Proof of F=ma from second law of Newton

[hasanhabibul]

from the second law of Newton we know... applied force is proportional to the change of momentum...that means F= k( mv-mu) where k is constant ...from here now proof F=ma

 

[Stonebridge]

The second law doesn't say that. Look at it again.
You need to start with the correct equation before you can prove F=ma

 

[saunderson]

I think i can help you, but with my own labelling ;)

F \sim m \cdot \frac{v^\prime - v}{\delta t}​

v^\prime is the speed after a short time interval \delta t related to the instant of time t. So the equation above can be written in a more descriptive way

\delta t \, F \sim m \cdot \Bigl( v(t+\delta t) - v(t) \Bigr)​

if the time interval \delta t is very small, we can approximate v(t+\delta t) according to Taylor (keyword Taylor expansion).

v^\prime = v(t + \delta t) \approx v(t) + \dot v(t) \, \cdot \, \delta t + \mathcal{O}\bigl((\delta t)^2 \bigr)​

putting it in the initial equation that yields

\delta t \, F \sim m \cdot \Bigl( ~ \bigl( v(t) + \dot v(t) \, \cdot \, \delta t \bigr) ~ - v(t) \bigr) = m \, \cdot \, \dot v(t) \, \delta t​

Here you see, that the time interval \delta t must be very small, otherwise the first term of the Taylor series isn't a good approximation any longer.

Hope that i could help you...