Motion of bodies of variable mass

The general equation

• When the mass changes with time (a body collecting dust or burning fuel) we must not write F = m a; the product must be differentiated: F = d(mv)/dt = m dv/dt + v dm/dt.
• The term v dm/dt appears only because the mass changes; it vanishes for constant mass and we return to F = m a.
• Procedure: write m as a function of t, get v by differentiating s, form p = m v, then differentiate p to get F.

Worked examples

1. A body of mass (2t + 1) kg moves on a line with s = t² metres. v = ds/dt = 2t m/s, so p = m v = (2t + 1)(2t) = 4t² + 2t and F = dp/dt = 8t + 2.
   At t = 3: p = 36 + 6 = 42 kg·m/s and F = 24 + 2 = 26 N.
2. A metal ball of mass 100 g moves with uniform velocity 10 m/s through dust that sticks to it at 2 g/s. Then m = 100 + 2t grams. Uniform velocity means a = 0, so F = v dm/dt. With v = 10 m/s = 1000 cm/s and dm/dt = 2 g/s: F = 1000 × 2 = 2000 dyne, constant and in the direction of motion.
3. Reading the phrase "collects dust at a constant rate" as dm/dt = 0 is wrong: it is precisely what makes dm/dt a positive constant.