I had a dream about this.

Would a cart going down a friction-less slope that went on forever eventualy reach the speed of light?

Mcloud
Welcome to sciforums :)

In response...

Possibly. But you would have to make more of the characteristics of the scenario ideal.
Frictionless bearings in the cart, absolute vacuu, and so on.

Interesting question :)

Hi Mcloud,

The experiment is still to be conducted, but from the theories we have the answer would be: no, the car will never reach the speed of light (in vacuum). It will go faster and faster (assuming there always is some gravitational interaction that pulls the car down the slope), and asymptotically approach the speed of light. However, it would take an infinite amount of time to actually reach the speed of light.

Bye!

Crisp

Correct.
Yet in the first post on the thread, we are told that the cart will descend for ever.
So t can be taken as being close enough to infinite as to not make a difference.

Hang on. No friction, no air resistance, continually shooting down. If we assume into a gravity well, it is continually accelerating. Regardless of what speed you set as your benchmark, it will reach it. It is continually accelerating. The leaps in speed in m/s are continually growing. How could it not reach lightspeed, or any given speed?

Originally posted by Adam
Hang on. No friction, no air resistance, continually shooting down. If we assume into a gravity well, it is continually accelerating. Regardless of what speed you set as your benchmark, it will reach it. It is continually accelerating. The leaps in speed in m/s are continually growing. How could it not reach lightspeed, or any given speed?

Very valid.

I concur.
Yet ~grins with anticipation~ to play devils advocate, what happens after that?
We reach c and keep accelerating? Do we go FTL?

Your reasoning for the initial question is flawless, I just want to know "And then...?"

Erm... it should reach velocity of light i presume...,continous acceleration,continous gravititional pull...why it shouldnt,Crisp.?

please explain...:confused:



bye!

I have a very Newtonian mind. *shrug*

Hi all,

Important remark: from a Newtonian point of view, and this is what our "daily intuition" pretty much comes down to, you would expect that the velocity increases towards infinity:

x(t) = x₀ + (1/2)*a*t²

Unfortunately, at high speeds, this formula no longer works out that well. At higher speeds you need to correct the reasoning above. I won't go too much into relativistic dynamics (that covers acceleration), simply because I never studied it a lot, but you can see it goes wrong in the following relativistic formulas:

We know the energy of a particle/object is given by:

E = m₀c² / sqrt( 1 - (v/c)²)

Solving this for v² gives:

v² = (1 - m₀²c⁴ / E²)*c²

What the gravitational interaction does is increase the energy. There is no upper limit to the energy, so its contineous increase can be seen as the limit E -> infinity. Taking the limit E -> infinity makes the second term between the brackets disappear and we find:

v = c (for E = infinity)

The limiting speed is hence the speed of light in vacuum, but to attain it we need to supply an infinite amount of energy. Since infinity is very (very, VERY) large, we'll need a long time to add that amount of energy... an infinite amount of time actually. So to conclude: the vehicle/object/particle will always increase its velocity towards the speed of light, but it will never actually reach that speed.

I hope this explains things a bit.

Bye!

Crisp

ok, don't now where, but i recall i've read something like the following:

when you reach speeds near the speed of light, you become much heavier (due to the enormous amount of energy you 'absorb' orso?????).....and then you need more energy to accelerate, you'll get heavier, more energy etc. etc.
this makes it impossible to reach the speed of light...

No I don't remember all of it....but his it about everything I know about it.
Am i totally wrong here or is there really some speed-weight theory??

the answer is no as terminal velocity with gravity is when the force of air resistance pushing up against the skydiver is equal to the force of gravity pushing her downward or roughly 120 miles per hour. Body weight , size ,and how you lay effect that. But if you go to space or a place with no gravity than no force would push you down.

Common sense will tell you that in order to exceed the speed of light you just accelerate two different objects to more than half the speed of light and have them travel towards one another, thus obtaining a relative speed greater than c. If for example an object is moving at 55 MPH in one direction and another object is moving at 55 MPH in the opposite direction, they're combined speeds will equal 110 MPH. But common sense does not prevail here because the universe doesn't work that way. The fabric of space and time is such that no one observer ever sees another object moving at speeds greater than c. Combined velocities of any two objects moving towards one another will always = < c.

As an object approaches the speed of light, its relativistic mass increases. That is not to say that the actual mass of the object (rest mass) begins to increase, or gets 'heavier' but instead certain properties of the mass increase. Those properties include the relativistic momentum and relativistic kinetic energy. The amount of energy required to accelerate mass to attain light speed reaches infinity. And the only known source of energy that is considered infinite enough is the energy contained in a 'Big Bang.'

The Einstein velocity addition expressions are used to show that the relative velocity of any two objects never exceeds the velocity of light. One would apply the Lorentz transformation to the different velocities to express the relative velocities as seen by the different observers. However, in most applications, Newtonian physics can be used to calculate velocities to about 85-90% the speed of light. Once the velocities or combined velocities exceed 90% the speed of light, the velocity addition expressions should be used. It is within this window of near speed of light velocities that most drastic changes occur; ie. time dilation, length contraction, relativistic mass increases and relative velocities.

I just want to make clear that the cart's own energy increases. It's energy requirements do not.

Originally posted by (Q)
If for example an object is moving at 55 MPH in one direction and another object is moving at 55 MPH in the opposite direction, they're combined speeds will equal 110 MPH. But common sense does not prevail here because the universe doesn't work that way. The fabric of space and time is such that no one observer ever sees another object moving at speeds greater than c. Combined velocities of any two objects moving towards one another will always = < c.

And the only known source of energy that is considered infinite enough is the energy contained in a 'Big Bang.'


ok Q, you're raising some questions now:
when we can calculate up to 85% of light-speed, why can't the observer see someone relatively going faster than c? As long as they both travel at 1/2 c, I can't of any phusic-law to 'forbid' it.

another note about the big-bang thing...I don't think you are right there, as long as e=mc^2 is valid....an infinite amount of energy at the big-bang will give us an infinite amount of energy in space, combined with an infinte amount of mass....that's impossible as far as I know:
an infinite amount of energy at the start will give the expansion of space an infinite acceleration, space will be infinite, with infinite mass....which will be impossible because of the acceleration....
phew, I know, I am making some mistakes here, but the main point is true: There is no such thing as an infinite amount of energy.

space ya!

why can't the observer see someone relatively going faster than c?

The speed of light is the ultimate speed limit in the universe. Neither of the observers can travel faster than c, in fact they can only approach c, at best.

As observer (A) approaches c, he sees observer's (B) clock slowing down and practically stopping. Observer (A) also sees the distance between himself and observer (B) in his direction of motion become infinitesimally small. From his (A) point of view he is able to get to observer (B) in practically no time. In pother words, if observer (A) were to hypothetically reach the speed of light, he would, from his point of view, get to observer (B) instantaneously. These are relativistic effects caused by length contraction and time dilation and are why observer (A) cannot see (B) moving faster than c.

All of the above effects are the same for observer (B) from his own point of view.

another note about the big-bang thing...I don't think you are right there

Probably not, that was more or less a tongue-in-cheek comment to help visualize the magnitude of energy required. In my opinion, the only infinite source of energy imaginable is that contained in a Big Bang.

It could arguably be infinitely finite. ;)

With the sliding-down-a-slope thing, the acceleration would not be constant, even with no friction. The acceleration would gradually decrease as the speed approached the speed of light, relative to a stationary observer.

Crisp,

can you solve the following limit:

M = LIMIT V->C Mo/(SQRT(1-V^2/C^2))??

tHIS IS SUPPOSE WILL ANSWER MOST OF THE THREAD ISNT IT?


bye!

Hi zion,

Yes, the answer is infinity.

But please don't mistake the "M" (which denotes relativistic mass in this formula) for the restmass. You don't get "heavier" when going faster in the sense that you or your spaceship gain extra matter.

Bye!

Crisp

Err...yeah M is relativistic mass i know,but whats the point Crisp?couldnt get you...

please explain ...:confused:


bye!

Hi zion,

Allow me to quote (Q)'s post from a few days back:


As an object approaches the speed of light, its relativistic mass increases. That is not to say that the actual mass of the object (rest mass) begins to increase, or gets 'heavier' but instead certain properties of the mass increase. Those properties include the relativistic momentum and relativistic kinetic energy. The amount of energy required to accelerate mass to attain light speed reaches infinity.

So the formula for M you gave is completely equivalent with the energy formulation: an infinite amount of energy is required, there is no such thing as adding an infinite amount of energy to an object, hence the vehicle will never reach the speed of light (it will approach the speed of light more and more though). The reason I repeated that M does not stand for restmass is simply because often people reason that you actually gain restmass when accelerating, while this is ofcourse not true.

Anyway, that is the answer Special Relativity gives us. One is ofcourse free to believe it or not :).

Bye!

Crisp

I'm not sure that I understand Crisp either.

I don't see what the atomic energy inside the object has to do with it's potential maximum speed (not to be confused with velocity...we are talking speed here?)

Gravity does not increase the potential energies of an object, just changes them from one to another.
So if you have an infinitly high slope in an infinite gravity field then the Ep is infinite. As the object moves down the slope, Ep changes to Ek eventually giving infinite Ek and so infinite speed.

But how does an infinite amount change to 0? Infinite means "has no end", how can a value with no end reach 0? Where do we draw the line, where does the Ep change to 0? As you can see, infinities bring up all sorts of problems, and the only way to solve this question is by using infinities. Im a firm believer in the idea that infinities in any equation mean we have either done something wrong, or the equation cannot be applied to that problem.

Also, if we have an infinite Ep, a finite Ek and an infinite time period in which the object could potentially reach c, it's obvious it will not. It's not as easy as dividing out infinitiy/infinity=1. If it's there for an infinite amount of time, it will never reach c because Ek will never reach infinity. Am I coming across ok here?

Infinities are often defined in terms of limits of infinite sequences. Dividing one infinity by another infinity implies dividing the terms of the first sequence by the terms in the second and then looking at the limit. The limit might turn out to be infinity, zero, or any other number depending on the sequences involved. (Or the limit might not exist. The “limit” value might not “settle” on any value.)

Some of the “mathematical” tricks in physics aren’t formally correct. They work only because of additional properties of the universe that are seldom formally stated. (The average physicist would have to spend too much effort studying arcane math to understand those other properties. And for what purpose? All the systems the physicist looks at have those properties. Why should the physicist care about the weird spaces that don’t have those properties?)

Physicist: “So is my answer correct?”
Mathematician: “Well…yes. But only because…”
Physicist: “Whatever.”

Hi all,

Esp,

"I don't see what the atomic energy inside the object has to do with it's potential maximum speed (not to be confused with velocity...we are talking speed here?)"

Sorry, in my native language there is no difference between "speed" and "velocity", that probably gives rise to some confusion as I mix the terms. I'm talking speed ofcourse (the size of the velocity vector).

Now, even a macroscopic object has energylevels just like an atom does. A collection of atoms has different energylevels than the individual atoms, but is has an energyspectrum nevertheless. However, that is not really relevant for the discussion: in Newtonian and Relativistic mechanics, energy is associated to an entire object, and is not immediatelly related to energylevels of the constituting atoms.

Both Newtonian and Relativistic mechanics have no upper limit to the energy a massive object can have. The major difference between the two is the limitation of speed Relativistic mechanics has (and we know from experiments that Relativistic mechanics from the Special theory of Relativity is closer to reality than Newtonian mechanics).


Xelios,

"But how does an infinite amount change to 0? Infinite means "has no end", how can a value with no end reach 0? Where do we draw the line, where does the Ep change to 0?"

Potential energy is always relative to a chosen energylevel. Most of the times the value Ep = 0 is associated at infinity. However, we don't need that either for the reasoning:
- As we all (seem to) agree on, an infinite slope with no friction, ... has the possibility to push the kinetic energy of the object (which in relativistic mechanics is actually the only "real energy", i.e. observable). As the object falls/glides down the slope, it gains more and more energy because it is contineously accelerating.
- This (kinetic) energy increases every second, but it will never get infinitely large (= the energy needed to gain lightspeed). It doesn't matter how long we wait, the energy will never reach the value "infinity"... It will get very large, but since we add a finite amount of kinetic energy every second, it takes an infinite amount of seconds to get an infinite amount of energy.


Imahamster2,

"Some of the “mathematical” tricks in physics aren’t formally correct. "

I think this is formulated in an unfortunate way: I'd personally say: "some of the mathematical tricks in physics aren't formally proven yet, but by experiment have been proven to work in the scenarios we can think up at the moment" ;). Also, there's nothing in the reasoning stated above that needs some dirty non-mathematical trick to work, it can be proven very nicely.


Bye!

Crisp

If we take another look at [b]Mcloud]/b] thought experiment, we know that the cart is moving down a frictionless slope. In other words, the cart is not powering itself. It is relying on a gravitational source. Therefore for the purpose of the thought experiment, we can remove the slope altogether and imagine the cart freefalling towards a source of gravity. The source must be massive enough to pull the cart towards it in order to reach near light speed. Does this sound familiar?

Essentially we are talking about an object accelerating towards something massive, a black hole for example.

well I havent read the entire thread due to laziness, but I thought I'd mention that if space could be folded then it would be possible to exceed the speed of light, but only relative to the outside because the actual object would still have mass....blah blah...blah....

and if you were in space, why would you want a frictionless surface to travel on?

and I think that if this frictionless incline plane where on a large enough body, like, oh, say a black hole, then maybe..........
well all I'm saying is there probably is an exeption even though I'm to lazy to chekc em all out right.

I'll be back with an idea:rolleyes: soon..........

Crisp, didn’t mean to imply you made any math “errors”. Was just remembering with amusement early hamster exposure to some mathematics in physics books where terms were pulled inside and outside integrals and re-ordered with no apparent concern for whether the operations were valid. They were valid but a physics student learning mathematics from those examples might believe such operations were always valid. Understanding the mathematical reason why those operations were valid was beyond the math level of the target audience of the physics book. Certainly the theoretical physicists who derive mathematical physics do understand the math far better than any hamster.

Would a cart going down a friction-less slope that went on forever eventualy reach the speed of light?

Since this can't happen... why bother worring about it?... :p

Interesting question though... ;)

Well... if the cart goes on forever, and it accelerates... it most likely will get to the speed of light, unless there is a limit for an actual speed... what I don't think so...
We can sucessfully accelarate a particle only until 99.99999...% of the speed of light... Perhaps that's the limit for your cart buddy... :D

If a little particle can, why not an object? ;)

Love,
Nelson

Q has raised a good point Here.Certainly under free fall we would approach velocity of light.true.



bye!