Variable-Mass Systems (Optional Extension)

1. Definitions (Must Know)

• Variable-mass system: a system whose mass changes with time because mass enters or leaves (e.g. a rocket ejecting fuel, sand falling onto a trolley).
• Momentum, vector p: vector p = m vector v (vector).
• Mass flow rate, m dot: rate at which mass changes, in kg s⁻¹.
• Exhaust speed, vₑ: speed of expelled mass relative to the rocket (direction matters).
• Thrust, T: the force produced by expelling mass; in simple models, magnitude T ≈ m dot vₑ (given in the question or derived from a model).

Most core mechanics questions assume constant mass, so you can use: ∑ F = ma

For variable mass, start from Newton’s 2nd law in momentum form:

∑ vector Fₑₓₜ = (d vector p)/dt

where vector p = m vector v.

See also: Momentum & Impulse.

2. Key Ideas (What Earns Marks)

• Use momentum ideas and state your system clearly (“rocket only”, or “rocket + fuel”, etc.).
• Most A Level questions will either:
  • give the thrust T, or
  • give m dot and vₑ so you can find T.
• Once you have a thrust force T (already accounting for the mass ejection), you can usually write the rocket’s equation of motion at that instant as: ∑ F = ma (for the rocket’s instantaneous mass m)
• Track signs carefully (choose a positive direction and apply it consistently).

3. Detailed Explanations

A. Why F = ma needs care when mass changes

Momentum is: p = mv

If both m and v can change: dp/dt = d(mv)/dt = mdv/dt + vdm/dt

So you cannot blindly replace dp/dt with mdv/dt unless mass is constant.

B. Thrust as “momentum per second”

When a rocket ejects mass backwards, it gives that mass momentum. By Newton’s 3rd law, the rocket gets an equal and opposite effect: a forward thrust.

In a simple model (constant exhaust speed relative to the rocket), the thrust magnitude is: T ≈ m dot vₑ

Many questions treat T as given, so you can focus on forces on the rocket (thrust, weight, drag).

4. Common Mistakes

• Writing ∑ F = d(mv)/dt and then replacing it with ∑ F = mdv/dt without stating “mass constant”.
• Mixing up speed relative to the ground with speed relative to the rocket (vₑ).
• Using m dot without a sign convention (is mass decreasing or increasing?).
• Double-counting thrust (e.g. using both T and a separate “momentum flux” term when T already represents it).

5. Exam Tips

• Start with: “take forward as positive” (or similar).
• Write down what mass is changing and why (ejected/collected).
• If thrust is given, treat it as an external force on the rocket and use ∑ F = ma with the rocket’s instantaneous mass.
• Always check units: m dot vₑ has units kg s⁻¹ · m s⁻¹ = N.

6. Worked Examples

7. Mind Stretchers