# B Whether constant acceleration and zero acceleration are the same

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1. Aug 11, 2017

### Mathivanan

In the equation v=u+at, u=5 m/sec and a=0, then v=5 m/sec. That means the body is moving with a constant velocity of 5 m/sec. In a velocity-time graph, the equation produces a straight horizontal line. Some texts say the body is moving with zero acceleration and others say it is moving with constant acceleration. Which is correct?

2. Aug 11, 2017

### scottdave

If the constant is equal to zero, then they both can be right. I usually see constant acceleration to mean a nonzero constant, though.

3. Aug 11, 2017

### Staff: Mentor

Isnt zero a constant? What is the confusion here?

4. Aug 11, 2017

### Ibix

We do tend to make a distinction between "not doing something" and "doing something" in natural language. But maths regards it as a distinction without a difference. So, as others have noted, zero is just a constant like any other.

5. Aug 11, 2017

### Mathivanan

The confusion is: what is the need for qualifying acceleration as constant; it is positive or negative or zero, in all these cases it's a constant.

6. Aug 11, 2017

### Mathivanan

Inaction and action are both actions, right!

7. Aug 11, 2017

### scottdave

There are formulas for constant acceleration. These formulas work just fine when acceleration is a constant zero - they can be written in simpler terms, because a=0.
Everything changes when you have a non-constant (changing) acceleration, though.

8. Aug 11, 2017

### vanhees71

Constant acceleration implies
$$\ddot{\vec{x}}=\vec{a}=\text{const}.$$
Integrating twice with respect to time, gives immediately
$$\vec{x}(t)=\frac{\vec{a}}{2} t^2 + \vec{v}_0 t + \vec{x}_0,$$
where $\vec{v}_0$ is the velocity of the body at time $t=0$ and $\vec{x}_0$ is its position at time $t=0$.

Nothing in the above calculation forbids me to apply it to the special case $\vec{a}=0$, where of course, you get uniform motion (i.e., motion with constant velocity), as it must be from Newton's 1st law:
$$\vec{x}(t)=\vec{v}_0 t + \vec{x}_0.$$

9. Aug 11, 2017

### Staff: Mentor

Acceleration need not be constant. A simple example is a spring and simple harmonic motion. The acceleration varies over time, it is not constant.

The "suvat" equations do not apply for such motion.

10. Aug 11, 2017

The word "zero" can be used more descriptively and more economically than the word "constant" because it can be more specific. Here's an example comparing the use of the two words in describing the motion of an object during a certain time interval:
1. During that time interval the acceleration of the object was zero.
2. During that time interval the acceleration of the object was constant.
For sentence two to be as specific as sentence one it would need to be added to to give the value of the acceleration.

11. Aug 11, 2017

### Mathivanan

What I thought is: a body is said to have zero acceleration when it is either at rest or comes to rest if it's moving; it is because acceleration due to gravity. In the equation v=u+at, when a=0, the body moves with a constant velocity. Can we say that the body is moving with zero acceleration? If it is how a body can move with constant velocity without acceleration. It should have a positive acceleration to move with constant velocity. Hence 'a' cannot be zero. Therefore, there is something wrong with my understanding. In the equation, 'a' is not simply acceleration but rather a change in acceleration or Δa. As a body cannot be moved without acceleration, it has an initial acceleration. When there is no change in the initial acceleration, that is Δa=0, the body moves with constant velocity. It seems the equation is v=u+Δa*t.

12. Aug 11, 2017

### ZapperZ

Staff Emeritus
Zero acceleration is a SPECIAL CASE of constant acceleration.

Zero acceleration is a member of constant acceleration set. However, a constant acceleration does NOT mean the acceleration is zero.,

There are numerous other examples of this. A uniform, zero potential is a special case of a constant potential. However, a constant potential does not imply a potential value of zero.

Zz.

13. Aug 11, 2017

### boringelectron

Constant acceleration means that the velocity with which an object is moving with is constantly changing with time
This gives us a horizontal line in acceleration/time graph.

If the acceleration is zero means that the object is moving at a constant velocity which is a straight line on a velocity/time graph.

Constant acceleration can not be zero

14. Aug 11, 2017

### Ibix

No. Constant acceleration means $da/dt=0$. That is satisfied by any $a$, zero or non-zero, that does not vary with time.

15. Aug 11, 2017

### Ibix

This isn't right. A body moving at constant velocity has zero acceleration, whether it's moving at zero velocity or at hundreds of metres per second.

Our every day experience is that stuff that is moving slows down and stops. But this is actually an oddity of being on the Earth in an atmosphere where there are always frictional forces (with the air or the ground) that try to make you move the same way the ground is moving. You need to separate two concepts in your mind: one is that you will always move in a straight line at a constant velocity unless forces act (that's Newton's first law). The other is that (in the circumstances that happen to be true on the Earth) there are always forces acting unless you are stationary with respect to the surface of the Earth.

16. Aug 11, 2017

### Mathivanan

Think of a car at rest. Unless a force acted upon, it cannot move. When you apply force it starts moving against acceleration due to gravity. Depending upon the change in this initial acceleration, the car keeps moving or comes to rest.

17. Aug 11, 2017

### ZapperZ

Staff Emeritus
This is wrong at the elementary level. A force that typically causes a car to move is PERPENDICULAR to the direction of gravity. So the force isn't applied to overcome gravity, but rather because of friction and to overcome inertia.

Now, if you start lifting the car up vertically, then yes, you are opposing the weight of the car, but this isn't how a car normally moves.

I still don't know if you're still asking your original question or whether you've moved on to a different topic. It isn't clear if you've understood the explanation you've been given on your original question.

Zz.

18. Aug 11, 2017

### Ibix

More precisely, it cannot start moving without a force if it is initially at rest.
Really? The acceleration due to gravity is directed vertically downwards. Unless you have a flying car, you are not correct here. The car has to work against friction to move relative to the Earth and air.
No. The car will always come back to rest, but this is because of friction. This is the point I was talking about separating in your mind: the car will continue to move at the same speed as long as no force acts on it. But on the Earth there will always be a force acting on it as long as ot moves with respect to the surface of the Earth.

19. Aug 11, 2017

### Mathivanan

Therefore, the car is at rest due to friction and not due to earth's gravity.

20. Aug 11, 2017

### Mathivanan

I began with v=u+at and still continuing with that.