Escape Velocity and Rocket Fuel Mass Calculation

Rocket Physics Problem: Escape Velocity and Fuel Requirement

Question:

a) If a rocket starts from rest and the exhaust speed of the propellant is 3 km/s, what percentage of its original mass must be fuel in order to achieve Earth’s escape velocity?

b) Plot v vs m(t) and v vs t. Is there a maximum velocity the rocket can reach?

Answer:

To solve this, we use the Tsiolkovsky Rocket Equation, which relates the change in velocity of a rocket to the exhaust velocity and the initial and final mass.

v = ve × ln(m₀ / m_f)

Where:

v = final velocity (escape velocity)
ve = exhaust velocity of the rocket = 3 km/s = 3000 m/s
m₀ = initial mass (rocket + fuel)
m_f = final mass (after burning all fuel)

The escape velocity from Earth is approximately 11.2 km/s = 11200 m/s.

Step-by-Step Calculation:

11200 = 3000 × ln(m₀ / m_f)

Divide both sides by 3000:

ln(m₀ / m_f) = 11200 / 3000 = 3.733

Now exponentiate both sides:

m₀ / m_f = e^3.733 ≈ 41.82

This means the rocket’s initial mass must be 41.82 times its final mass.

Fuel Mass Fraction:

m_f / m₀ = 1 / 41.82 ≈ 0.0239 → 2.39%

Therefore, the fuel must be 97.61% of the rocket’s total initial mass.

Graphical Interpretation:

v vs m(t): The graph shows a logarithmic increase in velocity as the mass of the rocket decreases during fuel consumption.
v vs t: The graph shows the velocity increases gradually over time until it plateaus, approaching the escape velocity limit.

Conclusion:

To reach Earth’s escape velocity with an exhaust velocity of 3 km/s, the rocket must have at least 97.61% of its original mass as fuel. This demonstrates the immense energy required to overcome Earth’s gravitational pull and explains why multistage rockets are used.

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