Relative Velocity in 2D and 3D
Okay, so in the last section I tried to get you a feel for relative velocity. As you get into higher dimensions, the “feel” won’t serve you as well as getting the math. So, let’s dig in.
Since you have the basic idea from last time, let’s check out how the book sets up these equations, since it’s handy.
Okay, that looks ugly, but once you get what each thing means it’s simple. Here’s the idea: Say you have 3 moving things. One is you (given by the letter a). You are charging a train because you’re drunk out of your mind and you think it slept with your boyfriend. The second is the train, given by the letter b. The third is a mild-mannered old lady pacing down the aisle inside the train.
The equation says: The passenger’s velocity relative to you is equal to the sum of the passenger’s velocity with respect to the train and the trains velocity with respect to you. That is, if the train is moving away from you at some velocity, and the old lady is going away from you down the aisle of the train, then the velocity at which she’s moving away from you is equal to the sum of her velocity and the train’s velocity.
Note that even though there’s a plus sign there, it doesn’t have to be addition. One or both of those vectors could be in the direction defined as negative for a given system.
If it still seems ugly, fortunately the choice of symbols simplifies things a little. You can pretend you’re doing stoichiometry with the subheadings on those vectors. The vector for p/a is gotten by the addition of p/b and b/a. So, it’s like you’re multiplying through to eliminate the b part. Of course, you’re not actually doing this, but the symbols help you remember what you’re up to.
The cute part is that, since it’s all in vectors, you can compute more complicated motion than just you running behind a train. For example, if you’re headed toward the train at an angle, your velocity vector will reflect that. All you have to do is set up the system, see what the moving parts are, and add it up.
I could belabor this a bit more, but I think it’s a relatively simple concept. Plus, it’s the kind of thing you’re better off learning by doing once you’ve got the gist of it.
Holy balls! We’re done with chapter 3! In the next chapter we get to Newton’s Laws. To me, this is where we really enter physics land. Everything heretofore has really been giving you the mathematics of motion. We’re about to get the concept that underlies all motion – force. Hopefully we’ll also get a lot of “use the force” jokes as well.
Next stop: NEWTON’S LAWS