A rocket in deep space has an empty mass of 150 kg and exhausts the hot gases of burned fuel at 2500 m/s. It is loaded with 600 kg of fuel, which it burns in 30 s. What is the r…

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asked Dec 2, 2021 112k views

1 Answer

Undine · Answer 1 · 2021-12-07T19:49:59+0000

Answer:

$v(10\,s) \approx 775.387\,(m)/(s)$

$v(20\,s)\approx 1905.350\,(m)/(s)$

$v(30\,s) \approx 4023.595\,(m)/(s)$

Step-by-step explanation:

The speed of the rocket is given the Tsiolkovsky's differential equation, whose solution is:

$v (t) = v_(o) - v_(ex)\cdot \ln (m)/(m_(o))$

Where:

v_(o) - Initial speed of the rocket, in m/s.

v_(ex) - Exhaust gas speed, in m/s.

m_(o) - Initial total mass of the rocket, in kg.

- Current total mass of the rocket, in kg.

Let assume that fuel is burned linearly. So that,

$m(t) = m_(o) + r\cdot t$

The initial total mass of the rocket is:

$m_(o) = 750\,kg$

The fuel consumption rate is:

$r = -(600\,kg)/(30\,s)$

$r = -20\,(kg)/(s)$

The function for the current total mass of the rocket is:

$m(t) = 750\,kg - (20\,(kg)/(s) )\cdot t$

The speed function of the rocket is:

$v(t) = - 2500\,(m)/(s)\cdot \ln (750\,kg -(20\,(kg)/(s) )\cdot t)/(750\,kg)$

The speed of the rocket at given instants are:

$v(10\,s) \approx 775.387\,(m)/(s)$

$v(20\,s)\approx 1905.350\,(m)/(s)$

$v(30\,s) \approx 4023.595\,(m)/(s)$