Energy Budget for a Rocket in Free Space: Recall the analysis of the rocket engine in Project I, Flight of the Rocket. Although the momentum of the rocket and its exhaust is conserved (constant), their energy is not; energy must be supplied by the combustion of fuel. WARNING: This is not the same as the energy requirement of the jet airplane in Project II, Nonstop Service to Anywhere. Using the same approach we used to derive the First and Second Rocket Equation ---- sequential “snapshots” of the rocket---obtain an expression for the total kinetic energy gained by rocket and exhaust during the engine burn, in terms of the initial and final rocket masses Mi and Mf , the nozzle speed U of the exhaust, etc. This is the energy that must be supplied by combustion. HINTS: You may need to use the First Rocket Equation in your calculation. You may also use the fact that squares and products of differentials are negligible compared to the differentials themselves.

Question

Energy Budget for a Rocket in Free Space: Recall the analysis of the rocket engine in Project I, Flight of the Rocket. Although the momentum of the rocket and its exhaust is conserved (constant), their energy is not; energy must be supplied by the combustion of fuel. WARNING: This is not the same as the energy requirement of the jet airplane in Project II, Nonstop Service to Anywhere. Using the same approach we used to derive the First and Second Rocket Equation ---- sequential “snapshots” of the rocket---obtain an expression for the total kinetic energy gained by rocket and exhaust during the engine burn, in terms of the initial and final rocket masses Mi and Mf , the nozzle speed U of the exhaust, etc. This is the energy that must be supplied by combustion. HINTS: You may need to use the First Rocket Equation in your calculation. You may also use the fact that squares and products of differentials are negligible compared to the differentials themselves.

Answer 1

To derive an expression for the total kinetic energy gained by the rocket and exhaust during the engine burn, we can consider the following steps:

1. Start by considering a small time interval dt during the engine burn, and let the mass of the rocket at the beginning and end of this interval be Mi and Mf, respectively.

2. According to the First Rocket Equation, the change in velocity of the rocket during this interval is given by:
Δv = U ln(Mi / Mf)
where U is the exhaust velocity.

3. The mass flow rate, ṁ, is the rate at which propellant is being consumed during this interval and can be calculated using the First Rocket Equation:
ṁ = -Mi / (U ln(Mi / Mf))

4. The change in kinetic energy of the rocket during this interval is:
ΔKE_rocket = (1/2) Mi (v_f^2 - v_i^2)
where v_i is the initial velocity and v_f is the final velocity.

5. The change in kinetic energy of the exhaust during this interval can be approximated as the kinetic energy of the exhaust at the end of the interval:
ΔKE_exhaust ≈ (1/2) (Mf - Mi) (U^2)

6. The total change in kinetic energy is the sum of the change in kinetic energy of the rocket and the exhaust:
ΔKE_total = ΔKE_rocket + ΔKE_exhaust

7. Finally, integrate the total change in kinetic energy over the entire engine burn to obtain the total kinetic energy gained by the rocket and exhaust:
KE_total = ∫ ΔKE_total dt

Note: This derivation assumes that the rocket is in free space, without any external forces acting on it, and neglects the effects of gravity and air drag.

By following these steps, you should be able to obtain an expression for the total kinetic energy gained by the rocket and exhaust during the engine burn, in terms of the initial and final rocket masses (Mi and Mf), the nozzle speed (U) of the exhaust, etc.