⦁ A rocket of Total mass 1.11 x 10s kg of which 8.70 x 10* kg ts fuel, is to be launched vertically. The fuel will be burned at constant rate of 820 kg/s. Relative to the rocket, what is the minimum exhaust speed that allows liftoff at launch

To determine the minimum exhaust speed required for the rocket to achieve liftoff, we can apply the principles of rocket propulsion, specifically using the Tsiolkovsky rocket equation. This equation relates the change in velocity of a rocket to the exhaust velocity of the propellant and the mass of the rocket before and after the fuel is burned. Let's break this down step by step.

Understanding the Variables

First, we need to identify the key variables in this scenario:

• Total mass of the rocket (m0): 1.11 x 10⁵ kg
• Mass of the fuel (mf): 8.70 x 10⁴ kg
• Burn rate of the fuel (ṁ): 820 kg/s
• Gravitational acceleration (g): approximately 9.81 m/s²

Applying the Rocket Equation

The Tsiolkovsky rocket equation is given by:

Δv = v_e * ln(m0/mf)

Where:

• Δv: Change in velocity needed for liftoff
• v_e: Exhaust velocity
• m0: Initial total mass of the rocket (including fuel)
• mf: Final mass of the rocket after burning the fuel

Calculating the Final Mass

To find the final mass of the rocket after burning the fuel, we can use:

mf = m0 - fuel burned

Assuming we want to find the minimum exhaust speed for liftoff, we need to consider that the rocket must overcome the force of gravity. The force due to gravity is:

F_gravity = m0 * g

For the rocket to lift off, the thrust (which is equal to the mass flow rate times the exhaust velocity) must be greater than or equal to the gravitational force:

Thrust = ṁ * v_e ≥ m0 * g

Finding the Minimum Exhaust Speed

Rearranging the thrust equation gives us:

v_e ≥ (m0 * g) / ṁ

Now, substituting the known values:

• m0 = 1.11 x 10⁵ kg
• g = 9.81 m/s²
• ṁ = 820 kg/s

Plugging these into the equation:

v_e ≥ (1.11 x 10⁵ kg * 9.81 m/s²) / 820 kg/s

Calculating this gives:

v_e ≥ (1.091 x 10⁶ kg·m/s²) / 820 kg/s

v_e ≥ 1330.49 m/s

Conclusion

Therefore, the minimum exhaust speed required for the rocket to achieve liftoff is approximately 1330.49 m/s. This speed ensures that the thrust generated by the rocket is sufficient to overcome the gravitational pull acting on it, allowing it to ascend into the atmosphere.