# Is the inertia of a mass depended of his speed?

If a ball (10 kg) falls from a height of 10 km towards an (air free) planet with an acceleration of 10 m/s² then it gains every second in speed. But is the inertia of the ball getting every second a bit less? With other words, is a moving object easier to accelerate?

And is this a linear function or can it go up and down as I think that his inertia will first go down but when close to $c$ it will gain inertia (but I not sure about this)?

In Newtonian Mechanics, it is exceedingly simple. The law that governs the motion of a particle is $F = m\dfrac{dv}{dt}$. Where, $m$ is the mass (or inertia if you wish) of the particle, $F$ is the force acting on the particle, $v$ is the velocity of the particle, and $t$ is the time. It doesn't matter how fast the object is already moving, it will always take the same force to produce a particular acceleration in a given particle. That is, its inertia doesn't change with speed. End of the story. (If you observe the motion of the falling ball carefully, you can see (of course, only when you do precise measurements) that the change in the particle's velocity will be the same in each equal interval of time. This is to say that in every second, the velocity of the particle will increase by $9.81$ $m/s$.)
In Relativistic Mechanics, the law of the motion of a particle is $F^{\mu} = \dfrac{dp^{\mu}}{d\tau}$; where $\mu$ runs from $0$ to $3$. Here $F$ denotes the Minkowskian four-vector force - a relativistic extension of the Newtonian 3-vector force. $p$ is the momentum four-vector - a relativistic extension of the Newtonian 3-vector momentum. $\tau$ denotes a parameter called the proper time. If you work out the relationship between the applied force and the rate of change of velocity then it turns out that it becomes harder and harder to increase the speed of the particle with increasing values of its initial speed. This makes one feel like the mass or inertia of the particle is increasing but (long story short) there doesn't exist any consistent formulation of mechanics in which one can make the mass variant with velocity to accommodate this fact. Rather, the approach is to accept that the mass still remains invariant and the law governing the dynamics of the particles change which creates this effect that with an increased initial speed, it becomes difficult to accelerate it. A more detailed description of why we don't consider the mass to be variable with velocity can be found in my another answer.