can the velocity of an object be negative when its acceleration is positive? what about vice versa?

Answer 1

I've already answered this for your other question: is it possible to have zero velocity and non-zero acceleration at the same time? But I'll go into detail here.

First off, the answer is yes, the velocity of an object/body/particle can be negative when its acceleration is positive. Velocity and acceleration are vector quantities, and so negative and positive for them indicate opposite directions.

When velocity and acceleration are opposite signs, then it just means that the body/object is slowing down. It's decelerating. The velocity can be positive and the acceleration negative. Or the velocity can be negative and the acceleration positive.

When velocity and acceleration are the same sign, then it means that the body/object is speeding up. It's accelerating. The velocity can be positive and the acceleration positive. Or the velocity can be negative and the acceleration negative.

When you slide a puck on the floor, it will come to a stop soon enough. Why? Because of friction. (Or precisely, the force of friction.) Friction is a force that always acts in the direction opposite the motion.

And just like any force, we can apply Newton's Second Law of Motion to it: F=ma, where m is the mass of the object and is a scalar quantity (therefore has no concern or regard for direction), F is the force applied, and a is acceleration.

Both F and a are vector quantities AND they both act along the same direction. In any given force, there is an associated acceleration in the same direction. (The acceleration may be zero, but we still have to assign a direction to it.)

In the case of the sliding puck, the force of friction acts in the direction opposite that of the motion of the puck. The acceleration associated with this force is also in this
opposite direction.

Now, I bring this up because it'll be important later in my explanation. Please remember: a force and the acceleration associated with it are both in the same direction.

When I answered your other question, I mentioned Simple Harmonic Motion. I'd like to show you what SHM is like for the purpose of showing you how velocity and acceleration can change signs as a body goes through simple harmonic motion.

An example of SHM is a swinging pendulum. But I won't use this example because it's two dimensional and involves gravity.

Instead, we'll look at a body/particle oscillating along the x-axis. We choose the horizontal axis because we don't want to have to figure in the effect of gravity. And we make it move along one axis so as to involve only one dimension. This way the analysis will be simple.

A real-life setup can be an object, like say a ball bearing, held in the middle of a spring.

This spring-mass system is laid horizontally on a slick surface. Both ends of the spring are anchored on to a couple of fixed supports.

When the spring is totally relaxed, the ball bearing is in the centre. Let's call this position x=0. If you pull it to the side, let's say, 5cm to the right, we'll call this x=5.

So any position right of centre is positive. Any position left of centre is negative.

And get this: any motion towards the right is positive velocity. Any motion towards the left side is negative velocity. We're talking here the direction of motion, just as i explained earlier above.

And now get this: any force towards the right is positive. Any force towards the left is negative. Again, we're talking about the direction of the force. And remember we said that acceleration acts in the same direction as the force it is associated with. So, if the force exerted by the spring is to the right (and therefore positive), then it follows that the acceleration provided by the spring is also to the right and positive.

Remember also that we could have chosen to the left to be positive, and to the right negative. But because we have already decided that to the right is positive, so everything else follows along. Get it? Whatever we choose, whatever's convenient to us.

OK so now, let's get the ball moving.

Pull the ball bearing to one side and let go. The ball bearing will go from one end to the other and back again, and back to the other end and so on till friction brings it to a halt. This is oscillation. You have just demonstrated simple harmonic motion.

(We can reduce the effects of friction by making slick the surface that the whole setup is resting on. But it doesn't really matter. We just want to see how this system moves and how velocity and acceleration change. Just one or two oscillations will do.)

Now let's look at the oscillation in detail and analyse it in terms of velocity and acceleration.

We begin by pulling the ball bearing to the right, say, 5cm. At this point, the left side of the spring is tensed (stretched) and so is exerting a pulling force, pulling the ball to the left. At the same time, the right side of the spring is compressed, and so is pushing the ball towards the left. So the spring is exerting a force towards the left.

Now, we have established that "to the left" means the negative direction, therefore the force being exerted here is negative. And since the acceleration is the same direction as the force, therefore acceleration is also negative.

What acceleration? you may ask. After all, the ball isn't moving. Well, let it go! It will accelerate from a standstill and move to the left.

And since it's moving to the left, its velocity is negative also. So here in the first phase of the motion, we have negative velocity and negative acceleration. Since both velocity and acceleration are the same sign, the ball bearing speeds up.

In the centre, the ball reaches its maximum speed. The spring is totally relaxed, so no force is being exerted, and so there is no acceleration. But the ball's speed is maxed out.

Notice I say "speed" and not velocity. The velocity is negative, but the speed is maximum. Speed is just the magnitude of the velocity.

Now, past the centre and moving left, the spring is getting tensed again, but this time it is the right side of the spring that is being tensed (stretched). And its left side is being compressed. Overall effect? Force being exerted to the right. Positive force, positive acceleration.

Because the ball bearing is moving to the left, it has negative velocity. And since positive acceleration is the opposite direction, the ball decelerates: it slows down. Until it comes to a complete stop at x = -5cm.

Here the spring is exerting the maximum force towards the right. And the ball bearing shoots off towards the right because it is being accelerated in that direction. So during
this phase the ball has positive velocity and positive acceleration.

Once again at centre (x=0) the spring is totally relaxed. Force and acceleration are zero. Speed of the ball is maximum.

Past centre and moving right, the ball bearing slows down because the spring is now both pulling and pushing it towards the left. It is exerting a force in the negative direction. Negative force, negative acceleration. But because the ball is moving towards the right, it therefore has positive velocity.

The force and acceleration reach their maximum values when the ball gets to its maximum position of x=5cm. And we're back to where we began. The oscillation has completed one cycle. (If left alone, it will go for another cycle and at least a few more, each time sweeping a smaller distance than the time before because of friction.)

Well, there you have it. We have looked at the four phases of a simple harmonic motion.

In the first phase, the ball bearing was moving towards the left, velocity was negative, and acceleration was negative, speeding up the ball.

In the second phase, the ball bearing was still moving to the left, therefore velocity was negative, but acceleration was positive, slowing down the motion.

In the third phase, the ball was moving to the right, therefore velocity was positive, and acceleration was positive, accelerating the ball towards maximum speed which it would achieve at centre.

In the fourth and final phase, the ball bearing was moving away from the centre, towards the right, therefore positive velocity. But the spring was pushing it back towards the left,
therefore negative acceleration. And the ball decelerated to a stop at the rightmost point.

So there you have it, all four possible combinations of positive and negative velocity and acceleration. Positive and negative just tell us the direction, that's all.

I've also answered your other question again: zero velocity and non-zero acceleration at the same time. This happens instantaneously at the left-most and right-most points of the ball bearing's motion.

I wanted to show you mathematical proof but I was not sure what your level of math was. You would at the very least need to know how to analyse a graph OR a trigonometic equation. I decided it would take up too much space here, so I hope anyone else reading this and who is still skeptical would just take a look at any college-level physics textbook which surely would explain vectors and simple harmonic motion. Really, you don't need to look at SHM to know that velocity and acceleration can have opposite signs; any discussion on vectors will show that to be true. It's just that SHM goes through all the combinations of velocity and acceleration, and also it's easy to visualize (isn't it?), and that's why I chose to use it.

I hope I've helped. Anyone else still not sure, please e-mail me.

Answer 2

Yes of course. That is how you bring something to a stop and sent if off in the other direction.
Positive acceleration means that its velocity is increasingly positive so if you throw a rocket off a bridge with the engine aimed downward and it starts to fall at -30 feet per second with the engine applying a force to give it +40 feet/sec/sec it will fall for a while (don't forget gravity trying to do -32 ) falling more and more slowly, then the velocity becomes positive and it passes us on the bridge going up.
Every time you throw a ball in the air, the velocity is positive (up) and the acceleration is negative (down) and it comes to earth.

Answer 3

An object which moves in the negative direction has a negative velocity. If the object is speeding up then its acceleration vector is directed in the same direction as its motion (in this case, a negative acceleration).
If the object is slowing down, then its acceleration vector is directed in the opposite direction as its motion (in this case, a positive acceleration).

So an object can be moving in a negative direction, but changing speed in a positive direction. Think of an object at the end of a coil spring shot from a gun. The object will move to the left, but be changing speed toward the right.

Answer 4

Yeah he the if a body was moving in a certain direction with a velocity Vo and you apply a force to accelerate it in the negative direction for a while the velocity will be in negative direction to the acceleration

Answer 5

Yes.

a(t) = dv(t)/dt

The velocity can be negative with positive acceleration. Suppose you are driving backwards down the street and apply your brakes...you are still going backwards, but your speed is less, but the key here is less negative, i.e. positive acceleration is being applied. In the other case, your velocity can be positive with a negative acceleration. Suppose you are now driving forwards and apply your brakes...you are still going forward but your speed is less, so you are applying a negative acceleration.

Answer 6

To answer the first question, yes, if velocity and acceleration are vector quantities. Otherwise, negative velocity doesn't make much sense in any case. And if acceleration is negative (deceleration), velocity can be positive, whether dealing with vectors or scalar quantities.

Answer 7

sure! imagine this scenario.. Taking forward displacement as positive, you engage your reverse gear and reverse your car.. you are now travelling at a negative velocity with respect to your point of reference.. if you hit the breaks and slow down to a rest, during the period at which you were slowing down, your acceleration would be positive while your velocity would still be negative.

Answer 8

Anytime an object is 'slowing down', it is experiencing acceleration of opposite sign to its velocity.

You must define 'positive' and 'negative' to answer your question directly.

For example, if you define +ve for 'upwards' w.r.t the earth, then imagine a parachutist, just after he/she opens their 'chute.

Their velocity is negative ('down'), but their drop rate (negative velocity) is slowing rapidly, as they experience positive acceleration due to drag forces on their chite.

If you define +ve and -ve the opposite way, then throw a ball up. For a while, it is rising (-V) but slowing down (+A)

Anytime an object is 'slowing down', it is experiencing acceleration of opposite sign to its velocity.

Answer 9

if the acceleration is a,the velocity is v= at+vo
both a and vo are vectors so even if the are colineal the sign of v depends on the sign of vo( initial velocity)

Answer 10

You can't go up and down at the same time.