Momentum confusion...

The net external force acting on an object can be evaluated as the rate of change ofmomentum. This turns out to be a more fundamental way of stating the force than the use ofNewton's second law. Using Newton's second law and momentum:






But this limited relationship can be generalized to




I'm not sure how this was mathematically achieved? Can someone explain how this generalization is found?

Thanks for any help in advance

I suspect the problem is that Newton never wrote F = ma. That is an interpretation that was made of his original statement. There is some debate about his words but he referred to the change of motion of a body being proportional to the force acting on it. This is closer to F = dp/dt than F = ma. That means that nobody generalised from Newton. F = ma is not the starting point and is not actually Newton's second law, although it is often (mis)represented as such. It is a specific formulation of Newton's general law for situations where the mass of a body is constant.

The net external force acting on an object can be evaluated as the rate of change ofmomentum. This turns out to be a more fundamental way of stating the force than the use ofNewton's second law. Using Newton's second law and momentum:






But this limited relationship can be generalized to




I'm not sure how this was mathematically achieved? Can someone explain how this generalization is found?

Thanks for any help in advance

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Force is change of rate of momentum.

Momentum is p=mv
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To work out the change of rate of momentum (and therefore the force) you can do:

1) ∆p/∆T (change in momentum over change in time) This is the same as change RATE of momentum

or

2) m∆V/∆T (because p=mv and mass doesn't change)

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If mass is changing (like a rocket ejecting fuel and becoming lighter) but velocity stays the same, the force will still change.
So we get the equation:

3) V∆m/∆T (velocity doesn't change)

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But this is only an average.
The momentum might increase quickly at the beginning of the time frame (∆T) and slower at the end.
This would mean the force is larger at the start and smaller at the end.
So the force is only an average.

(To make the force more accurate you make the ∆T smaller)
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Not sure if this answers you question but I hope it helps

so the last equation is not correct? because im wondering how can it be used if both mass and speed are changing in a situation?

The last equation is correct - F = ma is a specific version of that equation for systems where m is constant, which is the case for discrete bodies in non-relativistic motion. The point is that the last equation is the proper expression of Newton's second law as Newton himself expressed it - it isn't an extension of F = ma that was added after Newton. Statements that say F = ma is Newton's second law of motion are technically wrong although they are common.

Ahh ok I see what you mean but say if both speed and mass are changing would you just calculate the change in mass and change in velocity and multiply them together and then divide them by the time period it all occurred?

How would the mass be changing? If a body is shedding material (e.g. a rocket) or accreting it (e.g. a collapsed star orbiting a giant main sequence companion) then you will need to consider what is happening to this lost or gained mass - what is its momentum before and after it is attached to the body? That will have an impact on the residual body's momentum, in addition to any other force acting. Alternatively, is this a relativistic mass change? If so, you will need to invoke special relativity. The general form of Newton's second law will still hold but p = $\gamma$*mv, where m is the (constant) rest mass so you need to evaluate the rate of change of that quantity.

Thanks for your help but what if both mass and velocity are changing at the same time?

F=∆P/∆T
Change in momentum can be calculated by doing several things…



Method 3 is used when both mass and velocity are changing.
It calculates the initial moment and takes it away from the final momentum. This gives the change in momentum

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So ∆P=P2-P1
and,
F=(P2-P1)/∆T

ahh ok thanks for your help

The net external force acting on an object can be evaluated as the rate of change ofmomentum. This turns out to be a more fundamental way of stating the force than the use ofNewton's second law. Using Newton's second law and momentum:






But this limited relationship can be generalized to




I'm not sure how this was mathematically achieved? Can someone explain how this generalization is found?

Thanks for any help in advance

This is not always correct.
Force is defined as the rate of change of momentum.

$F = \dfrac{dP}{dt}$

$F = \dfrac{d(mv)}{dt}$

$F = m\dfrac{dv}{dt} + v\dfrac{dm}{dt}$(product rule)

$F = ma + v\dfrac{dm}{dt}$