# Possible for a body to have zero velocity and non-zero acceleration

1. Sep 21, 2008

### Mitchtwitchita

Is it possible for a body to have zero velocity and non-zero acceleration. I would have to say no because acceleration is the time rate of change of velocity. So, if velocity isn't changing, how can there be an acceleration. Is this answer accurate?

2. Sep 21, 2008

### ZapperZ

Staff Emeritus
Re: Kinematics

You're forgetting that a body can have an instantaneous zero velocity, and yet, it's velocity is also changing. Throw a ball up in the air. At the highest peak, it's velocity has a zero velocity for an instant. Yet, all through the motion, it still has an acceleration that is non-zero, which is g.

Zz.

3. Sep 21, 2008

### Mitchtwitchita

Re: Kinematics

Gravity!!! Then I guess, I would have been wrong. So, I would have to say now that a body is unable to travel with a constant acceleration and a time-varying velocity. Would you say that this is a correct assumption?

4. Sep 21, 2008

### atyy

Re: Kinematics

No. Let's take Zz's example of a ball thrown up in the air. We pretend the earth is so large that it doesn't move, and only the ball does. The ball moves only a short distance, so we approximate the gravitational force to be constant over the ball's trajectory.

Force of gravity between the ball and earth = mg,
where m is the mass of the ball
where g is a constant, which represents the "effect of the gravitational mass of the earth".

Newton's 2nd law, F=ma,
where F is the total force on the ball
where m is again the mass of the ball
where a is the acceleration of the ball in response to F.

Since gravity is the only force on the ball, we combine the force of gravity and Newton's 2nd law as follows: mg=ma.
Hence a=g.

So the acceleration of the ball is constant in magnitude and direction. When you throw the ball upwards, a is downwards and opposite to the velocity, so the ball "decelerates". At the top of the trajectory, the ball has instantaneous 0 velocity. Then it moves downwards, a is still downwards but now in the same direction as the velocity, and the ball "accelerates". (Sorry I used "accelerate" in two slightly different ways here, hence the bolding for the first technically correct use, and the quotes over the second colloquial use.)

Last edited: Sep 21, 2008
5. Sep 22, 2008

### LURCH

Re: Kinematics

Perhaps I have missunderstood your question, but I think that a body traveling with a constant acceleration is not only able to have a time-varying velovity, it absolutely must have one. Perhaps you meant that a body under constant acceleration is unable to have a velovity that is constant over time? I would agree with that (form an inertial reference frame, of course). An accelerating body may have an "instantaeous zero velocity," as Zz said, but that velocity cannot remain zero over time.