# Can acceleration be both the “rate of increase of velocity” and the “rate of increase of speed” in Physics?

A Dictionary of Physics (Oxford University Press) defines acceleration as:

The rate of increase of speed or velocity

However, from reading many other definitions it seems to me that acceleration in Physics is generally held to refer to the rate of increase of velocity (rather than speed), and that acceleration is generally held to be a vector quantity rather than scalar quantity.

How does this relate to the definition above of acceleration being able to be an increase in speed? What are the contexts in which acceleration is treated as an increase in speed? Do these tend to be more simplistic, less real-world treatments of acceleration or do they belong to another field such as Mathematics where the treatment may be more abstract and direction can be ignored?

I note that the SI unit for acceleration is metre per second squared (m/s2). Is it significant that this unit does not specify direction? Does this allow for accelertion to be a scalar or vector quantity depending on whether or not a direction is specified?

• Closely related question here. You're right that the two notions are not consistent. It's all semantics. Laymen usually use the first definition and physicists usually use the second. – knzhou Jul 31 '18 at 10:51
• There's no fundamental difference, people are just using a word in different ways. It's like debating over whether a sunset is yellowish-red or reddish-yellow. – knzhou Jul 31 '18 at 10:51
• Also, you really can't say much by looking at the units alone -- what would "specifying a direction" by units look like? Would the units be "meters times direction per second squared"? Units just don't give that much information about the quantity that carries them. – knzhou Jul 31 '18 at 10:54
• Thanks for replying knzhou. The issue I have is that the writers of dictionary definitions such as OUP's A Dictionary of Physics are generally very careful about the terminology they use. – PrettyHands Jul 31 '18 at 10:54
• @PrettyHands It's a bit ambiguous. I suppose you could say that scalar acceleration is "the rate of change in speed", and vector acceleration is "the rate of change in velocity". However, one could also sensibly define scalar acceleration to mean "the magnitude of the vector acceleration", which is different. The point is, there is no "official" dictionary out there that everyone must adhere to. People use words in different ways. In practice, this is never going to matter as long as you pay attention to context. – knzhou Jul 31 '18 at 12:01

## 1 Answer

No. In physics there is only one official definition : acceleration is the rate of change of velocity. Like velocity it is a vector quantity which has both direction and magnitude. Acceleration includes change of direction as well as change of speed.

The distinction is important because the Laws of Physics are usually written as equations in terms of vector quantities wherever appropriate : for example $\mathbf{F}=m\mathbf{a}$. The scalar equation $F=ma$ ignores changes of direction when speed is constant, as in uniform circular motion. Vector equations are often more compact and enable a consistent sign convention to be applied. Equations written in terms of scalar magnitudes and angular directions are more complex and cause confusion when the sign convention changes to avoid negative values.

Sometimes the magnitude or the components of the acceleration vector are referred to as "acceleration" because it is more convenient, but this is not strictly correct. For example, centripetal acceleration in uniform circular motion is often described by its constant magnitude, because the varying direction (always toward the centre) is implied and does not often affect calculations.

The dictionary definition only mentions increase in velocity/speed. Probably it uses deceleration for decrease in speed but this distinction causes confusion. In 1D motion a -ve acceleration describes a slowing down and reversal of the motion of a particle travelling initially in the +x direction. Its deceleration (decrease of speed) changes to acceleration (increase of speed) although the applied force and its direction do not change.