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JohnH
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If velocity is energy divided by momentum it seems like the difference in scale between them is v and yet E=pc suggests the difference in scale is C not v. Why is this?
Because ##E=pc## only applies to massless particles, and these always move with speed ##c##. The general relationship that you want is ##E^2=(mc^2)^2+(pc)^2##, which reduces to ##E=pc## for massless particles and to the famous ##E=mc^2## when ##p## is zero (massive particle at rest).JohnH said:If velocity is energy divided by momentum it seems like the difference in scale between them is v and yet E=pc suggests the difference in scale is C not v. Why is this?
I think that is backwards. I think it is ##\vec v=\vec p/E##JohnH said:If velocity is energy divided by momentum
The units don't match, do they?Dale said:I think that is backwards. I think it is ##\vec v=\vec p/E##
In units where c=1, they do.nasu said:The units don't match, do they?
jbriggs444 said:In units where c=1, they do.
A good sanity check is the fact that the left hand side is a vector quantity and the right hand side has a vector quantity in the numerator and a scalar in the denominator. The competing formula, ##\vec{v}=E/\vec{p}## fails that sanity check -- can't divide a scalar by a vector.
Have you seen this formula with vectors in some book? I never said it should be written this way (with vectors, and vector p in the denominator).jbriggs444 said:In units where c=1, they do.
A good sanity check is the fact that the left hand side is a vector quantity and the right hand side has a vector quantity in the numerator and a scalar in the denominator. The competing formula, ##\vec{v}=E/\vec{p}## fails that sanity check -- can't divide a scalar by a vector.
I saw it in this thread.nasu said:Have you seen this formula with vectors in some book? I never said it should be written this way (with vectors, and vector p in the denominator).
Velocity is a vector. Energy is a scalar. Momentum is a vector. It is worthwhile emphasizing that fact ifJohnH said:If velocity is energy divided by momentum
You are right, but you can always throw in factors of c to fix that. In units where c=1 it is fine.nasu said:The units don't match, do they?
Actually, I just worked it out for massive particles and you can make it right by switching the numerator and denominator. For a massive object in units where c=1 we have:$$m^2=E^2-p^2$$ $$p= \frac{m v}{\sqrt{1-v^2}}$$ Which you can solve for ##v## and eliminate ##m## to getnasu said:Nugatory already showed the OP that his equation is not right for massive particles. You cannot make it right by just switching the numerator and denominator.
nasu said:With c=1 we have ## E=\sqrt{m^2 + p^2} ## so ## \frac{\vec{p}}{E}=\frac{\vec{p}}{\sqrt{m^2 + p^2}} ##. Can you manipulate this to give the velocity in the end?
Dale said:I am not sure ##\vec v = \vec p/E## is right
E=pc and v=E/p are two different equations that represent the relationship between energy (E), momentum (p), and velocity (v). In E=pc, the energy is equal to the product of momentum and the speed of light (c). In v=E/p, the velocity is equal to the energy divided by the momentum. Essentially, these equations show the different ways in which energy, momentum, and velocity are related.
Both equations are equally accurate in their respective contexts. E=pc is used to describe the energy of a particle with mass, while v=E/p is used to describe the velocity of a massless particle such as a photon. Therefore, the accuracy of each equation depends on the specific scenario being studied.
No, these equations cannot be used interchangeably. As mentioned before, E=pc is used for particles with mass, while v=E/p is used for massless particles. Additionally, the units for each equation are different. E=pc has units of energy (Joules), while v=E/p has units of velocity (meters per second).
E=pc and v=E/p are both derived from Einstein's theory of relativity. They are used to describe the behavior of particles at different scales, and they demonstrate the concept of mass-energy equivalence, which is a fundamental aspect of relativity.
E=pc and v=E/p have a wide range of applications in fields such as particle physics, astrophysics, and engineering. They are used to understand the behavior of particles in high-energy collisions, the movement of objects in space, and the design of advanced technologies such as particle accelerators and spacecraft. These equations have also played a crucial role in the development of modern physics and our understanding of the universe.