The formula that defines acceleration is \(a = \frac{\Delta v}{\Delta t}\text{,}\) where \(\Delta v\) is the change in the velocity and \(\Delta t\) is the change in time. This mirrors the formula that defines velocity. The units of acceleration take a little bit of getting used to. Since the units of velocity are \(\frac{\text{Length}}{\text{Time}}\text{,}\) the units of acceleration are \(\frac{\text{Length} / \text{Time}}{\text{Time}} = \frac{\text{Length}}{\text{Time}^2}\text{.}\)

It can be tempting to try to over-interpret these units. When you see it in the latter form, you might be tempted to try to understand what "squared seconds" are as a unit. But that unit is not a very sensible object to talk about because we don’t ever really multiply time by itself. The reason we write it in the latter form is because it’s more compact, but the "more correct" way to see it is the former way. It’s measuring how velocity (which happens to be measured in units of \(\frac{\text{Length}}{\text{Time}}\)) changes over time.

Just as with velocity, the basic formula technically gives us average acceleration. This is because you still have to have a time interval over which you’re measuring the change of velocity, and it’s possible for nuances of the changes in velocity to get lost in that. Just as before, we can construct the idea of the instantaneous acceleration as the accelration at a specific moment in time, which runs into the same conundrum of there not being a single instant of time that is being observed. The basic resolution, as noted before, is to just take small enough time increments so that the change in velocity over that period of time is negligible.

Acceleration is more a more complicated value, and it requires more information for us to determine it from positions. Let’s imagine that we have a movie. If we look at any particular frame by itself, we won’t be able to measure either velocity or acceleration. That single instant of time does not give us the information we need. If we were to look one frame later, would could see changes in the position, which would allow us to get the velocity. However, that is only a single velocity value. In order to see a change in velocity, we have to look at a third frame so that we can measure the velocity between frames the second two frames, and then see how that velocity is changing.

While this multi-layered calculation is often required, we have other tools that measure acceleration. Such tools are known as accelerometers. There are digital devices that do this, but while those devices are great for data collection, they also hide all of the concepts inside of the circuitry, and so it’s harder to understand what’s happening. So we will think about the classic analog accelerometer. You should imagine a clear tube with a ball in the middle that is connected to the top and bottom of the tube by springs. If you hold the tube vertically and motionless for a while, the ball will eventually settle into what is known as the "equilibrium position", which is what gives us our reference point. If you lift the tube upward quickly, you should be able to intuitively see that the ball will initially shift downward relative to the tube. This is how it indicates an upward acceleration. Similarly, (after letting it reach equilirbrium again) if you lower the tube quickly, the ball will shift upward. This gives the basic idea for how these devices work.

A lot of people have a misconception of acceleration. They often think that "acceleration" means "fast." For example, "a car with great acceleration goes fast." But this is the wrong way to think about it. Instead, it makes more sense to think about acceleration in terms of how we can experience it.

But before we can do that, we need to establish something about velocity. It turns out that we don’t directly experience velocity. This may sound strange at first, since you have a lot of experiences of moving quickly and moving slowly, and you know the difference between them. The challenge of experiencing velocity is that we can only observe it visually. We look at things that are around us, and those things tell us whether we’re moving. You might be sitting in a room and think that you are stationary, but in reality, you’re sitting on a globe that’s spinning. In fact, if you were at the equator, you would be traveling at around 1600 km/hr. And if you zoom out a bit more, you’re on a planet that is orbiting the sun at around 30 km/s. But you don’t experience any of that velocity because all you have as reference points are the objects around you, and they are on the same ride you are.

Acceleration, on the other hand, can be experienced directly. If you’re in an accelerating vehicle, you will feel your body being pushed backwards slightly if you are speeding up, or you will feel your body being pushed forward if you are slowing down. You will also feel acceleration when riding an elevator, by feeling yourself being pushed down slightly if the elevator is accelerating upward and by feeling a little lighter if the elevator is accelerating downward. Notice that the feeling in the elevator is independent of whether the elevator is going up or down. All that matters is how the velocity is changing.

A common unit for measuring acceleration in is in \(g\)s, which is also called g-force. Even though it has "force" in the name, it’s really an acceleration, but in a confusing twist of physics, it is very closely connected to forces and so the name really does make sense. (If you know about \(F = ma\text{,}\) then you might be ablet to make sense of the idea that g-force is force per unit of mass.)) The numerical value of \(g\) is 9.8 m/s\(^2\text{,}\) which is the conventional value for the acceleration on earth due to gravity alone. (It turns out that the true acceleration changes depending on where you are on the planet.)

G-forces are used to help give us a relative sense of how much acceleration something is experiencing. For example, it is known that the g-force on the moon is about 0.16 g. (This is not to be confused with 0.16 grams!) What this means is that you weigh about 16% (about 1/6) of what you normally do. Alternatively, if you on the sufrace of the sun (and somehow not dead), the g-force is 28 g, and so it would be like you weigh 28 times heavier on there.

G-forces are also used to measure acceleration in other contexts. A stunt pilot might experience an acceleration of 9 g, which means that for that particular maneuver it’s as if they weighed 9 times more with gravity pushing them in some direction based on how they are turning.