The Acceleration Vector
The section on acceleration very naturally proceeds out of the section on velocity. Velocity is change in position over time, and acceleration is change in velocity over time. We’ve already gone over this for 1D, and 3.2 takes a look in 2 and 3D.
The math is a vector form of the math from back when we first studied acceleration. So, you need to combine what you learned about 1D acceleration with what you learned about vectors.
I don’t want to get to far into the math, since I think it’s pretty intuitive from what you already know, and whatever I say will just be redundant with the book. Suffice it to say, you should go over it and make sure you understand it.
Instead, I want to dive directly into movement along curved paths, which sometimes can be confusing.
Recall what we said earlier about acceleration: When you run forward at one speed, then slow down to a lower speed, you’re still moving forward even though you’re accelerating backward. With a little thought and practice, this concept makes perfect sense.
Now imagine this: You’re running from the end of a tennis court to the net because you want to punch your opponent in the face. If you run toward the net then slow to a stop, you’re accelerating backward. But now, say instead of running straight for the net, you run forward and then to one side along a curved path. For simplicity, let’s say the curve you’re running on is a quarter turn of a circle.
We know that the speed at which you’re approaching the net is slowing down, since you’re not moving forward. So, we know that whatever vector describes your motion along the curve can be partly defined by a vector pointing back toward the end of the court your started at.
But, we also know you’re accelerating to one side (let’s say to the right side) because you weren’t going that way at first, but you’re going that way faster and faster now. So, while you’re on this curve, we should be able to define your acceleration in terms of two vectors – how much you’re accelerating back and how much to the right. If you’re running on the curve of a circle, you’ll find the total vector always points toward the center of the circle.
To understand rigorously, visualize the vectors. As you run around the curve, at the beginning you’re velocitating straight for the net. A second later, you’re velocitating on more of a diagonal. Acceleration is just the change in velocity over time. So, we subtract one velocity vector from the other. Recall from earlier that this can be done by putting the two vectors “tail to tail” and drawing a new vector “tip to tip” starting with the earlier vector. When placed at your position, that vector will point right toward the center of the circle.
Neat, huh? It means, for example, if you run in a circle with your inner arm extended, your arm will always point in the direction of your acceleration.
It’s important to note that this is true even though your speed is the same throughout. It’s counterintuitive because in non-physics land acceleration is about how the speed on your car’s dial changes. And, indeed, if you drive your car in a circle on cruise control, the speed won’t change.
But, think about how driving in a circle feels different from driving the equivalent length at the same speed. When you go forward at the same speed, you never feel pressed against your seat. When you go around a circle, even though your speed is constant, you may feel pressed against the window. Why? Because your body is attempting to go in one direction, but the car is going another. That is, the car is going a different way from the way it was going a second ago. This information gets communicated to your body when you get scrunched against the side of the car during a sharp turn. So you see, even though you’re going the same distance at a single speed, there is a real difference in what’s actually happening.
In fact, you’re going to find that motion on a curve (and around a circle in particular) can produce all sorts of surprising results.
Here’s something to think about, which we’ll get to understand better later: Assuming constant speed, what do you think takes more energy? Going 100 meters in a straight line or going 100 meters in a circle.
Next up: PROJECTILES!