Derivative of Momentum: Why is dp/dt = F?

[MHD93]

Why is dp/dt = F even if the mass is changing, can it be derived from Newton's second law?

 

[tiny-tim]

Hi Mohammad!

Why is dp/dt = F even if the mass is changing, can it be derived from Newton's second law?

That is Newton's second law …

force = rate of change of momentum.​

We can only use "F = ma" when the mass is constant.

(constant mass is the exception )

 

[MHD93]

Newton's second law says that force is proportional to acceleration, meaning that mass is the proportionality constant

Isn't that true?

 

[olivermsun]

It doesn't really matter how you write it. The original formulation is F = dp/dt, but if mass is constant you can take it outside the derivative. If not, then you will need to account for the momentum carried into or out of the system by the mass (change).

 

[DrBloke]

Another may of looking at it, but reiterating what the others have said, is as follows:
p=mv

F=dp/dt (Newton's second law states force equals rate of change of momentum)
F=d(mv)/dt
=mdv/dt +vdm/dt (equation 1)

if the mass is constant then the rate of change of mass, dm/dt=0 so this becomes
F=mdv/dt
=ma (equation 2)

 

[dextercioby]

Why is dp/dt = F even if the mass is changing, can it be derived from Newton's second law?

As others may have stated, dp/dt = F_net IS the mathematical form of the 2nd law, thus it can't be derived from itself.