# Changing mass with velocity

I was wondering that a change in velocity can change the mass of an object slightly. I thought of this since p=mv so if we rearrange it to be m=p/v. Does this mean that a change in momentum or velocity can actually change the mass?

• Oct 19, 2020 at 14:40
• What do you have in mind that can change momentum but not change velocity (or vice versa)? Oct 19, 2020 at 19:38

You have this equation, $$p = mv$$. If you increase the velocity and keep $$p$$ constant, yes you must have had an increase in mass. The question is, what does it mean for $$p$$ to be constant? It is defined by the product of mass and velocity. So you have done nothing to make predictions, you have just stated what you would call mass if you increased v and kept p the same. OR since mass and velocity are more readily known as measured quantities to the layman, you have said increasing v and lowering m keeps mv the same. See? you've said nothing of any value

Note, for a small change in $$p$$ (and $$v$$, as their correlation is 1, or $$m$$ depending on normalization):

$$\delta m = \frac d {dp}({\frac p v})\delta p+ \frac d {dv}({\frac p v})\delta v$$

$$\delta m = \frac{\delta p} v - \frac{p\delta v}{v^2}$$

Since:

$$\delta p = m \delta v$$

we get:

$$\delta m = \frac{\delta p} v - \frac{p\delta p/m}{v^2}$$

$$\delta m = \delta p(\frac 1 v - \frac{v}{v^2})$$

$$\delta m = \delta p(\frac 1 v - \frac{1}{v}) = 0$$

So, no.