M038 · Draw, follow and friction throw

Billiards with Spin

Why can the cue ball come back after impact? Impact can transfer most of the forward motion while leaving rotation behind. Cloth friction then turns retained backspin into backward motion, or forward spin into follow. Friction between the balls can also deflect the object ball and transfer spin; its outgoing direction then differs from the line of centres.

Change cue offset, ball friction and cushion restitution. See draw, follow, friction throw, spin transfer and the change from sliding to rolling.

Cue ball forward velocity $-0.163\,\mathrm{m\,s^{-1}}$

Forward rim velocity $-0.163\,\mathrm{m\,s^{-1}}$

Bottom-point horizontal speed $0\,\mathrm{m\,s^{-1}}$

First-impact model throw angle $-1.2\,\mathrm{{}^{\circ}}$

Model assumptions

Two equal homogeneous rigid spheres, each 0.17 kg with radius 0.04 m, move on a 1.4 by 0.8 metre teaching table without pockets. The cue is an imposed horizontal impulse; the speed control sets the resulting ball-centre speed, not cue speed. Offsets set angular impulse, without cue deflection, tip compliance, squirt or miscues. Ball contacts use Newton normal restitution and a three-dimensional Coulomb tangential impulse capped at zero relative slip. Friction coefficients are constant, not fitted to real ball surface-speed data. Cloth friction acts only during sliding contact; free flight, gravity and perfectly inelastic landing are included. The ideal point contact has no rolling or twisting resistance, so pure rolling and vertical spin persist. Cushions are vertical rigid walls, not calibrated rubber noses. Instantaneous coincident impacts are resolved sequentially. Maintained wall contact uses a short-step unilateral contact solve; its step and convergence tolerance are in the exported record. Finite contact time, deformation and sound are omitted. Trajectories and records are simulated model outputs. The first throw angle is measured immediately after the first computed ball collision against its line of centres, before later cloth or cushion forces.

Physics tutorial

A collision can leave spin behind

BackgroundThe cue ball can give most of its forward motion to the object ball and still retain rotation. Cloth friction then connects that rotation to the next motion: backspin can bring the cue ball back, while forward spin produces follow.

Why it mattersDavid Alciatore’s technical proofs describe cue-offset angular impulse, the smooth equal-mass collision limit and friction throw. The three-dimensional rigid-impulse model here uses those balances, with constant coefficients and ideal vertical cushions.

Start with the essentials

Focus question: Why does the cue ball sometimes come back after striking another ball?
One-sentence intuition: The ball-centre velocity and the motion of its bottom point are different. Friction responds to the bottom point sliding over the cloth.

Core mathematical model

An offset impulse creates rotation

\begin{aligned}I&=\tfrac25mR^2\\I\omega_y&=bJ,\qquad I\omega_z=-aJ\end{aligned}

The initial shot points along the horizontal forward axis. The vertical and side offsets are lengths; the sliders display them divided by the radius. Ball speed after the impulse is controlled directly.

Look at the material point touching the surface

\boldsymbol v_c=\boldsymbol v+\boldsymbol\omega\times\boldsymbol r_c

The contact-point velocity includes both translation and rotation. During maintained cloth contact, its horizontal part sets the direction of sliding friction. During a hop there is no cloth friction.

Friction cannot reverse relative slip in one impact

\begin{aligned}J_n&=\tfrac12m(1+e_b)v_{\rm app}\\J_t&=\min\!\left(\mu_bJ_n,\tfrac m7|\boldsymbol u_t|\right)\end{aligned}

These expressions apply to the two identical homogeneous spheres. Tangential impulse acts against their relative surface slip and stops growing when that slip reaches zero. Its direction is fully three-dimensional.

A useful no-spin benchmark

t_r=\frac{2v_0}{7\mu_sg},\qquad v_r=\frac57v_0

For one nonspinning sphere launched on horizontal cloth, with no intervening collision, sliding ends at this time. Existing spin or a collision changes the result.

Common difficulties

Side spin is not the same as forward roll

Typical misconceptionAny visible spin must make the ball roll faster.

Better mental modelRotation about the vertical axis does not cancel bottom-point sliding in the ideal point-contact model. It can still change a ball or cushion collision.

A cushion is an external interaction

Typical misconceptionThe momentum of the two balls must stay constant through the whole shot.

Better mental modelCloth friction, gravity during hops and cushion impulses exchange momentum with the surroundings. The exported record includes the cloth, gravity and cushion contributions; each isolated ball–ball impulse preserves system momentum.

Run the experiment

01
Compare draw and follow
Use Come back after impact, then Follow the object ball.
What to observe: Both transfer forward motion at impact. The retained spin and subsequent cloth friction send the cue ball in different directions.
02
Switch throw off
Compare Sidespin throws the object ball with Remove ball friction.
What to observe: The first-impact angle changes, while the initial shot and target placement remain the same.
03
Meet a cushion
Use Sidespin meets a cushion and vary cushion restitution.
What to observe: The normal rebound changes. Cushion friction can also exchange tangential motion and spin; these are ideal vertical-wall calculations.