The goal for today is to better understand what we mean by terms such as velocity, speed, acceleration, and deceleration. Let’s start with an example, namely the motion of a ball thrown upward and then acted upon by gravity.
A major source of confusion in problems of this sort has to do with blurring the distinction between speed and velocity. The speed s is, by definition, the magnitude of the velocity vector: s := |v|. Note the contrast: When we throw the ball upward:
| – velocity – | – speed – |
| The change in velocity is uniformly downward. The velocity is becoming less upward and/or more downward, which is the same thing. | The speed is decreasing during the upward trajectory, and increasing during the subsequent downward trajectory. |
At this point we have a problem, because the rate-of-change of velocity is called acceleration ... but the rate-of-change of speed is also called acceleration! Obviously this creates enormous potential for confusion. We can begin to untangle things as follows:
| The rate-of-change of velocity is the vector acceleration. | The rate-of-change of speed is the scalar acceleration. |
| The laws of physics are most simply written in terms of velocity, not speed. In a physics context, the unadorned word “acceleration” probably refers to the vector acceleration, but this is not 100% guaranteed. |
| There is no special word for the opposite of the vector acceleration. The opposite of an acceleration in the +X direction is also an acceleration, namely an acceleration in the −X direction. | Scalar acceleration means speeding up. The opposite is called deceleration, which means slowing down. |
Do not confuse the vector acceleration with the scalar acceleration. The scalar acceleration can be considered one component of the vector acceleration, namely the projection in the “forward” direction (although this is undefined if the object is at rest).
To repeat: The vector acceleration is defined to be the change in velocity, per unit time. As the name suggests, it is a vector. This term applies no matter how the acceleration is oriented relative to the initial velocity. There are several possible orientations. The following table shows how to convert vector language to scalar language in each case:
| – Vector language – | – Corresponding scalar language – |
| Acceleration in the same direction as the velocity. | Speeding up. |
| Acceleration directly opposite to the velocity. | Slowing down. |
| Acceleration at right angles to the velocity. | Constant speed. |
| Note: Sideways acceleration corresponds to turning. In the case of uniform circular motion, the magnitude of the acceleration remains constant, and the direction of acceleration remains perpendicular to the velocity. This is a classic example of a situation where the scalar acceleration is zero even though the vector acceleration is nonzero. |
| Acceleration at some odd angle relative to the velocity. | No good way to describe it in terms of scalars. |
| Acceleration of an object at a moment when its velocity is zero. | No way to describe it in terms of scalars; the scalar acceleration formula produces bogus expressions of the form 0/0. |
No matter what terminology you use, it is almost always a bad idea to formulate physics problems in terms of speed, so you should stick to describing motion in terms of the vectors: position, velocity, and acceleration (namely the vector acceleration).
Specifically, in the case of the ball moving under the influence of gravity, it would be unwise and unhelpful (although possible) to formulate the problem in terms of scalar speed and scalar acceleration. Using these ideas, you could say the ball “decelerates” on the way up and “accelerates” on the way down. That’s literally true, but unhelpfully suggests that the physics is different on the way up and on the way down. (For more about the physics of weight and gravity, see reference 1.) It would be much better to analyze the problem in terms of velocity (not speed) plus vector acceleration. The vector acceleration is the same throughout the flight, i.e. it has a constant magnitude and a constant direction. It is always directed downward.
For a much more technical discussion of acceleration, including a discussion of how things accelerate when they are already moving at nearly the speed of light, see reference 2.