speed of light, MPH

Factual Questions
1

easy question for physics types.

say a rocket ship takes off out of my room, and starts flying away from me… it accelerates really really fast and ends up going very very close to C, not just 99% but 99.999999999% (add how ever many 9s you feel like) so that it is going as arbitarily close to C as you want, but not actually breaking the laws of physics by actually reaching it.

now my question is:

how many MPH is the rocket ship going

A) in one of its hours

AND

B) in one of my hours

I know time dialation means that a clock on the rocketship and my clock won’t be syncronized even nearly… since time passes ‘more slowly’ on the ship if you count my clock as the referance. is there some MPH it ends up at the end if I am the referance? or does it forever keep changeing every time another 9 is added to 99.9% ?

2

I don’t understand your question.

To the rocket ship, you will appear to be moving away at 99.99 whatever % of the speed of light, and to you the rocket ship will be moving away at the same speed.

3

An object doesn’t move at all relative to itself. Think about when you’re driving a car. How fast is the car moving relative to the road? Now how fast is it moving relative to you, and everything inside the car? It’s not moving at all. So you could sit in a car doing 150 down the Autobahn for an hour, and it would still be going 0 relative to you. The only thing that matters to the rocket ship in your experiment, is how fast is your room travelling away from it? The answer is obviouos: 99.99999999999% of C.

4

In one of our hours - 300,000,000 MPH (roughly)

At a complete guess - In one of it’s hours it would be going at

299,999,999,970 MPH

5

friedo I think he was asking how for he would travel after an hour has passed on the clock in his rocket ship. Not how far away from himself he’s traveled.

6

Well, first of all, MPH is “miles per hours”. So MPH in “one of its hours” is somewhat confusing.

But, as far as how fast it’s going in each inertial reference frame:

In your viewpoint, it’s going approximately 186,000 miles per second, or about 670,000,000 mph.

In its own viewpoint, it’s going 0 mph. It’s standing still with respect to itself. It’s just like if you were standing still. You’re actually hurtling around the sun at an incredible speed. But since you’re looking at the Earth as your frame of reference, you’re not moving.

7

Okay, if what you mean is, how far will it have traveled in your frame of reference after it has traveled for one hour in its own reference frame, then…

The Lorentz transformation for time is, I believe, t / (1 - v[sup]2[/sup]/c[sup]2[/sup]). So, if it was going, say 99.99% of the speed of light in your reference frame, then one of its hours would take 1/(1 - .9999[sup]2[/sup]) = ~ 5000 hours in your reference frame. So, it will have traveled, in your frame of reference, about 3,350,000,000,000 miles.

8

The speed of light is 670616629.3844 MPH. In your frame of reference, the rocket could be going, say, 670616629.3841 MPH. That’s the answer to part A. You may not like the answer to part B, but it’s 0. That is, the rocket is not moving with respect to itself, so it sees itself as being still.

Now, maybe what you meant, and I misunderstood, is this. Suppose the rocket starts out at the starting line in a laboratory, and goes at 0.9999…999 times the speed of light. After its clock registers 1 hour, it stops. How many miles from the starting line is it, in the lab frame?

The answer is this. Take the answer to part A. Multiply it by 0.7071, and then multiply it by 10[sup]N/2[/sup], where N is the number of 9’s in its speed. So if the rocket goes 0.999999 times the speed of light (N = 6) and stops after 1 hour, it will have gone 670616629 × 0.7071 × 10[sup]3[/sup] = 47419756713 Miles. A whole lot more than you’d expect. You will have seen it travelling for 0.7071 &times 10[sup]3[/sup] hours = almost a month. As you can see, a couple of extra 9’s makes a big difference. If you stick two more on, the answer increases by a factor of 10. So for 0.99999999 (N = 8), it will have gone for almost 10 months.

9

THANK YOU Achernar… that was what I was asking. I did word it badly.

so there is no upper limit on how many miles you can go in an hour… as long as you don’t care about anyone else’s clocks? correct?

10

Yup, Achernar is correct. The Lorentz transformation for x is x = (x’ + v t’)/sqrt(1 - v[sup]2[/sup]/c[sup]2[/sup]). I forgot the square root in my equation.

So, since x’ = 0, t’ = 1 hour, and v = .9999c, then x ~= 6.71e8 mi /1.43e-2, which is 4.74e10, just like he said.

11

…er, for .9999c, the answer would be 4.74e10. For .999999c, in Achenar’s example, the answer would be 4.74e11. It appears he missed one decimal place in his answer, though his equation is correct.

12

Yeah, I wouldn’t be surprised if I missed a decimal or two. I’ll take your word for it. owlofcreamcheese, even though it sounds like the rocket will think it’s going faster than the speed of light, that’s not true. Here’s what really happens.

Let’s say the rocket is moving at 99% the speed of light, so that v/c is 0.99. The so-called “Lorentz factor” is 1/sqrt(1 - 0.99[sup]2[/sup]) = 7 (approximately). This is a handy number, because it comes up a lot. It tells you how much dimensions stretch or shrink by. For instance, as we’ve already said, if the rocket’s clock goes for one hour, the lab’s clock will go for 7 hours.

Okay, now the lab has lines painted on the floor, one every mile. We want to know how many of these lines the rocket crosses, and that will tell us its speed. Say the rocket flies for 1 second by its clock, 7 seconds by ours. At 0.99c, it will cover 184,420 miles every second, so after 7 seconds, that’s 1,290,937 miles, and so it will cross 1,290,937 lines. (Someone correct me if I make an algebra mistake.) Now, in the rocket’s frame, it crossed those 1,290,937 lines, but it did it in 1 second! If the lines are 1 mile apart each, that means it’s going at 1,290,937 miles/sec, or 6.93c, faster than the speed of light!

Here’s the kicker, though - those lines aren’t one mile apart. You may have heard that when things go fast, they shrink. In fact, they shrink by this Lorentz factor. So if the rocket is 70 feet long when it’s at rest, when it’s travelling at 0.99c, it appears to us to be only 10 feet long. And from the rocket’s point of view, it’s stationary, and the lab is moving, so the lab shrinks! That means these lines are no longer 1 mile apart each - they’re 1/7 of a mile apart each. So if the rocket measures its speed by counting 1/7 of a mile for each line it’s crossing, it will measure itself as going 184,420 miles/sec, just like it should.

Is that clear?

13

crazy stuff, achernar

14

C’est Relativité.