Momentum, centre of mass and systems with variable mass

Momentum of a force

The momentum J of a force F applied between times t1 and t2 is defined as:

J = integral(F dt, from t1 to t2) = Δp = p(t2) - p(t1)

This is the impulse theorem (or momentum theorem), which is very useful when F is known only by its average value Fav over a short time interval Δt (e.g. a golf ball impact, a hammer blow):

J = Fav * Δt

Example

A 60 g tennis ball arrives at 20 m/s and leaves at 25 m/s in the opposite direction after 5 ms of contact with the racket.

Delta p = m * (v2 - v1) = 0.06 * (-25 - 20) = -2.7 kg·m/s

Favg = Delta p / Delta t = -2.7 / 0.005 = -540 N

The racket therefore exerts an average force of 540 N on the ball.

Centre of mass

For a system of material points, the centre of mass G satisfies:

P = M_total * v_G, where M_total = sum(mi)

If the system is isolated, v_G is constant: the centre of mass of an isolated system moves in a straight line at a constant velocity, or remains stationary, even if the internal parts undergo complex movements (explosion, internal collisions).

System with variable mass: the rocket

For a rocket that ejects gas at a relative velocity u with respect to itself, the fundamental equation (known as Tsiolkovsky’s equation) gives:

m*(dv/dt) = -u*(dm/dt)

from which Delta v = u * ln(m_initial / m_final)

Common pitfall

For a rocket, one cannot write F = m*a directly because the mass m varies over time: one must start from dp/dt = F_ext, taking into account the term related to mass ejection, otherwise one will obtain an incorrect result.