# How to calculate increase of mass in relation to increase in velocity?

#### stu dent

ok, so, I know e=mc2 is a way i can find what quantity of energy a mass has.

and i know that as objects increase in speed they become more massive, which i assume is also elegantly portrayed in this equation.

but, this kind poses a problem for me, because i am wondering how much velocity increases mass.

it would seem that as you put more energy into an object to speed it up, if i am taking the meaning correctly, then it becomes more massive, and requires more energy in order to accelerate it further.

this would also mean that the more massive an object is, the more of an effect Δv will have on Δm.

I for some reason, had the impression that it was more Δv of an object that would increase its mass, rather than the quantity of energy input.

or rather, and i suppose it still may be this way, if you have 2 objects. one having mass X, and one having mass Y, if you increased their velocity N times, you'd end up with each object having bX and bY mass, by some ratio of N which may even be N=b, but i suspect not.

so, i guess what i'm wondering, is how really, i would go about calculating the increase in mass of an object in relation to a change in velocity, or an increase in velocity.

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#### Vorde

That $E=mc^2$ which is so beautiful, is really just a special case of a much more general function: $E^2=(m_0c^2)^2+(pc)^2$. That special case only works when the particle is motionless (at least from the perspective of whoever is measuring it).

In cases where a particle (or a system of particles) is moving, you have to look at the larger equation. In the larger equation, $P$ (which stands for momentum) is dependent on velocity, so you can see how the energy would increase if the speed increased.

#### Nugatory

Mentor
The calculation you are looking for is $m=\frac{m_0}{\sqrt{1-v^2/c^2}}$.

Note the following however:
1) This is the mass of an object as observed by an observer moving relative to the object at a speed v. Because the object is at rest relative to itself, an observer moving along with the object sees nothing unusual at all, and everything out there is already moving at .99999c (or whatever speed you choose) relative to some observer somewhere.
2) As Vorde pointed out above, $E^2=(m_0c^2)^2+(pc)^2$ is a more powerful and general way of describing the physics, so this notion of "mass increase with velocity" is seldom used these days. It takes some care to use relativistic mass correctly, and you cn easily confuse yourself by using it incorrectly.

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