)=0$$ or if the acceleration as function of time is zero at all times $$a(t)=0$$.

Thus, as physics is concerned, it is perfectly valid to say that the acceleration is constant with a value of 0. In this case, the velocity will be constant (at some specific value which itself may be 0). The same argument apply to negative values of acceleration.

In everyday language the matter is a bit different. Zero acceleration will be considered rather “no acceleration”. Negative acceleration will commonly be called “braking”.

Think of the theory of limits.

You will agree with me that an acceleration of $$1 \mathrm{m/s^2}$$ is constant. Similarly, an acceleration of $$1\mathrm{ mm/s^2}$$ is constant. So is $$10^{-9} \mathrm{mm/s}$$, or $$10^{-n}\mathrm{mm/s^2}$$, for every value of n you can think of.

As you use larger and larger values for n, in all cases the acceleration is constant. It will still be constant at the limit, where $$n=\infty$$, and $$10^{-n}=0$$.

Of course , as explained in @gandalf61's answer, this whole exercise is only applicable if the acceleration stays at the same value, and isn't just momentarily zero.

Feeling sorry to start this post with "zero" : (

Zero acceleration means constant velocity. If you consider zero acceleration as a constant acceleration, it implies that constant velocity is constant acceleration, which makes no sense.

This is an ambiguous question, like 'zero' is an ambiguous number.

Regarding Comments: I didn't say zero acceleration is not a constant acceleration. I just said it makes no sense. Quoting from another answer, a beggar who has no money would never say "I have a constant amount of money".(not intended to condemn)

Another point: As steeven correctly defined 'constant':

If a physical property doesn't change (over time in this case), then it is constant. Regardless of its value.

This is the correct definition which I definitely agree. But think you are standing still. So your acceleration is zero, then call it constant acceleration. Your velocity is zero, then call it constant velocity. And your displacement is zero, then you should be able to call it constant displacement (because your displacement doesn't change with the time in this case). But this seems odd.

• That actually makes sense but we can also put it in a way where we see it as the numerical value of acceleration is not changing. Aug 6 '21 at 8:36
• Why should zero be an “ambiguous number” ? What does this mean ? Are there any other “ambiguous numbers” ? Aug 6 '21 at 9:11
• @gandalf61, can you say whether zero is a positive or negative number?
– ACB
Aug 6 '21 at 9:19
• @ACB Zero is neither positive nor negative. There is nothing ambiguous about that. But a constant value of zero is still constant. Would you describe a stationary object as not having a constant velocity ? Would you describe a stationary object that is at $x=0$ as not having a constant position ? Aug 6 '21 at 9:24
• @gandalf61 ,I didn't say it is wrong. I was saying it is not normal.(no sense). And I agree with what your answer says. And zero is great number, should not deserve 'ambiguous'. But that is a special number.I don't know any other number like zero.
– ACB
Aug 6 '21 at 13:12

Language is an imprecise tool. Very few words and phrases have entirely unambiguous meanings. Zero acceleration can indeed be considered constant. However, to some people the word acceleration conveys the notion of a change in speed, which might cause them to assume that a reference to a constant acceleration means a non-zero one.

Other nouns convey the same effect- growth, for example. If you mention that the economy has been in constant growth for the last three quarters, people will assume the rate of growth to be non-zero, even though zero growth is a constant rate of growth.

Given that the ambiguity unavoidably exists, it would always be wise to add the words 'non-zero' after the word 'constant' if that is what you mean.

This would be unusual. In most (all?) cases I've seen with constant acceleration, the acceleration has been some positive, nonzero number. Constant acceleration with a negative acceleration would also be unusual; usually people say constant deceleration in that case. The case with $$0$$ acceleration is usually referred to as "constant velocity".

• It maybe unusual but i am asking if it is impossible Aug 6 '21 at 8:21
• @SouravSingh I don't think I can answer that question. There's nothing stopping you from talking about constant velocity motion as "constant acceleration motion", but you're likely to confuse people. In other cases it's for the person making the statement to clarify what they mean by constant acceleration. The default assumption would be $a > 0$. If they later say they also call $a = 0$ as "constant acceleration", then it's something you accept; they are making the statement after all. Aug 6 '21 at 8:26
• And yes i also believe that the case with 0 velocity(which you agreed is constant velocity) will also have 'constant acceleration'? Aug 6 '21 at 8:29
• Actually negative acceleration is not uncommon. Aug 6 '21 at 8:48
• Your answer may be true in every day language but not in physics. This is physics stack exchange. Aug 6 '21 at 9:23

Acceleration means change in velocity over change in time. So as in this case constant acceleration means constant change of velocity w.r.t. time but zero acceleration means no change in velocity over time. So it is inappropriate to say zero acceleration as constant acceleration. Zero acceleration simply means no acceleration at all. Now, consider a case in which an object moves constantly by a small distance in a very large time interval than acceleration is close to zero and constant but not exactly zero. I think if we are talking about constant acceleration than it is obvious that acceleration is non zero.