Question

at what speed the mass of a particle will double?​

Answer 1

Answer:

The rest mass of particles doesn't change, even when they move. You can, however, consider how fast a particle needs to be going for its total relativistic energy to double. This energy includes both the energy from the rest mass and the kinetic energy of the particle.

The relativistic energy of a particle is given by the following expression:

E=p2c2+m2oc4−−−−−−−−−−√

In this case, p is the relativistic momentum of the particle, c is the speed of light, and mois the rest mass of the particle. We can set this expression equal to twice the energy from the rest mass and then solve to find the speed we're looking for:

E=p2c2+m2oc4−−−−−−−−−−√=2moc2

p2c2+m2oc4=4m2oc4

p2c2=3m2oc4

p2=3m20c2

p=3–√moc

Now, we need to convert from momentum to speed. At low speeds, momentum is approximately just the product of an object's velocity multiplied by its mass, but at relativistic ones it's given by:

p=movγ=mov1−v2c2√

Substituting this in, we get:

p=mov1−v2c2√=3–√moc

v=3–√c1−v2c2−−−−−√

v=3–√c2−v2−−−−−−√

v2=3(c2−v2)

4v2=3c2

v2=34c2

v=3/4−−−√c≈0.866c

So, an object traveling at a speed of about 86.6% of the speed of light will have twice as much total relativistic energy as it does when it is stationary. Note that this answer doesn't depend on what object we're talking about. The answer is the same for other massive particles as well, not just electrons.

Explanation: