What is F_R?

Question

The gravitational force of attraction between two masses m and M at a distance apart of R is
F=(GmM)/R²

Where G is the universal gravitational constant. Suppose that m is the mass of a rocket which decreases as it burns up its fuel, and that M is constant.

What is F_R?

I thought it would be (-GmM)/(R^3) but I'm getting marked wrong. Please help?

(This is university level math)

Answer 1

This seems tricky and possibly some things are missing?

 

For F = GMm/R² , and ignoring the "mass decreaeses" part, we'd just have:

 

dF/dR = F_R = -2 GMm/R³

 

It isn't just the "2" that you were missing, is it?

 

- - -

If it is really more complex as the "mass decreases" suggests, then the rocket mass is modelled by a function m(R) and the dF/dR calculation would now use the product rule and there would be a dm(R)/dR = m_R term as well.

Another assumption we'd want to make is that the rocket is traveling radially away from the mass M (e.g., not orbiting).

Are there any initial conditions?  Starting at rest a R₀ ?  Assume the constant rocket thrust is greater than the gravitational force F(R₀) ?

For a constant burn rate we'd expect dm/dt to be a negative constant (i.e., losing so much mass of fuel per second); if instantaneously the rocket is at some velocity (dR/dt) then the constant dm/dt is equivalent to an instantaneous mass loss with distance: dm/dR ~ negative constant = m_R .

 

So we could use the product rule to get from:

   F   =  GM (  m(R)  1/R² )

to:

  F_R(R₀) = GM (  m₀ (-2/R₀³) +  m_R(R₀)  1/R₀² )

This is the rate of change of the force when the rocket is at R₀ with mass m₀ and losing mass at a rate of m_R(R₀)(which is negative).

 

What do you think of this ?