Variable Mass System: Rocket Motion

In classical mechanics, a rocket is an example of a variable mass system. The mass of the rocket changes continuously as fuel is expelled in the form of high-speed gases. Since Newton’s second law is strictly applicable only for constant mass systems, we apply the modified equation for such cases.

1. Principle of Rocket Motion

Rocket propulsion is based on the law of conservation of linear momentum. When fuel burns inside the rocket, it ejects gases at high speed in the backward direction. As a result, the rocket gets an equal and opposite forward momentum, making it move upward.

2. Upward Force on the Rocket

The upward force (or net thrust) acting on a rocket is given by:

\(F={V_e}\frac{dm}{dt}\) where,
  • F = Thrust (upward force)
  • \(V_e\) = Velocity of exhaust gases relative to the rocket
  • \(\frac{dm}{dt}\) = Rate of change of mass (rate of fuel ejection)

3. Acceleration of the Rocket

The acceleration of the rocket at any instant is:

If acceleration of the rocket is "a" then the upward force is F=m(g+a)

so,\(m(g+a)={V_e}\frac{dm}{dt}\)

\((g+a)=\frac{V_e}{m}\frac{dm}{dt}\)

\(a=\frac{V_e}{m}\frac{dm}{dt}-g\)

Where:

  • m = Instantaneous mass of the rocket
  • g = Acceleration due to gravity

This shows that the rocket accelerates only when the thrust force exceeds the weight.

4. Instantaneous Velocity of the Rocket

The instantaneous velocity v of the rocket after time t is:

\(\frac{dV}{dt}=-u\frac{dm}{dt}\)

\(dV=-u\frac{dm}{m}\)

\(\int_{v_0}^VdV=-u{\int_{m_0}^m}\frac{dm}{m}\)

\([v]_{0}^{V}=-u[lnm]_{m_0}^{m}\)

\(V-V_{0}=-u[{lnm-lnm_{0}}\)]

\(V=V{0}+u[{lnm_{0}}-lnm\)],

\(V=V_{0}+uln\frac{m_0}{m}\)

if at t=0 \(v_0=0\) then, \(V=uln\frac{m_0}{m}\)

  • m0 = Initial mass of the rocket
  • m = Mass at time t
  • ln = Natural logarithm

This equation is derived from integrating the modified Newton's law for variable mass systems.

5. Important Points

  • The rocket moves upward due to conservation of momentum.
  • As the mass decreases, the velocity increases.
  • The velocity is not uniform—it changes with time.
  • Air resistance and gravity affect real-world rocket motion.

6. Short Questions and Answers

  1. Q: What is a variable mass system?
    A: A system whose mass changes with time, such as a rocket ejecting fuel.
  2. Q: On what principle does a rocket work?
    A: A rocket works on the law of conservation of linear momentum.
  3. Q: What is the formula for thrust on a rocket?
    A: Thrust, F = ve * (dm/dt)
  4. Q: Why does the rocket velocity increase with time?
    A: Because its mass decreases as fuel burns, and the thrust remains high.
  5. Q: What is the role of exhaust velocity in rocket motion?
    A: Higher exhaust velocity provides greater thrust to the rocket.
  6. Q: Write the expression for rocket velocity.
    A: v = ve * ln(m0/m) - gt