What would qualify as a deceleration rather than an acceleration if speed is unchanged? The instantaneous acceleration $\textbf{a}(t)$ of a particle is defined as the rate of change of its instantaneous velocity $\textbf{v}(t)$: $$\textbf{a}(t)=\frac{\mathrm{d}}{\mathrm{d}t}\textbf{v}(t).\tag{1}$$ If the speed is constant, then $$\textbf{a}(t)=v\frac{\mathrm{d}}{\mathrm{d}t}\hat{\textbf{n}}(t)\tag{2}$$ where $\hat{\textbf{n}}(t)$ is the instantaneous direction of velocity which changes with time.
Questions:


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*According to the definition (1) what is a deceleration?

*In case (2), when will $\textbf{a}(t)$ represent a deceleration? For example, in uniform circular motion, why is it called the centripetal acceleration and not centripetal deceleration?
 A: Acceleration is the correct technical term for the physical quantity you mentioned in the equations you posted (i.e. a).
The term deceleration doesn't describe a rigorously-defined standard physical quantity, it's just a term used differently in different situations that means "handwavily" that the velocity or speed is decreasing.
Sometimes it could be clear that it refers to some precise quantity (e.g. the absolute value of a scalar acceleration along a curve, like when you are driving a car and keep an eye on the odometer), but without further context it has no rigorous meaning.
A: Acceleration is the general term for a changing velocity. Deceleration is a kind of acceleration in which the magnitude of the velocity is decreasing. The reason this might be confusing is because the word 'acceleration' is sometimes used to mean that the magnitude of the velocity is increasing, to contrast it with deceleration. One cannot go wrong, however, if one always takes acceleration to mean simply 'changing velocity'. In that case, circular motion corresponds to acceleration (because the velocity is changing) but not deceleration (because its magnitude is not decreasing).
A: According to this definition, "deceleration" is undefined.
A: Acceleration's the rate of change of a body's velocity.

*

*Deceleration refers to the decrease in the absolute value of the velocity. Definition (1) defines a case of deceleration when $a(t)<0$ when $\mathbf v(t)>0$, or $a(t)>0$ when $\mathbf v(t)<0$. It's just a fancy word to describe a body which is slowing down.

*In definition 2, $v$ will always be positive, because speed is a scalar quantity. But the direction of change of velocity can be negative.
Consider a body moving with a constant speed in a curved path on the following plane.

Even though the vertical component of velocity is positive, it is constantly decreasing, so we can call this deceleration along the y axis, even though we have simultaneous acceleration along the x axis. $\mathbf v_\mathbf y=v\times cos(kt)$, where k is some constant.Putting this in equation 2,$\textbf{a}(t)=-vk \times sin(kt)$, which is negative for the range of $t$ values we're considering.

*For circular motion, we call it centripetal acceleration because it's always positive. the formula, which is derived here: A simple derivation of the Centripetal Acceleration Formula?, is $a=v^2/r$. $v$ is speed, which is a scalar and is thus always positive, and $r$ is radius, which is positive again, so $a>0$.

However, in physics (from what I've seen), we don't use the word deceleration frequently, because as I showed earlier, it's simple when the velocity is positive, but when we're dealing with a body with a negative and changing velocity, it gets messy. It's more suitable for explaining things we see, and isn't easily compatible with math.
