High priority problem in Physics: real equation of force is F = ma and not F = dp/dt

[martillo]

I consider the subject of this thread of high priority in Physics because it profound implications in both the theoretical areas as in the applications in the practical areas. It must be analyzed by real physicists (not an Electric Engineer as I am) and channelized in Physics Science in the appropriated way.

The problem is in the formulation of the 2nd Newton's law. It is found that the real equation of force is F= ma and not F = dp/dt even for thee case of variable mass. I have already begun to treat the problem threads here at the SFN forum in 2019 and in 2023:

A new evidence surged from the reference of a book published by Dr. Cowan (London University). The participant @studiot posted a photo with the passage of the book involving the subject: 

I have to abandon the thread because of the lot of days discussing with several ones at the same time without sleeping properly. now I return to the subject with some more expertise and renewed energies.

I present the problem in a small manuscript I post here now:

 

The real equation of force is F = ma

Today's Physics is stating that the Equation of Force is F = dp/dt.

We will analyze the equation of motion of rockets to see that the real Equation of Force is:

F = ma

A rocket has variable mass in its trajectory and it's important to see its motion equation.

Let m be its variable mass at any instant in its movement composed by the mass of the rocket plus the mass of its contained fuel.

I have made a search in the internet about rocket motion equations and all the sites agree in the equation: ⁽⁶⁾

F = m(dv/dt) = –v_e(dm/dt) where v_e is the speed of the expelled fuel relative to the rocket. 

They all agree that the force acting on the rocket is due to the expelled mass and is F = –v_e(dm/dt) and that the equation of motion is F = m(dv/dt) = ma.

I assume the equation has been completely verified experimentally with enough precision from a long time ago.

It is evident that in the development it is used covertly the equation:

F = ma

 for the force and not:

F = dp/dt

By definition p = mv and dp/dt = m(dv/dt) + v(dm/dt).

They derive the rocket's equation of motion based on the principle of conservation of momentum but considering the momentum of the rocket with the totality of the fuel (the contained plus the expelled fuel) at any moment and stating dp/dt = 0. After that they derive the equation of motion of the rocket as:
m(dv/dt) = –v_e(dm/dt) and specifically say that  the force on the rocket is:

F = m(dv/dt) = –v_e(dm/dt)

m(dv/dt) = ma then it is clear that what is finally applied to the rocket to determine its movement is the equation F = ma and not F = dp/dt.

 

This indicates that today's Physics is wrong stating the Equation of Force as F = dp/dt.

The right Equation of Force is F = ma even when mass varies.

 

Note that the natural derivation of the famous equation E = mc² by Relativity Theory has no sense since it is based in the wrong relation F = dp/dt.

Relativity Theory becomes a wrong theory since it is based on a wrong law.

 

By definition:

p = mv

With partial derivatives:

dp/dt = m(dv/dt) + v(dm/dt)

Now as:

F = m(dv/dt)

Then, the valid Equation of Momentum and Force is

dp/dt = F + v(dm/dt)

Then, the principle of conservation of the momentum p = mv must determine that dp/dt = 0 when no forces are applied and when there's no variation on the considered mass.

It can be observed that this principle can be applied to the rocket as was applied in the cited cases giving the same result. Considering the mass m' of the composed body of the rocket and the total fuel (the contained plus the expelled fuel) which does not vary:

F’ = 0 and dm'/dt = 0

Then, the thrust equation can be derived (as below):

m(dv/dt) = –v_e(dm/dt)

where m is the mass of the rocket with its contained fuel.

Finally:

F = ma = m(dv/dt) = –v_e(dm/dt)  is the force exerted on the rocket.

F = –v_e(dm/dt)

 

Rocket thrust force derivation

The thrust equation of the rocket is derived here considering the approximation that the mass of the expelled fuel of the rocket is negligible compared with the mass of the mass of the rocket plus the mass of its contained fuel:

Momentum and Force equations:

p = mv, F = ma = mdv/dt

dp/dt = mdv/dt + vdm/dt = F + vdm/dt

Considerations:

a) masses equations:

m = mass of the rocket with its contained fuel

m_e = mass of the expelled fuel

m’ = m + me = constant = total mass of the system rocket with total fuel

dm’/dt = dm/dt + dm_e/dt = 0

Then: dm_e/dt = -dm/dt

b) velocities equations (one dimension):

v = absolute velocity of the rocket

v_e = velocity of the expelled fuel in relation to the rocket assumed constant

u = absolute velocity of the expelled fuel

v_e = v – u = constant

dv_e/dt = dv/dt – du//dt = 0

Then: du/dt = dv/dt

 

Derivation:

Total momentum of the system rocket with total fuel: p’ = mv + m_eu

dp’/dt = d(mv + m_eu)/dt = mdv/dt + v/dm/dt + m_edu/dt + u/dm_e/dt

Considering: du/dt = dv/dt and dm_e/dt = -dm/dt

dp’/dt = (m + m_e)dv/dt + (v-u)dm/dt

As v – u = v_e and considering m_e << m (m + m_e ≈ m) then:

dp’/dt ≈ mdv/dt + v_edm/dt and as dp’/dt = 0 then:

mdv/dt ≈ -v_edm/dt

Finally F = ma = mdv/dt ≈ -v_edm/dt under the approximation m_e << m

Then, the rocket thrust force is: F ≈ -v_edm/dt

 

[swansont]

!

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