Question

Calculate the maximum rate at which a rocket can expel gases if its acceleration cannot exceed seven times that of gravity. The mass of the rocket just as it runs out of fuel is75800 kg, and its exhaust velocity is 2.00 ? 10³ m/s. Assume that the acceleration of gravity is the same as on the earth's surface (9.80 m/s²).

Answer 1

Let's name some variables:

m_e = mass of the rocket just as it runs out of fuel = 75800kg

v_g = exhaust velocity (velocity at which gas is expelled)

m_g = mass of expelled gas

m = instantaneous mass of the rocket

u = Rocket's velocity

Let's solve the movement equation for the rocket using the same rocket as the refference system

Net force on the rocket F will be:

But the last term is zero because the speed of the rocket seen from the rocket is zero (remember de refference system)

So

That forces comes from the back expelled gass, thus

(because v_g is constant)

Finally:

But the rate at which the mass of gas is expelled is the same at which the rocket loss its mass, so

dm_g = - dm

So

If you consider the influence of the rocket weigth:

You want to calculate the rate

where a = 7g.

and for the maximun rate will be limited by the rate at which the rocket is almost empty (m = m_e), for that time, the rocket will get the max aceleration because it will has the lower mass

so