Section 5.6: Rocket Propulsion and Variable Mass Systems

For rockets or any system losing mass, the motion is governed by the variable mass system equation. The general form:

  • \( \vec{F}_{ext} = \frac{d\vec{P}}{dt} = M\frac{d\vec{v}}{dt} + \vec{v}_{rel} \frac{dM}{dt} \)

For a rocket in free space (ignoring gravity and drag):

  • \( M \frac{dv}{dt} = -u \frac{dM}{dt} \) → Tsiolkovsky Rocket Equation
  • Integrating: \( v_f - v_i = u \ln \frac{M_i}{M_f} \)

Where:

  • \( M_i, M_f \) = initial and final mass of the rocket
  • \( u \) = exhaust speed relative to rocket
  • \( v_i, v_f \) = initial and final velocity of rocket

Example 1

A rocket of mass 2000 kg (including fuel) expels 500 kg of fuel at 400 m/s relative to rocket. Find its velocity increase in free space.

\( \Delta v = 400 \ln \frac{2000}{1500} = 400 \ln(1.333) \approx 115 \text{ m/s} \)

Example 2

A rocket initially at rest burns 1000 kg of fuel at 300 m/s. Initial mass 3000 kg. Find final velocity.

\( v_f = 300 \ln \frac{3000}{2000} = 300 \ln(1.5) \approx 122 \text{ m/s} \)

Practice Problems

  1. A 500 kg rocket expels 200 kg of fuel at 250 m/s. Find velocity increase.
  2. Rocket mass 1000 kg, fuel 400 kg, exhaust speed 350 m/s. Find final velocity in free space.
  3. Rocket expels 600 kg of fuel, exhaust speed 500 m/s, initial mass 2000 kg. Compute Δv.
  4. A space probe mass 1500 kg burns 300 kg of fuel at 400 m/s. Find CM velocity.
  5. Rocket with initial velocity 50 m/s burns 200 kg fuel, exhaust 300 m/s. Compute final velocity.
  6. Two-stage rocket: Stage 1 Δv = 150 m/s, Stage 2 Δv = 100 m/s. Find total velocity change.
  7. Rocket in vertical flight (ignore gravity). Compute Δv after burning 25% of mass.
  8. A rocket expels fuel at 600 m/s. Find Δv if final mass = 0.7 initial mass.
  9. Variable mass system: Rocket of 1000 kg loses 200 kg in 5 s. Find acceleration at midpoint.
  10. Rocket burns fuel in vacuum. Calculate final velocity using Tsiolkovsky equation.
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