# In Newton's 2nd Law, does $m$ stand for rest mass?

I just learned that almost %99 percent of an atom's mass comes from Strong Force Field Energy. But this energy is counted as my rest mass nevertheless. So in $F=m.a$, should i take $m$ as rest mass (as standard description)? Or should i take $m$ as total energy that an object has? To make it more clear; if an object's kinetic energy increases, would it be harder to accelerate?

$F = ma$ is the approximation for when an object's rest mass is much greater than it's kinetic energy. This approximation is good for anything traveling less than 50 million mph.

For objects moving 10% of the speed of light or more, you have to worry a little more about the 'total energy' of the particle. Then it's better to use the definition of force:

$F = \frac{\partial}{\partial t} p$ Where $p$ is the momentum; $p = \gamma mv$.

$m$ is the rest mass of the particle, so it has no time dependence. Note that $\gamma$ and $v$ both have time dependence, so you have to use the product rule.

For slower objects (relative to the speed of light), $\gamma \approx 1$ and this reduces to $F = m\frac{\partial}{\partial t}v = ma$, the familiar result.

• Thank you for your answer. So the correct thing to say is; we cannot move at the speed of light because as we speed up our momentum increases and we would need more and more energy to speed up, not our mass increases right? – Tuna Oct 4 '17 at 1:51
• And one more question if i may, what about gravitational potential energy? What if my gravitational potential energy increases enormously, how does it apply with all this? – Tuna Oct 4 '17 at 1:54
• I'm not sure it's conclusive to say it's because our momentum increases (cause of course it'll increase as we speed up), but it's definitely related to the fact that you relativisitic momentum would have to go to infinity to travel the speed of light. – Señor O Oct 4 '17 at 1:55
• Some people do pretend like "mass increases" and they call it "relativistic mass" but most physicists find that interpretation inaccurate. Gravitational potential energy will not affect any of this (except, of course, that it could be used to accelerate you a lot) – Señor O Oct 4 '17 at 1:57

In Newton's 2nd law, written in terms of force, mass and acceleration, m stands for the mass in the original Newton's work, i.e. the quantity of the substance which makes up a material body. This quantity is absolute, i.e. will not change if a body is at rest or moves (which is fine, because a body could be at rest in some Galilean inertial frame while moving with constant velocity in another).

In the theory of special relativity, $\vec{F} = m\vec{a}$ is derived as the low-velocity approximation of $f^{a} = m \dot{u}^{a}$, where "a" is a generic index (in the sense of Wald's book), f is the 4-force, u is the 4-velocity, the dot stands for the derivative with respect to proper time, and m is the invariant mass (or simply mass).