Paradox - An Ant on a Rubber Rope

I am sure you know this paradox quite well that's why you are here to read this post but I think you are not able to grasp the solution of the first order differential equation.

Let's forget the calculus solution and do it with intuitive way. It might not provide you the solution but we may have great discussion.

To solve the complex problems like this, break the problem in chunks so start with the shorter version.

An ant is crawling on a stretchable rope at a constant speed of 1 cm/s. The rope is initially 4 cm long and stretches uniformly at a constant rate of 2 cm/s. When will the ant reach to the initial end point (Corner of the 4 cm rope)?

Initially the ant is at the starting point of the 4 cm rope.

After 1 second, the ant travels 1 cm on the rope. Now here consider the rope does not stretch continuously. It means the rope is stretched 2 cm after the end of every second (The stretching time is negligible i.e. a Femtosecond). This thought might change the whole approach but it can be quite easy to understand because if the rope would stretch continuously then it would stretch all the smallest slots of the second as the ant moves ahead. 

So After 1 second, the ant has traveled 1 cm, and the rope is stretched 2 cm uniformly. It means every one cm of the rope is stretched 0.5 cm so complete 4 cm is stretched to 6 cm.

It makes ant's traveled distance 1 cm to 0.5 cm more stretched so after 1 second the ant has traveled 1.5 cm.

After 2 seconds, the ant travels 1 more cm. So the ant is at the 2.5 cm from the start but the rope is stretched again 2 more cm after the end of 2 seconds. This time this increased 2 cm distance will be distributed between existing 6 cm uniformly. It means every one cm of the rope is stretched 0.3333 cm so complete 6 cm is stretched to 8 cm.

 

It makes ant's traveled distance 2.5 cm to '2.5 x 0.33333 = 0.8333'  more stretched so after 2 seconds the ant has traveled 2.5 + 0.83333 = 3.3333 cm and so on

By this way, the ant will reach to the initial end point between 9 and 10 Seconds.

But if you solve this question using calculus, the answer is 12.77 Seconds

L=4 cm (Initial Length of the rope)

v=2 cm/sec (Rope stretching Speed)

u=1 cm/sec (Ant's speed)

 

There is difference in the answers, this is because we took the different approach but it will give you an idea how you will try next time to solve this question for your own satisfaction.

Approach : 2

Initially the ant is at the starting point of the 4 cm rope.

a) After 1 second, the ant travels 1 cm on the rope.

So After 1 second, the ant has traveled 1 cm, and the rope is stretched 2 cm uniformly. It means every one cm of the rope is stretched 0.5 cm. This 1 cm has been traveled by the ant so stretching distance will be (1x0.5)/2=0.25.

Ant will travel 1.25 cm after 1 second.

b) After 2 seconds, the rope is stretched again 2 more cm. This time this increased 2 cm distance will be distributed between existing 6 cm uniformly. It means every one cm of the rope is stretched 0.3333 cm so complete 6 cm is stretched to 8 cm.

If ant would stand still at 1.25 cm then it would have been moved to 1.25x0.3333=0.41666 cm more that is 1.25+0.416666=1.666666667 cm but ant is moving with speed 1 cm/sec so this 1 cm distance traveled by the ant will be stretched with (1x0.3333)/2=0.16666 so the total distance moved by ant after 2 seconds will be 1.666666+1+0.166666 =2.833332 cm

c) After 3 seconds, the rope is stretched 2 more cm. This time this increased 2 cm distance will be distributed between existing 8 cm uniformly. It means every one cm of the rope is stretched 0.25 cm so complete 8 cm is stretched to 10 cm.

If ant would stand still at 2.83333 cm then it would have been moved to 2.83333x0.25=0.7083325 cm more that is 2.8333333+0.7083325=3.5416658 cm but ant is moving with speed 1 cm/sec so this 1 cm distance traveled by the ant will be stretched with (1x0.25)/2=0.125 so the total distance moved by ant after 2 seconds will be 3.5416658+1+0.125 =4.6666658 cm

d) After 4 seconds, the rope is stretched 2 more cm. This time this increased 2 cm distance will be distributed between existing 10 cm uniformly. It means every one cm of the rope is stretched 0.2 cm so complete 10 cm is stretched to 12 cm.

If ant would stand still at 4.6666658 cm then it would have been moved to 4.6666658x0.2=0.93333316 cm more that is 4.6666658+0.93333316=5.59999896 cm but ant is moving with speed 1 cm/sec so this 1 cm distance traveled by the ant will be stretched with (1x0.2)/2=0.1 so the total distance moved by ant after 2 seconds will be 5.59999896 +1+0.1 =6.69999896 cm and so on.

By this way, the ant will reach to the initial end point between 12 and 13 Seconds.

This is an almost same time calculated by calculus solution.

The difference between approach 2 and previous approach is the stretching distance calculation of ant's traveled distance. We took half of the stretching distance in approach 2 and it gets us near the exact answer.

Case II

Now Let's discuss one easy condition in this question. Suppose the ant is at the fixed distance from the initial point. Let's say 1 cm from the initial point but the ant does not move a bit. Only stretching factor of the rope makes it move further.

One thing is clear in this condition, the ant will never reach to the end point.

But it will never be at the 1 cm always so how much will it move.

At 0th second, it is 1 cm from the start.

After 1 second, rope is stretched 2 more cm. This 2 cm is distributed between existing 4 cm uniformly that makes the total length 6 cm. Every cm is stretch to 0.5 cm so the ant will be at 1.5 cm from the start. That is 25% of the total length.
75% remaining.

After  2 seconds, rope is stretch 2 more cm. This 2 cm is distributed between existing 6 cm uniformly that makes the total length 8 cm. Every cm is stretch to 0.3333 cm so the ant will be at 2 cm from the start. That is 25% of the total length. 75% remaining.

After  3 seconds, rope is stretch 2 more cm. This 2 cm is distributed between existing 8 cm uniformly that makes the total length 10 cm. Every cm is stretch to 0.25 cm so the ant will be at 2.5 cm from the start. That is 25% of the total length. 75% remaining.

It means if the ant does not move, The 'distance to cover' ratio does not change and that's why if the ant starts moving very slowly, it will always reach up to end,  if sufficient time is given and stretching speed is constant.

Case III

Now there is another condition. Suppose the rope gets doubled after every second. Speed of the ant is same i.e. 1 cm / second and initial length of the rope is also 4 cm.

What will happen now ? Will it reach up to the end point?

After 1 second, length of the rope is 8 cm. It means increased 4 cm is distributed between existing 4 cm uniformly. Every one cm of the rope is stretched to 1 cm. The ant will be at 2 cm but length of the rope will be 8 cm.

After 2 seconds, length of the rope is 16 cm. It means increased 8 cm is distributed between existing 8 cm uniformly. Every one cm of the rope is stretched to 1 cm. The ant was at 2 cm and now it travels 1 more cm in 2nd second, as the rope is stretched, the ant will be at 6 cm but length of the rope will be 16 cm.

After 3 seconds the ant will be at 14 cm from the start and length of the rope will be 32 cm.

After 4 seconds the ant will be at 30 cm from the start and length of the rope will be 64 cm.

The stretching speed of the rope is not constant, it is accelerating and that's why the ant will never reach up to the end point.

 

 

Interstellar (2014) - Calculations and Analysis

                                             Interstellar (2014)
                                              Welcome to the Fountain
                                           Quench Increase your Thirst

Fact : 1
Miller's Planet to outside observers orbits Gargantua every 1.7 hours. On Miller's Planet, that means the planet orbits ten times a second around Gargantua , which is normally faster than the speed of light. But since the spin from Gargantua caused space to whirl around it similar to wind, Miller's Planet does not travel faster than light relative to its space as the laws of physics say you cannot travel faster than light relative to space, but space itself is not bound by the speed limit. As such, faster than light travel is possible by bending and twisting space. However, Gargantua would have to fill half the sky in order for it to be so close.

Fact : 2
The time dilation on Miller due to the gravitational forces of Gargantua would be tantamount to the planet moving through empty space at roughly 99.99999998% the speed of light. 

Fact : 3
Gargantua’s mass must be at least 100 million times bigger than the Sun’s mass. If Gargantua were less massive than that, it would tear Miller’s planet apart. The circumference of a black hole’s event horizon is proportional to the hole’s mass. For Gargantua’s 100 million solar masses, the horizon circumference works out to be approximately the same as the Earth’s orbit around the Sun: about 1 billion kilometers.

Fact : 4
Miller’s planet is about as near Gargantua as it can get without falling in and if Gargantua is spinning fast enough, then one-hour-in-seven-years time slowing is possible. But Gargantua has to spin awfully fast. There is a maximum spin rate that any black hole can have. If it spins faster than that maximum, its horizon disappears, leaving the singularity inside it wide open for all the universe to see; that is, making it naked—which is probably forbidden by the laws of physics

Fact : 5
Einstein’s laws dictate that, as seen from afar, for example, from Mann’s planet, Miller’s planet travels around Gargantua’s billion-kilometer-circumference orbit once each 1.7 hours. This is roughly half the speed of light! Because of time’s slowing, the Ranger’s crew measure an orbital period sixty thousand times smaller than this: a tenth of a second. Ten trips around Gargantua per second. That’s really fast! Isn’t it far faster than light? No, because of the space whirl induced by Gargantua’s fast spin. Relative to the whirling space at the planet’s location, and using time as measured there, the planet is moving slower than light, and that’s what counts. That’s the sense in which the speed limit is enforced.


QUERIES 
1. How old is Miller’s planet? If, as an extreme hypothesis, it was born in its present orbit when its galaxy was very young (about 12 billion years ago), and Gargantua has had its same ultrafast spin ever since, then the planet’s age is about 12 billion years divided by 60,000 (the slowing of time on the planet): 200,000 years. This is awfully young compared to most geological processes on Earth. Could Miller’s planet be that young and look like it looks? Could the planet develop its oceans and oxygen-rich atmosphere that quickly? If not, how could the planet have formed elsewhere and gotten moved to this orbit, so close to Gargantua?

2. What is the gravitational time dilation equation for Miller's planet? As there is 60000 ratio between time on earth and time on Miller's planet , to balance the equation what should be the distance of planet from Gargantua, angular momentum as it is revolving around very fast spinning object and Mass of Gargantua ?

3. We all know Gravitational Time Dilation does not affect Mann's planet as it is far from Gargantua’s vicinity. But we also know, almost immediately after the Endurance’s explosive accident in orbit around Mann’s planet, the crew find the Endurance being pulled toward Gargantua’s horizon. From this it appears when crew leaves Mann’s planet, the planet must be near Gargantua. Following diagram is the orbit of Mann's Planet.

According to this, what should be the orbital period and orbital velocity of Mann's planet for a person on Earth and a person on Mann's planet.

4. How long Dr. Mann spent time on Mann's planet according to him and according to an observer on Earth(Keep its orbital path in mind)? How long did he spend in hibernation for both observers?

5. Why was Endurance able to receive signal from Earth but Earth was not able to receive signal sent by Endurance.

6. When Cooper left Earth, he was 35 Years old and when he returned, He was 124 years old for Murph. 35 + 2 years for Saturn Journey + 23 years for Miller's Planet Journey + 51 years for Black Hole Journey = 111, Where are 13 Years Missing? How Long was Cooper out for himself?

8. What is the Orbital velocity of Miller's planet for a person on Earth and a person on Miller's planet?

9. What is the age of Brand when Cooper arrives at Edmund's Planet?

10. How long Dr. Laura Miller spent time on Miller's planet according to her and according to an observer on Earth? Lets keep in mind that she died minutes ago before Cooper and Brand reached there.

11. Miller’s planet travels around Gargantua’s billion-kilometer-circumference orbit once each 1.7 hours. Could Rom see it moving very fast from Mothership?

12. If Coop and team would try to communicate with Rom from Miller's planet, how would their communication appear? According to Coop how fast would they get response from Rom and similarly how long would Rom get response from Coop & team?

Space Time

Frequently Asked Questions

What is light?

Light is a phenomenon that has particle and wave characteristics. Its carrier particles

are called photons, which are not really particles, but massless discrete units of

energy.

What is the speed of light?

The speed of light is 299,792,458 m/s in a vacuum. The symbol used in relativity for

the speed of light is "c", which probably stands for the Latin word "celeritas",

meaning swift.

Is the speed of light really constant?

The speed of light is constant by definition in the sense that it is independent of the

reference frame of the observer. Light travels slightly slower in a transparent

medium, such as water, glass, and even air.

Can anything travel faster than light?

No. In relativity, c puts an absolute limit to speed at which any object can travel,

hence, nothing, no particle, no rocket, no space vehicle can go at faster-than-light

(=superluminal) speeds. However, there are some cases where things appear to move

at superluminal speeds, such as in the following examples: 1. Consider two spaceships

moving each at 0.6c in opposite directions. For a stationary observer, the distance

between both ships grows at faster-than-light speed. The same is true for distant

galaxies that drift apart in opposite directions of the sky. 2. Another example:

Consider pointing a very strong laser on the moon so that it projects a dot on the

moon's service and then moving the laser rapidly towards earth, so that it points on

the floor in front of you. If you accomplish this in less than one second, the laser dot

obviously traveled at superluminal speed, seeing that the average distance between

the Earth and the Moon is 384,403 km.

What is matter?

The schoolbook definition would be: Matter is what takes up space and has mass.

Matter as we know it is composed of molecules, which themselves are built from

individual atoms. Atoms are composed of a core and one or more electrons that spin

around the core in an electron cloud. The core is composed of protons and neutrons,

the former have a positive electrical charge, the latter are electrically neutral. Protons

and neutrons are composed of quarks, of which there are six types: up/down,

charm/strange, and top/bottom. Quarks only exist in composite particles, whereas

leptons can be seen as independent particles. There are six types of leptons: the

electron, the muon, the tau and the three types of neutrinos. The particles that make

up an atom could be seen as a stable form of locked up energy. Particles are extremely

Spacetime, small, therefore 99.999999999999% (or maybe all) of an atom's volume is just empty

space. Almost all visible matter in the universe is made of up/down quarks, electrons

and (e-)-neutrinos, because the other particles are very unstable and quickly decay

into the former.

How fast does an electron spin?

An electron in an hydrogen atom moves at about 2.2 million m/s. With the

circumference of the n=1 state for hydrogen being about 0,33x10-9 m in size, it

follows that an n=1 electron for a hydrogen atom revolves around the nucleus

6,569,372 billion times in just one second.

Are quarks and leptons all there is?

Not really. Fist of all, quarks always appear in composite particles, namely hadrons

(baryons and mesons), then there is antimatter, and finally there are the four

fundamental forces.

What is antimatter?

The existence of antimatter was first predicted in 1928 by Paul Dirac and has been

experimentally verified by the artificial creation of the positron (e+) in a laboratory in

1933. The positron, the electron's antiparticle, carries a positive electrical charge. Not

unlike a reflection in the mirror, there is exactly one antimatter particle for each

known particle and they behave just like their corresponding matter particles, except

they have opposite charges and/or spins. When a matter particle and antimatter

particle meet, they annihilate each other into a flash of energy. The universe we can

observe contains almost no antimatter. Therefore, antimatter particles are likely to

meet their fate and collide with matter particles. Recent research suggests that the

symmetry between matter and antimatter is less than perfect. Scientists have

observed a phenomenon called charge/parity violation, which implies that antimatter

presents not quite the reflection image of matter.

What are the four fundamental forces?

The four fundamental forces are gravity, the electromagnetic force, and the weak and

strong nuclear forces. Any other force you can think of (magnetism, nuclear decay,

friction, adhesion, etc.) is caused by one of these four fundamental forces or by a

combination of them.

What is gravity?

Gravity is the force that causes objects on earth to fall down and stars and planets to

attract each other. Isaac Newton quantified the gravitational force: F = mass1 * mass2

/ distance2. Gravity is a very weak force when compared with the other fundamental

forces. The electrical repulsion between two electrons, for example, is some 10^40

times stronger than their gravitational attraction. Nevertheless, gravity is the

dominant force on the large scales of interest in astronomy. Einstein describes

gravitation not as a force, but as a consequence of the curvature of spacetime. This

means that gravity can be explained in terms of geometry, rather than as interacting

forces. The General Relativity model of gravitation is largely compatible with Newton,

except that it accounts for certain phenomena such as the bending of light rays

correctly, and is therefore more accurate than Newton's formula. According to

Spacetime,General Relativity, matter tells space how to curve, while the curvature of space tells matter how to move. The carrier particle of the gravitational force is the graviton.

What is electromagnetism?

Electromagnetism is the force that causes like-charged particles to repel and

oppositely-charged particles to attract each other. The carrier particle of the

electromagnetic force is the photon. Photons of different energies span the

electromagnetic spectrum of x rays, visible light, radio waves, and so forth. Residual

electromagnetic force allows atoms to bond and form molecules.

What is the strong nuclear force?

The strong force acts between quarks to form hadrons. The nucleus of an atom is hold

together on account of residual strong force, i.e. by quarks of neighboring neutrons

and protons interacting with each other. Quarks have an electromagnetic charge and

another property that is called color charge, they come in three different color

charges. The carrier particles of the strong nuclear force are called gluons. In contrast

to photons, gluons have a color charge, while composite particles like hadrons have

no color charge.

What is the weak nuclear force?

Weak interactions are responsible for the decay of massive quarks and leptons into

lighter quarks and leptons. It is the primary reason why matter is mainly composed of

the stable lighter particles, namely up/down quarks and electrons. Radioactivity is

due to the weak nuclear force. The carrier particles of the weak force are the W+, W-,

and the Z particles.

How are carrier particles different from other particles?

Carrier particles, such as the photon, gluon, and the graviton are hypothetical. They

are thought to be massless and having no electrical charge (except W+ and W-). Force

carrier particles can only be absorbed or produced by a matter particle which is

affected by that particular force. They allow us to explain interactions between

matter.

How old is the universe?

Today's most widely accepted cosmology, the Big Bang theory, states that the

universe is limited in space and time. The current estimate for the age of the universe

is 13.7 billion years. This figure was computed from the cosmic microwave

background (CMB) radiation data that the Wilkinson Microwave Anisotropy Probe

(WMAP) captured in 2002.

What came before the Big Bang?

The Big Bang model is singular at the time of the Big Bang. This means that one

cannot even define time, since spacetime is singular. In some models like the

oscillating universe, suggested by Stephen Hawking, the expanding universe is just

one of many phases of expansion and contraction. Other models postulate that our

own universe is just one bubble in a spacetime foam containing a multitude of

Spacetime, universes. The "multiverse" model of Linde proposes that multiple universes

recursively spawn each other, like in a growing fractal. However, until now there is no

observational data confirming either theory. It is indeed questionable, whether we

will ever be able to gain empirical evidence speaking in favor these theories, because

nothing outside our own universe can be observed directly. Hence, the question can

currently not be answered by science.

How big is the universe?

The universe is constantly expanding in all directions, therefore its size cannot be

stated. Scientists think it contains approximately 100 billion galaxies with each galaxy

containing between 100 and 200 billion star systems. Our own galaxy, the Milky

Way, is average when compared with other galaxies. It is a disk-shaped spiral galaxy

of about 100,000 light-years in diameter.

What is the universe expanding into?

This question is based on the popular misconception that the universe is some curved

object embedded in a higher dimensional space, and that the universe is expanding

into this space. There is nothing whatsoever that we have measured or can measure

that will show us anything about the larger space. Everything that we measure is

within the universe, and we see no edge or boundary or center of expansion. Thus the

universe is not expanding into anything that we can see, and this is not a profitable

thing to think about.

Why is the sky dark at night?

If the universe were infinitely old, and infinite in extent, and stars could shine

forever, then every direction you looked would eventually end on the surface of a star,

and the whole sky would be as bright as the surface of the Sun. This is known as

Olbers's paradox, named after Heinrich Wilhelm Olbers [1757-1840] who wrote about

it in 1823-1826. Absorption by interstellar dust does not circumvent this paradox,

since dust reradiates whatever radiation it absorbs within a few minutes, which is

much less than the age of the universe. However, the universe is not infinitely old,

and the expansion of the universe reduces the accumulated energy radiated by distant

stars. Either one of these effects acting alone would solve Olbers's paradox, but they

both act at once.

If the universe is only 13.7 billion years old, how can we see objects that

are 30 billion light-years away?

This question is essentially answered by Special Relativity. When talking about the

distance of a moving object, we mean the spatial separation now, with the positions of

us and the object specified at the current time. In an expanding universe, this

distance is now larger than the speed of light times the light travel time due to the

increase of separations between objects, as the universe expands. It does not mean

that any object in the universe travels away from us faster than light.

Source:- thebigview.com