A scrum has two distinct physics phases. First the hit: both packs sprint the engagement gap and collide almost inelastically — they lock and share a common velocity, exactly like a perfectly inelastic collision:
Hit: p_A = m_A·v_A, p_B = m_B·v_B (v_B taken as negative, opposing)
v_combined = (p_A + p_B) / (m_A + m_B)
Impulse J = |p_A| + |p_B| (momentum removed from each pack at contact)
Then the sustained drive: each pack keeps applying a roughly constant push force (legs driving through the hips), opposed by the other pack's force and by friction between boots and turf. The whole locked mass obeys Newton's second law along the scrum axis:
F_net = F_A − F_B − μ·N·sign(v), N = (m_A+m_B)·g
a = F_net / (m_A + m_B)
v(t+dt) = v(t) + a·dt, x(t+dt) = x(t) + v(t)·dt
- Drive power sliders — the steady force (N) each pack's forwards can sustain; a heavier, stronger pack (more power) generates more thrust but the simulation keeps mass fixed per pack (8 forwards each) so it's a pure force contest.
- Ground friction μ — higher μ (dry turf, good studs) resists motion more, so both packs need more surplus force to actually shift the scrum; low μ (wet/muddy pitch) lets even a small force advantage translate into rapid ground gained.
- Engage Scrum — runs the hit (computing the inelastic collision above) then the drive phase in real time until the scrum reaches the sideline or stalls.
- The push bar shows the instantaneous force balance; ground gained is the net displacement of the locked scrum along its axis — a real "scrum penalty" or "won strike" is exactly this quantity crossing a threshold.
Real-world relevance: this is the same momentum/impulse and net-force reasoning that explains any rugby collision — tackles just compress the same physics into a single instant instead of a sustained push.