3.5 Relative Velocity
Man, I gotta admit – I think I understand relative velocity pretty well, but the way the book explains it gave me a headache. So, I’m gonna try to do this my own way.
Relative Velocity in One Dimension
You may have at some point during the physics textblogging here thought to yourself “in what sense can I say that I’m walking at a kilometer per hour? What if I’m walking forward, but the Earth is spinning in the opposite direction. And what if all that is insignificant, since the Earth is speeding through space around the sun, and the Sun is bobbing up in down in the galactic plane, and space is expanding and entropy is increasing and love is a lie?!”
Well, first off, love is only a lie for you. Second, all velocity is relative.
We’re not really talking about the relativity you’ll learn later, where things get funky at high speeds. We’re talking about what is sometimes called “Galilean” relativity, which should give you a sense of how long it’s been understood. Here’s the basic idea: When you say “velocity is change in position over time,” you’re really saying “velocity is change in position with respect to something else over time.”
Think about it. Any time you talk about someone’s velocity, you could end your sentence by saying “with respect to ______.”
So, say you’re in a train. In this train are two things that are currently on your mind: (1) a bratty redhaired 10 year-old who is running up and down the aisle screaming. (2) a small but heavy football clenched on your good hand, ready to be thrown at the little ginger fucker.
Now, say you get a call from your mom, saving the little bastard for a few minutes. Your mom says “How fast are you going?” Without thinking, you interpret her question as “How fast are you going, relative to the land around you.” You discard all sorts of potential interpretations, such as “How fast are you going with respect to the international space station” or “With respect to other nearby trains.” Interestingly, you also assume she isn’t asking how fast you’re going with respect to the train.
It occurs to you that, with respect to the train, you’re moving at 0 miles per hour. You know that just by looking around. The train isn’t moving away from you. That is, no particular point on the train seems to be changing position with respect to you. However, you look outside, and see lots of particular points (that tree, that bridge, that lake) zipping away.
You reply “about 60 miles per hour,” and hang up so you can focus on the pintsize bastard. You look at him. He’s running from the front of the traincar to the back, where you’re sitting. You grip the ball tightly and try to estimate how fast the kid is going. But, note, the velocity estimate here is NOT with respect to the outside. It’s with respect to the train (and ergo, you). The kid looks to be running in your direction at about 5 miles per hour.
You whip the ball at him, but miss. DAMMIT. And now he’s singing the theme song from Hannah Montana. You desperately need him to die. So, you call in a favor – a friend in a town you’re approaching has a tranquilizer gun with a laser sight. You text him the number of your car, but warn him that the kid will be moving.
As you approach the sniper point, your friend requests to know the speed of the kid with respect to him. Well, you know the train is moving down the track at 60. As you pass the city, the kid is once again running at you at 5. How fast is the kid moving with respect to the sniper?
Intuitively, you guess it’s some combo of 60 and 5. If the kid were moving faster than the train, he should be moving past it (that is, toward the front). However, he’s moving backward. He’s losing ground to the train. So, he must be going more slowly. You radio your friend and say “He’ll be traveling 55 mph with respect to your position.”
Unfortunately, your friend is drunk. He shoots at the little bastard, misses, and hits an old lady who keels over dead. The kid finds this funny, and increases his speed to 10 mph in his exuberance. Fuck.
Okay, not a problem. Your drunk friend has a helicopter, which he hops into.
The helicopter has to catch up with the train, so the helicopter flies down the line of the track at 80 miles per hour relative to the track. That means it’s going 20 miles per hour relative to you. And, if that little ginger fuckwad is once again headed toward you (and therefore the helicopter) at 10 mph, the helicopter is moving at 30 mph relative to the kid.
The helicopter pulls alongside the train and slows to match the train’s speed. The helicopter is traveling at 0 mph relative to you, so it must be at 60 mph relative to the track. At this point, the kid is running away from you, toward the front of the train, at 10 mph.
From the sniper’s perspective then, the kid is running at 10 mph. Your drunk friend can’t hit a moving target, so while the kid is running forward, he increases his speed to match the kid’s movement. So, he moves at 70 mph. He’s now moving at 10 mph relative to you, 70 mph with respect to the track, and 0 mph with respect to the kid.
He takes aim and fires. Happily, he shot from too close, and the kid’s head explodes suddenly and gruesomely, underscoring the strange combination of brutality and meaninglessness that characterizes human existence.
There, now you have a flavor for how this stuff works.
So, if you wanted to work out that last part mathemtically, you could say something like this:
“The velocity of the kid relative to the track is equal to the velocity of the train relative to the track plus the velocity of the kid relative to the train.
This works out nicely. If the kid is moving back in the train, his speed relative to the track is lower. If the kid is moving forth in the train, his speed relative to the track is higher.
An equation might look like this:
The nice thing you get from this equation is that you can also calculate everything else from it. If you want to know the velocity of the train with respect to the track, it’s just the velocity of the kid with respect to the track minus the velocity of the kid with respect to the train.
You’ll perhaps note that the train example is a little convenient, since the train sits on a track going in a straight line. But, how would the sniper calculate his velocity with respect to the kid if he were approaching the train along a diagonal (i.e. in 2 dimensions).
Next stop: Relative velocity in 2 or 3 dimensions!