M035 · Closed-system momentum

Recoil & Explosion Designer

Can internal releases steer the centre of mass? Internal releases cannot steer the centre of mass. A payload gains momentum and the remaining carrier takes the opposite change; an off-centre release also turns it. With no external horizontal force, all the parts together keep the original centre-of-mass velocity, whether they separate together or one at a time.

Arrange up to four payloads on a frictionless carrier. Change each release direction, energy and timing; compare recoil and spin with the unchanged centre-of-mass motion.

COM speed $0.25\,\mathrm{m\,s^{-1}}$

Carrier speed $0.215\,\mathrm{m\,s^{-1}}$

Carrier angular velocity $-0.14\,\mathrm{rad\,s^{-1}}$

Released spring energy $0.06\,\mathrm{J}$

Model assumptions

The carrier and payloads are uniform disks in one horizontal plane. Gravity is balanced by ideal frictionless supports. Arms, guides, feet and springs are massless. Attached parts form one rigid body. Releases are instantaneous internal impulses along each nozzle; equal reaction impulses include carrier rotation. Simultaneous releases share one impulse-progress parameter and each source supplies its stated positive work. Zero-energy unlocking retains the local rigid-body velocity and spin. Free disks then coast without drag. The record ends before the first free-body recontact; subsequent impacts are not modeled. Rotation is positive counterclockwise in the top view; gold denotes the mass-weighted centre-of-mass projection, not another body.

Start with the essentials

Focus question: Can changing internal release directions make the gold centre-of-mass marker turn?
One-sentence intuition: Internal impulses occur in opposite pairs. They cancel in total momentum; their angular effects also balance when orbital motion and disk spin are both counted.

Core mathematical model

Centre of mass from actual positions

\mathbf R=\frac{\sum_i m_i\mathbf r_i}{\sum_i m_i},\qquad \mathbf P=\sum_i m_i\mathbf v_i=M_{\mathrm{tot}}\dot{\mathbf R}

The gold marker is evaluated from the carrier and every payload, including those already far away. Its projected trail remains straight when the initial drift is nonzero.

An internal impulse and its reaction

\Delta\mathbf p_i=J_i\mathbf n_i,\qquad \Delta\mathbf P_{\mathrm{rem}}=-J_i\mathbf n_i,\qquad \Delta L_{\mathrm{rem}}=-\mathbf r_i\times J_i\mathbf n_i

The lever arm is measured from the remaining rigid group’s centre of mass. Opposite radial shots can cancel recoil; an off-centre nozzle can turn the carrier.

No horizontal external force

\dot{\mathbf P}=\mathbf0,\qquad \mathbf R(t)=\mathbf R(0)+\frac{\mathbf P(0)}{M_{\mathrm{tot}}}t

The support balances gravity but supplies no horizontal force or torque. Changing timing does not change this straight-line law.

Energy and angular momentum

E=\sum_i\left(\frac12m_iv_i^2+\frac12I_i\omega_i^2\right)+U_{\mathrm{spring}},\qquad L=\sum_i\left(\mathbf r_i\times m_i\mathbf v_i+I_i\omega_i\right)

Uniform disk inertia is used for each physical disk. The energy plot separates the remaining rigid group’s translation and rotation, and includes the intrinsic spin of each free disk.

Common difficulties

The carrier is only one part

Typical misconceptionIf the carrier bends away, the centre of mass must bend too.

Better mental modelThe payloads carry the compensating momentum. The mass-weighted sum of all trajectories retains the original centre-of-mass motion.

Unlocking need not add energy

Typical misconceptionA zero-energy payload must stop when released.

Better mental modelBefore release, a spinning assembly already gives each payload a tangential velocity and spin. Unlocking preserves those instantaneous velocities.

Run the experiment

01
Turn the carrier
Watch Off-centre recoil. Drag a payload or its aiming handle and replay the result.
What to observe: The carrier recoils and turns after each release; the gold trail keeps its direction.
02
Program multiple releases
Select each payload and set its mass and energy. Open the release program to choose separate times or add a fourth payload.
What to observe: Simultaneous sources act on the same remaining carrier. Separating them in time changes the later launcher orientation and fragment paths.
03
Release without a spring push
Choose Unlock a spinning assembly, then compare its trails with Opposite simultaneous shots.
What to observe: The unlocked disks leave tangentially with no added spring energy. The opposite-shot preset instead converts spring energy into balanced outward motion.