Motor Daddy:
Clearly, trying to discuss pendulums in accelerating cars is way too complex a problem for you to cope with at this stage. So, let me just go back over the whole relative-velocity problem you were having earlier in the thread.
Saying "one car is traveling 20 m/s faster than the other" means there is two separate velocities. What are the two velocities and what are those velocities relative to?
First, let's dispense with the confusion about whether we're talking one or two velocities. To specify a relative velocity, we
always need two velocities. The velocity of object B relative to object A is
always defined to be:
$$v_{rel} = v_B - v_A$$
So, if we say that car B is travelling 20 m/s faster than car A we have:
$$20 = v_B -v_A$$
Notice that this tells us
nothing about the velocities of cars A and B, other than that one is 20 m/s greater than the other. Thus, it is possible that
$$X: v_A = 0, v_B=20$$
or
$$Y: v_A=80, v_B=100$$
or whatever.
What about this:
$$Z: v_A=-10, v_B=10$$
?
This is also fine.
"What's with the X,Y, and Z?", I hear you ask. X,Y and Z are what we call
reference frames. I'm sure you remember the definition of a reference frame from our many previous discussions (think sticks and clocks etc. - remember?)
When we say "Car B is travelling 20 m/s faster than Car A" we haven't explicitly specified any particular reference frame. And until we do, we can't assign numbers to the velocities of cars A and B. One important caveat, of course, is that we
do require a pre-existing concept of what a metre and a second are, or else the expression "20 m/s faster" is itself meaningless. But that's basic common sense.
So all you know is that the relative velocity is 20 m/s. That could mean that each rocket is traveling away from each other at the same rate (10 m/s), or that each rocket is traveling in the same direction at either 80/60, 29/9, 21/1.
Yes. Each of these situations corresponds to a different choice of reference frame.
So you have no basis for claiming that one rocket is traveling 20 m/s faster, since they could actually be traveling in opposite directions at 10 m/s each, 20 m/s relative velocity between them.
This raises a slight complication.
$$W: v_A=-15, v_B=5?$$
In that case, B's velocity relative to A is still 20 m/s, but can we say that B is travelling faster than A? Clearly, in this particular choice of reference frame, the
magnitude of the velocity (i.e. the
speed) of B is less than the speed of A, but that does not change the fact that the
velocity of B relative to A is still 20 m/s.
So, if you're having issues with "B is going faster than A", then I suggest you do
not choose a frame such as frame W in the above example. You're better off working in a simple frame, and the simplest one to choose is the one where:
$$v_A = 0, v_B = 20$$
That way, you won't tie yourself in mental knots trying to work it all out.
It's not semantics. Saying one object is faster than another object means each object has it's own speed. A relative velocity is not of one object's speed but of a closing speed, which is nothing more than measuring the distance between the objects at points in time and stating that information in terms of the units for distance and time. The distance and time is not of one object's motion but of the space between the objects.
Right. A relative velocity assumes that we have already established a notion of what we mean by the term "velocity", because relative veloocity is just the difference between two velocities.
I hope that's sorted now to your satisfaction. If you have any questions, please ask.
Absolute rest is when the center of a sphere emits light and the expanding light sphere remains centered on the outer sphere at all times.
There's nothing absolute about that. In the rest frame of the sphere that always happens, whether or not the sphere is moving in some other reference frame.
Two rockets in space and the distance between them is increasing at the rate of 20 m/s. How do you know which direction of travel they are each traveling?
Where did this "direction" idea of yours come from? It sounds like you're assuming some frame of reference prior to the information you have given. In a particular given frame, we presumably know the velocity of at least one of the rockets. Then, observing that the distance between the two is increasing allows us to deduce the direction of travel of the other rocket in that given frame.
If at t=0 the distance between them is 100 meters, at t=1 the distance between them is 120 meters. Do you think it's possible that each rocket could be traveling in different directions, possibly even at the same speed (gasp!)?
Sure. It's just a matter of picking the right frame of reference.
Hope this helps!