Answer:
![v(10\,s) \approx 775.387\,\frac{m}{s}](https://tex.z-dn.net/?f=v%2810%5C%2Cs%29%20%5Capprox%20775.387%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
![v(20\,s)\approx 1905.350\,\frac{m}{s}](https://tex.z-dn.net/?f=v%2820%5C%2Cs%29%5Capprox%201905.350%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
![v(30\,s) \approx 4023.595\,\frac{m}{s}](https://tex.z-dn.net/?f=v%2830%5C%2Cs%29%20%5Capprox%204023.595%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
Explanation:
The speed of the rocket is given the Tsiolkovsky's differential equation, whose solution is:
![v (t) = v_{o} - v_{ex}\cdot \ln \frac{m}{m_{o}}](https://tex.z-dn.net/?f=v%20%28t%29%20%3D%20v_%7Bo%7D%20-%20v_%7Bex%7D%5Ccdot%20%5Cln%20%5Cfrac%7Bm%7D%7Bm_%7Bo%7D%7D)
Where:
- Initial speed of the rocket, in m/s.
- Exhaust gas speed, in m/s.
- 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](https://tex.z-dn.net/?f=m%28t%29%20%3D%20m_%7Bo%7D%20%2B%20r%5Ccdot%20t)
The initial total mass of the rocket is:
![m_{o} = 750\,kg](https://tex.z-dn.net/?f=m_%7Bo%7D%20%3D%20750%5C%2Ckg)
The fuel consumption rate is:
![r = -\frac{600\,kg}{30\,s}](https://tex.z-dn.net/?f=r%20%3D%20-%5Cfrac%7B600%5C%2Ckg%7D%7B30%5C%2Cs%7D)
![r = -20\,\frac{kg}{s}](https://tex.z-dn.net/?f=r%20%3D%20-20%5C%2C%5Cfrac%7Bkg%7D%7Bs%7D)
The function for the current total mass of the rocket is:
![m(t) = 750\,kg - (20\,\frac{kg}{s} )\cdot t](https://tex.z-dn.net/?f=m%28t%29%20%3D%20750%5C%2Ckg%20-%20%2820%5C%2C%5Cfrac%7Bkg%7D%7Bs%7D%20%29%5Ccdot%20t)
The speed function of the rocket is:
![v(t) = - 2500\,\frac{m}{s}\cdot \ln \frac{750\,kg -(20\,\frac{kg}{s} )\cdot t}{750\,kg}](https://tex.z-dn.net/?f=v%28t%29%20%3D%20-%202500%5C%2C%5Cfrac%7Bm%7D%7Bs%7D%5Ccdot%20%5Cln%20%5Cfrac%7B750%5C%2Ckg%20-%2820%5C%2C%5Cfrac%7Bkg%7D%7Bs%7D%20%29%5Ccdot%20t%7D%7B750%5C%2Ckg%7D)
The speed of the rocket at given instants are:
![v(10\,s) \approx 775.387\,\frac{m}{s}](https://tex.z-dn.net/?f=v%2810%5C%2Cs%29%20%5Capprox%20775.387%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
![v(20\,s)\approx 1905.350\,\frac{m}{s}](https://tex.z-dn.net/?f=v%2820%5C%2Cs%29%5Capprox%201905.350%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)
![v(30\,s) \approx 4023.595\,\frac{m}{s}](https://tex.z-dn.net/?f=v%2830%5C%2Cs%29%20%5Capprox%204023.595%5C%2C%5Cfrac%7Bm%7D%7Bs%7D)