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Suppose that we have a circular path which has a radius of $r$ and constant velocity $v$ that is tangent to the circle that the object moving around, I know that centripetal acceleration is expressed like this: $$a_{c}=\frac{v^2}{r}$$ And I know that the direction of $a_c$ is to the center (hence, the name.).


But the question is: is the direction of $a_c$ constant? one of my friends said that yes it constant to the center but the other one said no, the direction is changing that's why the velocity direction keeps changing.

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First, let's make sure we are viewing this scenario from an inertial frame.

But the question is: is the direction of $a_c$ constant? one of my friends said that yes it constant to the center but the other one said no, the direction is changing that's why the velocity direction keeps changing.

It depends on what you mean by "direction", and both can be correct, I suppose. If you express the acceleration using polar coordinates / vectors then you can write it as $$\mathbf a=-\frac{mv^2}{r}\,\hat r$$

The magnitude is constant, and it always points in the $\hat r$ direction, so in terms of polar coordinates it doesn't change directions.

However, I would say this is a confusing way to talk about this. The direction of the unit vector $\hat r$ depends on where on the circle you are looking at, so if you were drawing the acceleration vector at various points in time, you would be drawing vectors that point in different directions. In other words, it is better to use a coordinate system, like Cartesian, where the unit vectors themselves do not change. Then you would have something like $$\mathbf a=-\frac{mv^2}{r}(\cos\theta\,\hat x+\sin\theta\,\hat y)$$

which you can see does change directions as $\theta$ changes.

Therefore, ultimately I would say that the direction of the acceleration is changing, but I can understand the point saying that "towards the center" is a single direction, as long as we all agree on what we mean by this. I just think the latter might be a little confusing to introductory students.

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One of my friends said that yes it constant to the center but the other one said no, the direction is changing that's why the velocity direction keeps changing.

Both are saying the correct thing but using different words.

The centripetal acceleration vector $\mathbf{a_c}$ always points to the center and that means it always changes direction too. The tangential velocity vector $\mathbf{v}$ also changes direction, which is why the centripetal acceleration is needed in the first place, because:

$$\mathbf{a_c}=\frac{\text{d}\mathbf{v}}{\text{d}t}$$

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No, because the line along r will always intersect with any line along any instance of r that is not zero or 2π from inital r forming an angel at the center, thus,pointing at different directions.

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