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Fundamental Velocity Question

  1. Jul 25, 2006 #1
    I hope you can follow my train of thought (I'm not being elitist here, I am open to the possibility that my post may be unintelligible)

    1. Velocity is a parallel rate.

    The distance/time formula with which we are familiar is an expression of the number of meters laid end to end alongside one second. For example, another parallel rate would be the number of traffic signs alongside a kilometer of roadway.

    2. Parallel rates have reversible numerators and denominators.

    1 meter per second is the same as 1 second per meter. Looking back at my example of the roadway, we can see that 4 signs per kilometer is the same as .25 kilometers per sign.

    So the fundamental issue of velocity is how we address the inevitable increase of one component with respect to the other.

    Imagine that an object's velocity increases from 1 m/s to 2 m/s.

    All of a sudden there is twice as much distance alongside the same 1 second of time.

    Was this accomplished by the squeezing of 2 meters into one second?

    Or was it accomplished by the stretching of that second out to a distance of 2 meters?
  2. jcsd
  3. Jul 31, 2006 #2
    No takers?

    Perhaps I am not explaining it adequately.

    Look at the figure below.



    Red = time
    Black = distance

    In the top figure, the velocity is 6 units per second. The bottom figure shows 3 units per second.

    In terms of metrics, however, the distance units of the two velocities are different. With respect to a third measurement (pixels in this case) the time units are the same in length. The higher velocity of the upper figure was accomplished by shrinking the distance unit so that more could fit within one second.

    There is an alternate version.

    Look at the figure below.


    In this illustration, the distance units are standardized. The mechanism which differentiates between the velocities here is the number of distance units one second is 'stretched' to cover.

    Which is the mechanism that distinguishes one velocity from another?

    Or are they both correct?
    Last edited: Jul 31, 2006
  4. Aug 1, 2006 #3
    I think I figured out the answer myself.

  5. Aug 2, 2006 #4


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    Staff: Mentor

    Moving clocks run faster.

    I deal with satellites and GPS systems. One of the major concerns is that the clocks on the satellites run faster than stationary ground clocks. The solution has been to program the satellite clocks to run slower. They still need to be re-synchronized periodically.

    According to the IEEE "At an altitude of. 20 184 km, a satellite clock runs fast relative to a clock on the geoid at a rate of 45.7 µs per day."

    Unfortunately you have to be a member to view the document.

    I found this though. This should explain why moving clocks go faster.

  6. Aug 2, 2006 #5

    Moving clocks run slower according to that paper. The data for GPS that shows the clocks
    running faster is a result of the gravitational red shift time dilation (faster) overshadowing
    the velocity time dilation (slower).The satellites
    at lower elevation is less redshifted for example and shows a slower
    moving clock. Basically they assumed they could add the time dilation effects together.
    Last edited: Aug 2, 2006
  7. Aug 2, 2006 #6
    I think I figured out the answer myself.

    "Moving clocks run slow, moving meter sticks are shortened..."

    Tim Halpin-Healy
    Theoretical Physicist
    Barnard College, Columbia University

    Well, let's apply that explanation to a test experiment with
    two or three or more cars racing from one stop light to the next.
    One car has twice the velocity as the other. It's apparent the
    kilometer distance between stop lights doesn't change during the race.
    It's also apparent that the car traveling twice as fast doesn't
    shrink to half the first car's size. So this should be enough to falsify
    the part about moving sticks shortening. Now, if
    the drivers were wearing stop watches, they could also time their
    race. Afterwards they compare the stop watches, they would indeed
    find the faster car to have recorded half the time as the slower car.
    So empirically, it would seem like a change in velocity is better
    viewed as a change in time rate- or real time dilation without space

    Another analogy would be like a movie. A 2 and a half hour reel of
    film for example could be played at normal speed or at fast forward.
    At all times , if matter is conserved, that reel of film is one length.
    Played fast forwarded, it would finish playing earlier than at normal
    speed. So again, time dilation is apparent but not length or space
    Last edited: Aug 2, 2006
  8. Aug 2, 2006 #7


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    Staff: Mentor

    No, moving clocks run faster. "The results of the first flight test, shown in Figure 8, are representative of our results in general. The
    aircraft flew for nearly 4 hours at an altitude of 11,000 m (35,000 ft) at an average speed of 790 km/h
    (425 knots). Our analytical results suggest that the flight clock should have run fast by 11.46 ns due to
    the difference in gravitational potential, slow by 3.03 ns due to velocity, and slow by 0.01 ns due to the
    net Sagnac effect, resulting in a total flight clock change of 8.42 ns (fast). Preliminary measured test
    results, after calibrating the estimated clock drift between the two clocks, showed that the flight clock ran
    fast by 8.3 ns relative to the ground clock.
    The difference between the predicted and measured values is
    consistent with clock noise on the order of a few parts in 1014 over the flight time interval. The
    anomalous dip of about 3 ns at time 9,000 s is not inconsistent with typical cesium clock behavior."

    There are multiple examples all showing the clocks running faster. This is a fact, we have to modify clocks orbiting the earth because they run fast when compared to the ground clocks.
  9. Aug 2, 2006 #8

    On page 204, the ISS and TopEx satellites have greater velocity
    time dilation than gravitational red shift so their clocks run slower.
    SR is an observer dependent theory and is kind of nonsense though so not
    sure how that velocity time dilation comes about. Could be some mechanical
    attribute of the clock or could be real time dilation.
    Last edited: Aug 2, 2006
  10. Aug 3, 2006 #9


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    Gold Member

    What is the difference? This sounds to me like two ways of stating exactly the same result. You can probably see this by stating it more accurately, because neither the second nor the meter are doing anything. They are simply units of measurement and not physical actors. What you should really be asking is "Does X travel 2 meters over the course of 1 second or does X experience the passage of 1 second over the course of travelling 2 meters?" The answer is both; each statement is equivalent to the other.
  11. Aug 3, 2006 #10
    I agree that both statements are equivalent to each other.

    My question concerned the mechanics of velocity.

    For example, let's say we have a long length of rubber tubing. This tubing will represent one second. We are on a city block and there is a fire hydrant midway between every intersection. The tubing is currently stretched along a length of one block, from one intersection to the next. It can therefore be said that the 'rate' at which we encounter fire hydrants is one hydrant per second (or one second per hydrant). Now let's say we want to double the rate at which we encounter fire hydrants. This can be accomplished by stretching the rubber tubing out to the next intersection, thereby placing another fire hydrant alongside the same one second. Or we can call city hall and have another fire hydrant erected alongside the rubber tubing as it currently is.

    Both actions have the same result - the rate is increased to 'two hydrants per second,' but they are different mechanically. I originally envisioned that velocity was like the former, where if we are dealing with meters instead of city blocks, a doubling of velocity is a stretching of one second over twice the distance. I hoped that this would have built in relativity, as the stretching of one second would effectively apply to observers. If a stretched second for a rocketship is three times longer with respect to an observer's then, from the observer's point of view, he ages three seconds for every one the spaceship crew does. I then read that time dilation is not a linear relation.

    The only other possibility I could think of was that a shrinking of distance units accompanied a stretch in time (with respect to an observer). This was confirmed by Dr. Halpin-Healy.
    Last edited: Aug 3, 2006
  12. Aug 3, 2006 #11
    I made a drawing to show what I am referring to:


    As before, black is distance and red is time.

    Figure 1 is the base velocity, used for comparison, one distance unit per second.

    Figure 2 is the velocity increased to 2 units per second. The time unit is constant but distance is compressed.

    Figure 3 is also 2 units per second. In this figure the distance unit is constant but time is stretched.

    As you can see, figures 2 and 3 both represent 2 units per second, but are fundamentally different.

    Both retain one component from the base velocity but need to alter the other to double the velocity.
    Last edited: Aug 3, 2006
  13. Aug 13, 2006 #12
    They are only different "mechanically" (to use your expression) because you have an in-built frame of reference against which you can measure both the length of tubing and the separation of fire hydrants - the city blocks. Take away that frame of reference and you are left with just tubing and hydrants - and now there is absolutely no difference between "doubling the number of hydrants per length of tubing" and "stretching the tubing to extend along twice as many hydrants" - the two operations are equivalent.

    They only become inequivalent when you insert another reference frame, such as the city blocks, against which you can compare either or both hydrants and tubing.

    Best Regards
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