Why we take $k=1$ in $F=ma$?

Question

I know that $F$ is directly proportional to mass and acceleration but why in formula we take $k=1$ not other constant?

Answer 1

I know that F is directly proportional to mass and acceleration but why in formula we take k=1 not other constant?

This depends on your system of units. For SI units $$\Sigma \vec F = m \vec a$$ but for US customary units $$\Sigma \vec F = k \ m \vec a$$ where $$k=\frac{1}{32.2}\mathrm{\ \frac{lb_f \ s^2}{lb_m \ ft}}$$

This is not limited to Newtons laws. Any time that you see a dimensionful universal constant in a law of physics, that constant can be removed by choosing appropriate units. For example, Gauss’ law has $\epsilon_0$ in SI units, but it goes away in Heaviside Lorentz units. Similarly Coulomb’s law has $k_e$ in SI units, but it goes away in Gaussian units.

Dimensionful universal constants describe your units, not nature. That is why the SI is now explicitly defined in terms of universal constants.

Answer 2

The force was found to be proportional to the acceleration, not $ma$. This means that $F=ma$ with $m$ the constant of proportionality.