1.2 Speed and Acceleration

Critical Questions:

  • What happens in a car when you push on the gas pedal or step on the brake?

As a physics teacher, I blame a lot of my problems on cars. People spend hours every day in cars of one kind or another, and they’ve developed strong ideas about the relationships between the gas pedal, the engine, and the car’s movement. I will soon try to convince you to think differently about those relationships, but for now I’m going to make use of what you already know.

The Batmobile
Fig. 1: The Batmobile, which doesn't break the laws of physics, but certainly bends them.
Speed is the easy one. If you want to know how fast a car is going, just glance at its speedometer. Unfortunately for high school physics students, however, it is quite easy to complicate even such a simple concept as speed. This practice goes all the way back to pre-Socratic Greece, when the philosopher Zeno asked how a flying arrow could both occupy a space and yet also be in motion, a question which eventually forced mathematicians to give up and invent calculus. Nevertheless, for the purposes of this site, “speed” (or “velocity”) means nothing more than how fast a thing is travelling.

We’re going to have to be a bit more careful about acceleration, though. Not only is the physics definition of acceleration slightly different from the everyday one, it also represents our first tricky concept — one that you might find difficult to wrap your head around.

There are three kinds of acceleration. The first is the one you already know about, which is what happens to your car when you push down on the accelerator: the car speeds up.

The second kind of acceleration is the one that non-physicists might call deceleration. We don’t use that word in physics; slowing down is not any special kind of acceleration, it’s just speeding up backwards.

The third kind of acceleration occurs when something changes direction. That’s right, on this site we are going to say that a car rounding a bend in the highway is accelerating, even if its speedometer stays pointing at exactly 44 miles per hour.

The reason for this is that when physicists talk about velocity, they are also very much interested in the direction of that velocity. For an air traffic controller, a plane travelling at 900 km/h is rather different from one going the other direction; the same is usually true in physics. (In fact, from now on I will often use the word “velocity” instead of “speed” just to remind you that direction is important.) And if we define acceleration as any change of velocity, then it makes sense that changing direction counts as acceleration. There is, in fact, a very good reason for lumping these three types of acceleration together, but I don’t want to start talking about it yet.[1]

So at this point, we have two connections to a car: speed is the number the speedometer is pointing at; acceleration is how quickly the speedometer needle is moving (or how quickly the car is changing direction).

Another important idea to discuss here is that of “constant” acceleration. One trick to understanding this concept is simply to replace the word ‘acceleration’ with its definition: change of velocity. To imagine an object with a constant acceleration, then, we just have to imagine an object whose velocity is constantly changing. The simplest example of this would be an object whose speed is increasing over time; if the object keeps moving faster and faster at a consistent rate, we can say it has a constant acceleration.

The other trick that can help you feel comfortable about this idea is to fully separate in your mind the concepts of velocity and acceleration. For example: imagine two cars, a red one and a blue one. The red car is ahead of the blue car, moving at a constant velocity. The blue car is moving at a slower speed, but it has a forward acceleration.

Can the blue car ever pass the red car? Of course. Aren’t you glad I didn’t put numbers into that question and ask you when the two cars would meet if one left from Los Angeles and the other left from Cleveland? Frankly, so am I. But try this: imagine the blue car’s speedometer steadily climbing up, and confidently tell yourself that it has a constant acceleration, while the red car has a constant velocity. And if you want to get really good at this, imagine different combinations of accelerations and velocities, and watch the results play out in your imagination.

So what happens when something doesn’t have a constant acceleration? This is something we’re all familiar with — it happens every time we change the pressure we’re applying to the gas pedal. With a small push, you get a low acceleration (the speedometer needle moves slowly), and if you push harder, you can increase your acceleration or at least spin your tires. Acceleration is a change of velocity; the technical term for a change of acceleration is jerk (seriously). In the real world, cases of constant acceleration are actually very rare. Things are jerking all over the place, literally and figuratively. But in order to understand jerk, you need to first understand acceleration and what causes it, so we will usually be dealing only with situations of constant acceleration, and if I’m ever describing an object’s motion and forget to say whether its acceleration is constant or not, you can just go ahead and assume that it is.

Steve Martin, "The Jerk," 1979
Fig. 2: Non-constant acceleration, or "Jerk", 1979, starring Steve Martin.

There is another assumption I’m going to make for a while (right up until Chapter 10), which is that the things we’ll be talking about are all moving at normal, everyday speeds. This is an assumption that people didn’t even know they were making throughout most of history, but in 1905 we learned just how weird things can get when you start moving faster. That is the year when Einstein published his Special Theory of Relativity. His work showed that in situations of very high speeds or energies, bizarre things happen: time takes longer to pass by, lengths expand and contract, and the very fabric of space changes its shape.

But as I said, all of that can wait until Chapter 10. For now, please relax and enjoy the comfortable simplicity of the regular or “non-relativistic” velocities.

Big Ideas:

  • Velocity is the number the speedometer points at, although direction is also important.
  • Acceleration is any change of velocity, either in amount or direction.
  • Constant acceleration occurs when an object’s velocity changes constantly (e.g. the speedometer needle moves at a constant speed).
  1. Here’s a hint: each one is a possible effect of the same cause.

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