Relative momentum and relative mass

[Nugatory]

It's also saying that total momentum is not conserved if we use classical law of momentum. I think I need an example for this

An object of mass m is approaching you from the northwest, moving to the southeast at a speed of .5c relative to you. A second object, also of mass m, is approaching you from the southeast, moving to the northwest at a speed of .5c relative to you. They collide, and the first object rebounds to the northeast at a speed of .5c while the second one rebounds to the southwest at that speed. Both $mv$ and $\gamma{m}v$ are conserved in this collision (they're both zero before and after).

Now, consider this exact same momentum-conserving collision as described by an observer traveling due east at a speed of $\sqrt{2}c/4$ relative to you. He will describe the first object as approaching him directly from the north and rebounding back to the north, while the second object approaches him from a direction somewhere between east and southeast, and rebounds in a direction between southwest and west. Use the relativistic velocity addition law to calculate the velocities of the two objects as described by the second observer - you will find that quantity $mv$ is not conserved but the quantity $\gamma{m}v$ is conserved.

 

[Amr Elsayed]

you will find that quantity mvmv is not conserved but the quantity γmv\gamma{m}v is conserved.

Thank you for response, but there is sth concerning axis and angular directions in the example above. I will try to apply it on a simpler example, and I hope you will tell me where I have mistaken
Assume that those 2 objects with the same mass are both moving with a speed of 0.5C according to the origin, First object is coming from west and second one is coming from east.According to point of origin: both objects will crash then rebound in opposite direction with a speed of 0.5C and mv is conserved
According to a third object who is moving due east with speed of 0.5C according to" with respect to" origin : first object is stationary, and second object is coming from west with speed of 299999999.8 m/s (using relativity addition law) , after collision the second object is stationary and first object is moving due west with a speed of 299999999.8 m/s.

For both frames momentum will be observed the same before and after collision , but momentum itself will differ, and there is nothing about that 'cause momentum is frame dependent property
When we apply m*v*gamma we will also conserve momentum since change will be zero, but again momentum of each frame is different
I want to know what was wrong about that .
I want to know why we use distance measured by an observer watching but the time measured by the object itself to calculate velocity

Thank you, and I'm grateful for your care

 

[Amr Elsayed]

I can conclude the increase in mass if I consider time dilation, for example a spacecraft moving from our perspective with 0.9 C will shoot with a right angle from the direction of motion. Since the action of shooting will slow down but when the shot ball hits another ball , from my perspective it will go faster " from my perspective" since it's not affected by time dilation and I conclude that I should observe more mass from the moving ball for instance. Is that right ? but with shooting in the same direction of motion we shall need no increase in mass to keep the conservation of momentum.
regards

 

[Nugatory]

I can conclude the increase in mass

You can conclude an increase in mass, but we've spent the best part of two pages now trying to convince you that this is a bad idea, that you should not be thinking of momentum as $m_Rv$ where $m_R$ is something that increases with speed ($m_R=\gamma{m}$), but rather as $\gamma{m}v$ where $m$ does not change with speed.

 

[Nugatory]

or both frames momentum will be observed the same before and after collision , but momentum itself will differ, and there is nothing about that 'cause momentum is frame dependent property
When we apply m*v*gamma we will also conserve momentum since change will be zero, but again momentum of each frame is different
I want to know what was wrong about that .

Nothing is wrong with that, but it also doesn't tell you anything - a conservation law has to work always, not just for carefully selected special cases, or it's not a conservation law. It's easy to construct special cases in which both $mv$ and $\gamma{m}v$ happen to be conserved for some observers in some situations, and the case of two equal masses moving on the same axis as the observers is one of those special cases. (If you try it with different masses you will find that $mv$ isn't conserved; the reason I didn't use that example is that the calculations are much more difficult).

 

[Amr Elsayed]

where mRm_R is something that increases with speed

Well, mass is not increasing, but what I see as a moving observer for that mass is that its bigger. just illusion ? The left thing I need to know is about reality of mass change
When I see it changing" and it's not because it's sth like the total energy" how can I experience this relativistic increase ? And for a force that is not at rest according to the object. Does it have more force and more work to be accelerated ?? Gravity for example, when an object is in a free fall and after a while, what about force that is affecting it ?
I know Earth is not a frame of reference, but I mean ground
Another point, Isn't the gravitational force or work infinite ?
Thank you and sorry for bad English

 

[Orodruin]

but what I see as a moving observer for that mass is that its bigger.

No, you do not see this. As has been explained to you repeatedly in this thread, relativistic mass is only a convention for making the expressions for total energy and momentum look more like the classical expressions. It has little to do with actual resistance to acceleration and in order to figure that one out you need to differentiate momentum with respect to time and you will get the result that the inertia depends on the direction (but always grows to be infinite when approaching the speed of light). This inertia is not what physicists call "mass".

And for a force that is not at rest according to the object.

A force is not a thing which is at rest, it is a change of momentum of an object.

Gravity for example, when an object is in a free fall and after a while, what about force that is affecting it ?

As already pointed out to you, gravity is a very bad example when it comes to special relativity.

 

[Amr Elsayed]

As has been explained to you repeatedly in this thread

Thank you for explaining it repeatedly, but in deed I didn't mean it. All I meant was the question : how I will experience relativistic mass . My bad English doesn't help me choose the most suitable words to explain. Any way, you tell me I shall not experience it, and it's just mathematical
Thank you

 

[harrylin]

Thank you but it's not my problem, I want to learn about relativistic mass, I heard sth like we know it changes because of relativity and conservation of momentum [..]

If you are near to a library, you may check out for example Feynman's lectures on physics. He explained rather well the usefulness of "relativistic mass". Also Alonso&Finn's physics books are good for that.

Note however that in the end it's just a matter of definitions, and our preferences are related to our philosophy.

There are also old threads on that topic worth reading, such as https://www.physicsforums.com/threads/relativistic-mass-in-six-not-so-easy-pieces-feynman.808533/

 

[Amr Elsayed]

harrylin, Thank you, dear

 

[harrylin]

[..] The left thing I need to know is about reality of mass change
When I see it changing" and it's not because it's sth like the total energy" how can I experience this relativistic increase ? And for a force that is not at rest according to the object. Does it have more force and more work to be accelerated ?? [..]
Thank you and sorry for bad English

Too many questions, I take here the simplest and most straightforward one. What you are really asking, is if one needs to put more work into accelerating an object at high speed, so that it truly carries additional kinetic energy that cannot be explained by speed increase alone. Yes, that is the case, and it has been demonstrated as an educational experiment by Bertozzi :

- http://iopscience.iop.org/0031-9120/33/3/019
- https://en.wikipedia.org/wiki/Tests_of_relativistic_energy_and_momentum#Bertozzi_experiment

As you can read, this additional kinetic energy was measured as generated heat from impact.
Alternative theories that explain the maximum speed of c by saying that simply less and less energy is transferred to a fast moving particle, were thus disproved.

PS. I now see that the more recent paper to which I linked, only briefly discusses that pertinent feature of Bertozzi's experiment. I do have a copy of the original paper.

 

[Amr Elsayed]

that is the case

yeah, I needed a confirmation that more work is needed to keep accelerating an object, but mass is not really changing. I wonder why more work is needed

 

[harrylin]

yeah, I needed a confirmation that more work is needed to keep accelerating an object, but mass is not really changing. I wonder why more work is needed

Once more, that is a matter of interpretation / definitions. According to what you call mass, mass changes. According to what most other people here call mass, mass does not change. What is useful is not to linger on definitions (except for clarification) but on physics.

 

[Amr Elsayed]

I know that, but I was talking about mass which is resistance to be accelerated. I think it does change ,but not sure

 

[harrylin]

I know that, but I was talking about mass which is resistance to be accelerated. I think it does change ,but not sure

Yes, the resistance against acceleration increases with speed.

 

[Amr Elsayed]

Thanks so much

 

[jtbell]

mass which is resistance to be accelerated

That is one definition of mass. In Newtonian mechanics, there are various ways to define mass, which all give the same value. In relativistic mechanics, they do not all give the same value, so we have to choose one.

Most physicists today use a definition of mass which makes mass a property of an object that is invariant between different reference frames, i.e. does not change with the object's speed. They agree that "resistance to acceleration" increases with an object's speed; they simply do not use it as the definition of mass.

 

[Noctisdark]

I know that, but I was talking about mass which is resistance to be accelerated. I think it does change ,but not sure

I'm just going to repeat what all of you have said, according to Einstein's mass-energy equivalence, mass is energy, on the other hand, mass depends on how much matter are there, which when going at high speed won't change so the "real" mass won't change, yet when going at high speed, the object start to resist acceleration, it has more inertia, this might seem to anyone in the universe like it's mass have changed by the fact F = ma, but NOO !The mass doesn't change, moving objects have some special type of energy, which is kinetic energy, this energy be added to the system's energy and returning to E = Mc^2, you notice that the "Mass" have changed, but it's the universe laughing on you ! and it happens that M = m/sqrt(1-(v/c)^2), so the relativistic momentum in what it is p = (ym)v = Mv, because different frames can see you moving different speeds, momentum, energy, time, length, are all frame dependent, and mass is energy !

 

[PeterDonis]

the resistance against acceleration increases with speed.

But it also depends on direction: the resistance to force applied parallel to the direction of motion is different from the resistance to force applied perpendicular to the direction of motion.

And, all of this is "resistance" to "force" as viewed from a reference frame in which the object is moving. But to the object itself, there is only one "resistance" to force, which is its rest mass. In other words, if you view things from a frame in which the object is momentarily at rest--which, physically, represents the force the object itself feels, for example, the force the crew of a spaceship feels as the ship's engines fire--then the object's rest mass is what determines its "resistance" to force. So the whole business about resistance increasing with speed, and being different in different directions, is just unnecessary confusion; you don't need any of it to analyze problems and make predictions.

 

[harrylin]

[..] the whole business about resistance increasing with speed, and being different in different directions, is just unnecessary confusion [..]

I wonder what you find confusing about it? Understanding the relationship between inertia and energy is helpful for physical understanding.

 

[Amr Elsayed]

PeterDonis ,I think direction is not that important. If you affect a moving object with a right angle to its motion, this has nothing to do with its movement. Just will move in 2 direction. And you will find only rest mass as a resistance. Resistance increases to acceleration in its direction when speed increases, and that's because of inertia. Only if there is a relative velocity between the force and the object.
I hope you correct any wrong point of mine
regards,

 

[harrylin]

PeterDonis ,I think direction is not that important. If you affect a moving object with a right angle to its motion, this has nothing to do with its movement. Just will move in 2 direction. And you will find only rest mass as a resistance. Resistance increases to acceleration in its direction when speed increases, and that's because of inertia. Only if there is a relative velocity between the force and the object.
I hope you correct any wrong point
regards,

Hi Amr, if you are moving an object in a right angle to its motion then you measure not its "rest mass" m₀ but its "relativistic mass", γm₀. That is a measure of its already increased total energy, which doesn't change by that action as the speed remains constant. A fast moving object should thus feel "heavier" even when pushing sideways.

However, if you accelerate an object in the direction of its motion then you need to do much extra work as you are not only measuring its energy but also increasing its kinetic energy. This doing of extra work is felt as additional resistance to acceleration.

 

[Nugatory]

I think direction is not that important. If you affect a moving object with a right angle to its motion, this has nothing to do with its movement. Just will move in 2 direction. And you will find only rest mass as a resistance.

If you apply a force at right angles to the direction of movement of an object, the object's speed will not change but its velocity will change. The change in velocity is given by $\vec{F}=\gamma{m}_0\vec{a}$.

If you apply a force in the direction of movement of an object, the object's speed and velocity will both change. The change in velocity is given by $\vec{F}=\gamma^3{m}_0\vec{a}$.

Thus, the direction is important; one way you get a factor of $\gamma$ and the other way you get a factor of $\gamma^3$. In neither case is the resistance equal to the rest mass.

 

[Amr Elsayed]

if you are moving an object in a right angle to its motion then you measure not its "rest mass" m0 but its "relativistic mass", γm0. That is a measure of its already increased total energy,

Hi , yeah and that's also because of inertia which gets bigger for a moving object even when you try to change its direction ?? " that's a question"

Thus, the direction is important; one way you get a factor of γ\gamma and the other way you get a factor of γ3\gamma^3. In neither case is the resistance equal to the rest mass

Velocity as a component will of course change, but velocity in the first direction will stay same, right ??
Would you please tell me how to get those equations. I mean resistance as gamma^3 times mass or gamma times mass

 

[Amr Elsayed]

By the way, I meant " correct any wrong point of mine "

 

[Nugatory]

Velocity as a component will of course change, but velocity in the first direction will stay same ??

I have no idea what you mean by "velocity as a component", but the component of the velocity vector in the direction of motion does not stay the same. This is elementary circular motion from classical physics.

Would you please tell me how to get those equations. I mean resistance as gamma^3 times mass or gamma times mass

If you google for "longitudinal transverse mass" you will find plenty of good derivations.

 

[harrylin]

Hi , yeah and that's also because of inertia which gets bigger for a moving object even when you try to change its direction ?? " that's a question" [..]

Yes, once more: the already increased energy can be detected as increased inertia.
If I recall correctly, that is the Feynman procedure to measure relativistic mass.

 

[Amr Elsayed]

but the component of the velocity vector in the direction of motion does not stay the same. This is elementary circular motion from classical physics.

I mean the force will not be affecting it all the time. Like a projectile moving forward under effect of gravity. my push has nothing to do with vertical component downward. And so, If an object is moving and I affected it by a force perpendicular to its movement, I shall experience a resistance bigger than its rest mass because of inertia. This means that mechanics which I was talking about is just an approximate of realty The thing I don't know is how both resistances are different but I will be searching after it. I needed to know that direction really affects it as PeterDonis said and he was right

Thank you all