91Ó°ÊÓ

The total impulse \( I \) is calculated by the relation \( I = Ft \), where \( F \) is the average thrust and \( t \) is the thrust duration. Substituting the respective values from the problem statement: \( I = 5.26 N * 1.90 s = 10 Ns \). Note that we need not convert to kilogram or meter as the unit of force in SI system is Newton(N) and time is in seconds(s).

The average exhaust speed can be calculated using the equation for conservation of momentum in a rocket: \( v_{e} = \frac{I}{m_{Fuel}} \), where \(v_{e}\) is the exhaust velocity and \(m_{Fuel}\) is the fuel mass. We convert the fuel mass from gram to kilogram (as the SI unit for mass is kilogram). So the mass of the fuel is \(0.0127 kg\). Substituting the respective values we get: \(v_{e} = \frac{10 Ns}{0.0127 kg} = 787.4015748 m/s\). The average exhaust speed is approximately \(787.4 m/s\).

The final velocity of the rocket fired in space can be found using Tsiolkovsky's rocket equation (equation of motion): \(v_{f} = v_{e} * ln(\frac{m_{i}}{m_{f}})\), where \(m_{i}\) is the initial mass of the spacecraft and \(m_{f}\) is the final mass of the spacecraft. The initial mass (m_{i}) is total of rocket body mass and the initial mass of the engine fuel: \(m_{i} = m_{rocket} + m_{fuel_{initial}} = 0.0535 kg + 0.0255 kg = 0.079 kg\). The final mass \(m_{f}\) is the rocket's mass after the fuel is burnt which is the rocket body mass \(m_{f} = 0.0535 kg\). Substituting the respective values we get \(v_{f} = 787.4 m/s * ln(\frac{0.079 kg}{0.0535 kg}) = 316.867394 m/s\). The final velocity of the rocket is approximately \(316.8 m/s\).