Naked Science Forum
Non Life Sciences => Physics, Astronomy & Cosmology => Topic started by: thebrain13 on 29/06/2006 19:37:19
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Does anyone know einsteins equation for relative velocity based on the force applied to an object with a specific mass?
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In other words I mean, if you applied enough force to accelerate an object to 2c according to newton, how fast would it be moving according to einstein. Whats the equation for that?
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Velocity is not a function of force, it is a function of energy.
Acceleration is a function of force, but acceleration is only a measure of the rate at which velocity changes, it is not a measure of the velocity itself.
The relationship between energy and velocity is:
http://en.wikipedia.org/wiki/Special_relativity#Mass.2C_momentum.2C_and_energy
quote:
Mass, momentum, and energy
In addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.
There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.
Given an object of invariant mass m0 traveling at velocity v the energy and momentum are given (and even defined) by
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fa%2F4%2F5%2Fa4592f0c78a3265eca17dc18ac8fb04e.png&hash=1e36a20663c5262e9466a0efffc75be6)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fe%2F0%2F3%2Fe03af76e2c4f8ec30ca743cf02f2d2de.png&hash=92f1030c5e8bf12d4556f41c7c1786f4)
where #947; (the Lorentz factor) is given by
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fa%2F0%2F6%2Fa06c67a58d69b67566199fc2fb684cc1.png&hash=8c00e2b0db3aca14ab8541202216bf8e)
and c is the speed of light. The term #947; occurs frequently in relativity, and comes from the Lorentz transformation equations.
Relativistic energy and momentum can be related through the formula
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fe%2F7%2F1%2Fe7112bab562122702df6ab027b85bc4b.png&hash=fec4d91d8441c7c30dbb805fef318af7)
which is referred to as the relativistic energy-momentum equation.
For velocities much smaller than those of light, #947; can be approximated using a Taylor series expansion and one finds that
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2F1%2F7%2F3%2F173298d64109390794efa8a6a21557e4.png&hash=18651e10518e16307ddcbdd5309a8fc2)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fd%2F2%2Fe%2Fd2ec9d43aff248411fd82177e6478372.png&hash=064074aa54166f9af49ffe13a5f268f8)
Barring the first term in the energy expression (discussed below), these formulas agree exactly with the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities.
Looking at the above formulas for energy, one sees that when an object is at rest (v = 0 and #947; = 1) there is a non-zero energy remaining:
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fb%2F8%2F3%2Fb834331a23ab23b15d15a6905dedc7dd.png&hash=0c48ba2fad40510926568ab3151dbfdd)
This energy is referred to as rest energy. The rest energy does not cause any conflict with the Newtonian theory because it is a constant and, as far as kinetic energy is concerned, it is only differences in energy which are meaningful.
Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:
Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy.
This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy which can be released by nuclear reactions, providing important information which was useful in the development of the nuclear bomb. The implications of this formula on 20th century life have made it one of the most famous equations in all of science.
George
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I guess im missing something, what is the equation? shouldn't this equation start with v?
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quote:
Originally posted by thebrain13
I guess im missing something, what is the equation? shouldn't this equation start with v?
The v is in the Lorenz factor.
George
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Im sorray but I still dont understand, what is the equation?
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The two equations that matter are:
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fa%2F4%2F5%2Fa4592f0c78a3265eca17dc18ac8fb04e.png&hash=1e36a20663c5262e9466a0efffc75be6)
and
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fa%2F0%2F6%2Fa06c67a58d69b67566199fc2fb684cc1.png&hash=8c00e2b0db3aca14ab8541202216bf8e)
Which, when combined, give us:
E = m0 C^2 / sqrt(1 – v^2 / c^2)
which translates to:
E = m0 c^3 / sqrt(c^2 – v^2)
Note that this is both the kinetic energy and the mass energy combined – hence, when v=0, one gets:
E = m0 c^3/sqrt(c^2 - 0)
Which becomes E = m0 c^2
George
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could you possibly simplify this because I am an idiot. what is the one actual equation, which gives velocity based on the amount of energy applied.
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The key equation is: E = m0 c^3 / sqrt(c^2 – v^2)
Where:
m0 = rest mass
c = speed of light
v = velocity
E = energy
George
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what does v equal?
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thebrain13 The information you have been given is complete and accurate you just need to rearrange the equation to extract the velocity if you are happy to work in energy terms if you don't know how to do this you really need to learn a bit more algebra if you want to understand the question you have asked properly.
Going back to your original question.
You have to remember that the velocity of an object does not depend on strictly on the force that is acting on it it depends on the force AND the length of time that it is acting. If I give you a brief push with a force you wont move very far or fast but if I push you steadily for a long time you will go fast and a long way.
So to make your original question accurate you need to specify for example that the force is constant and always in the same direction and that the body starts from rest relative to some point.
Then you need to rearrange the equation to get what you want
The fundamental newton equation is, force equals mass times acceleration, so the acceleration equals the force divided by the mass.
If then you assume that you start with zero velocity the velocity you get to is the acceleration times the time that it has been operating.
so the velocity is the time, multiplied by the force and divided by the mass.
To convert this equation to a relatavistic form is not all that easy because as the velocity gets significantly close to the velocity of light the mass is increasing continuously by the lorenz factor (stated above) and a fixed force (like that produced by a rocket motor) gradually produces less acceleration and you need to do some integrating to get to the final velocity.
However if you are thinking about a rocket accelerating there is an added complication. The rocket is using up fuel by chucking mass out the back, so the mass of the main object is decreasing for all the time that the force is applied and this means that the relatavistic decrease in final velocity is not quite as much as you might expect.
Learn, create, test and tell
evolution rules in all things
God says so!
-
If you want to know how fast something would be going if it had the same amount of energy as something that would be moving at 2c according to Newton, the answer is a lot easier, and possibly good enough?
the amount of energy according to Newton you would have En = 1/2 m (2c)^2 = 2mc^2
we take soul surfer's nice equation which relates energy to velocity according to einstein
E = m0 c^3 / sqrt(c^2 – v^2)
rearrange it so that it relates velocity to energy - from how you are talking you are not very comfortable with this so:
sqrt(c^2-v^2) = m0 c^3/E -> multiply both sides by sqrt(c^2-v^2) and divide by E to get everything on different sides
c^2 - v^2 = (m0 c^3/E)^2 -> square both sides
v= sqrt(c^2 - (m0 c^3/E)^2 ) -> move the subtract c^2 from both sides then multiply by -1
You can now feed En into E in this equation and find out what v is.
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thank you, so if you had enough energy to accelerate an object to 2c according to newton, it would travel at just under .97c according to einstein, correct?
-
In other words I mean, if you applied enough force to accelerate an object to 2c according to newton, how fast would it be moving according to einstein. Whats the equation for that?
-
Velocity is not a function of force, it is a function of energy.
Acceleration is a function of force, but acceleration is only a measure of the rate at which velocity changes, it is not a measure of the velocity itself.
The relationship between energy and velocity is:
http://en.wikipedia.org/wiki/Special_relativity#Mass.2C_momentum.2C_and_energy
quote:
Mass, momentum, and energy
In addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.
There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue the idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.
Given an object of invariant mass m0 traveling at velocity v the energy and momentum are given (and even defined) by
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fa%2F4%2F5%2Fa4592f0c78a3265eca17dc18ac8fb04e.png&hash=1e36a20663c5262e9466a0efffc75be6)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fe%2F0%2F3%2Fe03af76e2c4f8ec30ca743cf02f2d2de.png&hash=92f1030c5e8bf12d4556f41c7c1786f4)
where #947; (the Lorentz factor) is given by
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fa%2F0%2F6%2Fa06c67a58d69b67566199fc2fb684cc1.png&hash=8c00e2b0db3aca14ab8541202216bf8e)
and c is the speed of light. The term #947; occurs frequently in relativity, and comes from the Lorentz transformation equations.
Relativistic energy and momentum can be related through the formula
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fe%2F7%2F1%2Fe7112bab562122702df6ab027b85bc4b.png&hash=fec4d91d8441c7c30dbb805fef318af7)
which is referred to as the relativistic energy-momentum equation.
For velocities much smaller than those of light, #947; can be approximated using a Taylor series expansion and one finds that
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2F1%2F7%2F3%2F173298d64109390794efa8a6a21557e4.png&hash=18651e10518e16307ddcbdd5309a8fc2)
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fd%2F2%2Fe%2Fd2ec9d43aff248411fd82177e6478372.png&hash=064074aa54166f9af49ffe13a5f268f8)
Barring the first term in the energy expression (discussed below), these formulas agree exactly with the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities.
Looking at the above formulas for energy, one sees that when an object is at rest (v = 0 and #947; = 1) there is a non-zero energy remaining:
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fb%2F8%2F3%2Fb834331a23ab23b15d15a6905dedc7dd.png&hash=0c48ba2fad40510926568ab3151dbfdd)
This energy is referred to as rest energy. The rest energy does not cause any conflict with the Newtonian theory because it is a constant and, as far as kinetic energy is concerned, it is only differences in energy which are meaningful.
Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:
Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount of energy.
This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy which can be released by nuclear reactions, providing important information which was useful in the development of the nuclear bomb. The implications of this formula on 20th century life have made it one of the most famous equations in all of science.
George
-
I guess im missing something, what is the equation? shouldn't this equation start with v?
-
quote:
Originally posted by thebrain13
I guess im missing something, what is the equation? shouldn't this equation start with v?
The v is in the Lorenz factor.
George
-
Im sorray but I still dont understand, what is the equation?
-
The two equations that matter are:
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fa%2F4%2F5%2Fa4592f0c78a3265eca17dc18ac8fb04e.png&hash=1e36a20663c5262e9466a0efffc75be6)
and
(https://www.thenakedscientists.com/forum/proxy.php?request=http%3A%2F%2Fupload.wikimedia.org%2Fmath%2Fa%2F0%2F6%2Fa06c67a58d69b67566199fc2fb684cc1.png&hash=8c00e2b0db3aca14ab8541202216bf8e)
Which, when combined, give us:
E = m0 C^2 / sqrt(1 – v^2 / c^2)
which translates to:
E = m0 c^3 / sqrt(c^2 – v^2)
Note that this is both the kinetic energy and the mass energy combined – hence, when v=0, one gets:
E = m0 c^3/sqrt(c^2 - 0)
Which becomes E = m0 c^2
George
-
could you possibly simplify this because I am an idiot. what is the one actual equation, which gives velocity based on the amount of energy applied.
-
The key equation is: E = m0 c^3 / sqrt(c^2 – v^2)
Where:
m0 = rest mass
c = speed of light
v = velocity
E = energy
George
-
what does v equal?
-
thebrain13 The information you have been given is complete and accurate you just need to rearrange the equation to extract the velocity if you are happy to work in energy terms if you don't know how to do this you really need to learn a bit more algebra if you want to understand the question you have asked properly.
Going back to your original question.
You have to remember that the velocity of an object does not depend on strictly on the force that is acting on it it depends on the force AND the length of time that it is acting. If I give you a brief push with a force you wont move very far or fast but if I push you steadily for a long time you will go fast and a long way.
So to make your original question accurate you need to specify for example that the force is constant and always in the same direction and that the body starts from rest relative to some point.
Then you need to rearrange the equation to get what you want
The fundamental newton equation is, force equals mass times acceleration, so the acceleration equals the force divided by the mass.
If then you assume that you start with zero velocity the velocity you get to is the acceleration times the time that it has been operating.
so the velocity is the time, multiplied by the force and divided by the mass.
To convert this equation to a relatavistic form is not all that easy because as the velocity gets significantly close to the velocity of light the mass is increasing continuously by the lorenz factor (stated above) and a fixed force (like that produced by a rocket motor) gradually produces less acceleration and you need to do some integrating to get to the final velocity.
However if you are thinking about a rocket accelerating there is an added complication. The rocket is using up fuel by chucking mass out the back, so the mass of the main object is decreasing for all the time that the force is applied and this means that the relatavistic decrease in final velocity is not quite as much as you might expect.
Learn, create, test and tell
evolution rules in all things
God says so!
-
If you want to know how fast something would be going if it had the same amount of energy as something that would be moving at 2c according to Newton, the answer is a lot easier, and possibly good enough?
the amount of energy according to Newton you would have En = 1/2 m (2c)^2 = 2mc^2
we take soul surfer's nice equation which relates energy to velocity according to einstein
E = m0 c^3 / sqrt(c^2 – v^2)
rearrange it so that it relates velocity to energy - from how you are talking you are not very comfortable with this so:
sqrt(c^2-v^2) = m0 c^3/E -> multiply both sides by sqrt(c^2-v^2) and divide by E to get everything on different sides
c^2 - v^2 = (m0 c^3/E)^2 -> square both sides
v= sqrt(c^2 - (m0 c^3/E)^2 ) -> move the subtract c^2 from both sides then multiply by -1
You can now feed En into E in this equation and find out what v is.
-
thank you, so if you had enough energy to accelerate an object to 2c according to newton, it would travel at just under .97c according to einstein, correct?
-
I found the equation, in much a more simplified versio
v=k/sqrt(k^2+1)
k is newtonian velocity and v is relative velocity.
So if you have enough energy to accelerate to 2c according to newton, you would travel at .895c according to einstein. not .97c, I must of errored somewhere using your equation.