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For a jet arriving horizontally at a target displaced horizontally by w and upward by h, the target lies at the projectile apex. Vertical kinematics gives $u_y^2=2gh$ and flight time $u_y/g$; the horizontal travel then fixes the displayed $u_x$. The launch speed is $\sqrt{u_x^2+u_y^2}$ and $\tan \theta =2h/w$. The ideal ballistic approximation neglects air drag, breakup and internal jet stresses after launch.

Motion is a change of position with time, described relative to a reference frame. A book resting on a train has constant position in the train's frame while its position changes in a frame attached to the track. Neither description is complete until the frame and the measured time have been specified.

A record of successive positions gives a trajectory. Comparing nearby measurements gives velocity, and changes of velocity give acceleration. A camera recording a rolling ball at known time intervals provides a simple measurement: unequal gaps between successive images reveal a changing speed. The description of this record is kinematics; explaining the change using forces belongs to particle dynamics.

Use the fundamental SU(3) normalization $\mathrm{tr}(T^a{T}^b)={\delta }^{ab}/2$, with Hermitian generators. The average over the $3\times 3$ incoming colours and sum over the outgoing colours givesThe complete spin and colour averaged result is consequentlyFor massless two-body centre-of-momentum kinematics, $t=-s(1-\cos \theta )/2$ and $u=-s(1+\cos \theta )/2$, so the same answer is $2g_s^4(1+{\cos }^2\theta )/9$. This checks positivity, the symmetry under exchanging the outgoing directions, and the absence of a forward $t$-channel singularity in this particular process.

This is the charge-and-parity-conjugate reaction of part (i). There are $t$-channel Z boson and crossed W boson contributions, with all fermion arrows reversed relative to part (i). For matched unpolarized kinematics and negligible CP violation,Combining all four results in the regime specified in part (i),In particular, conjugating both initial particles restores the part-(i) rate; conjugating the neutrino alone does not. Initial spin polarization and energies near a weak-boson mass require a separate comparison.

For a parent at rest, momentum conservation gives equal and opposite daughter and alpha particle momenta. In the nonrelativistic recoil approximation,Approximating nuclear masses by their mass numbers, $M_D\simeq (A-4)m_u$ and $m_{\alpha }\simeq 4m_u$, givesThe printed phrase “atomic number $A$” must here mean mass number. Atomic number alone does not determine the recoil fraction. For example, ${}^{210}\mathrm{Po}$ and ${}^{212}\mathrm{Po}$ have the same atomic number $84$, but the approximate fractions are $206/210$ and $208/212$, respectively.

For completeness, exact relativistic two-body decay kinematics giveswhere $M_P=M_D+m_{\alpha }+Q/c^2$. Expanding for $Q\ll M_P{c}^2$ recovers the recoil-sharing expression.

The three on-shell spatial momenta have nine components. Four conservation constraints leave five, and an unobserved overall spatial rotation removes three for the scalar decay, leaving two internal kinematic variables. The pair invariants obeyThus two determine the third. They also determine each rest-frame daughter energy,and hence all momentum magnitudes and pairwise angles. The momentum triangle is fixed up to overall orientation. The Dalitz plot describes this internal kinematics, rather than claiming to record three independently oriented Four-vectors. Polarized parents or additional measured spin information can require orientation variables beyond the two invariant axes.

The charged two-pion and three-pion channels can be assigned to the same spin-zero kaon by their parent mass and lifetime. For two spinless pions, total angular momentum zero forces their relative orbital angular momentum to zero, so the product of their intrinsic parities givesFor three spinless pions, use Jacobi orbital angular momenta $l,L$. Coupling them to total zero requires $l=L$, and henceThus both channels cannot conserve parity in decays of one parent with a definite parity. The weak decay interaction permits the observed coexistence.

Dalitz plots identify the three-body events, determine their internal kinematics, and allow amplitude and angular-momentum analyses through their populations and resonance structures. Together with the spin and parent-identity evidence, they support the parity assignment. They are not a standalone parity-odd observable: reversing all three momenta leaves every $s_{ij}$, and therefore every point on the plot, unchanged. A flat distribution by itself cannot establish parity violation without the channel and spin information.

Neglecting the electron mass in the centre-of-mass kinematics, take $k=(E^{\ast },0,0,E^{\ast })$, $p=(E^{\ast },0,0,-E^{\ast })$, and $k^{\prime }=(E^{\ast },E^{\ast }\sin {\theta }^{\star },0,E^{\ast }\cos {\theta }^{\star })$. Then $p\cdot k=2E^{\ast 2}$ and $p\cdot k^{\prime }=E^{\ast 2}(1+\cos {\theta }^{\star })$. HenceThe constant Jacobian $|d\cos {\theta }^{\star }/dy|=2$ givesIncluding both Standard Model chiral channels in the contact approximation instead gives $d\sigma /dy\propto g_L^2(1-y{)}^2+g_R^2$. The right-electron term is flat in $y$.

In the electron-rest laboratory frame $p=(m_e,0)$, the invariant definition immediately givesThis last equality is exact. With electron mass retained in the centre-of-mass relation, the angular formula becomes $y=(s-m_e^2)(1-\cos {\theta }^{\star })/(2s)$, approaching the boxed massless relation at high energy.

The literal $t\to qZ\gamma$ final state has a three-vertex loop using the Standard Model quartic vertex $W^{+}{W}^{-}Z\gamma$. The two other vertices are $t\to d_i{W}^{+}$ and $d_i{W}^{+}\to q$, with $q=u,c$. The internal down-type quark joins those two vertices, while two charged-vector propagators join them through the quartic vertex emitting both neutral bosons. The bottom panel of the figure shows this topology, satisfying the three-vertex hint without dropping the photon. Its additional electromagnetic coupling is contained in the four-leg vertex; counting vertices alone does not determine coupling order.

The companion single-$Z$ penguin diagram illustrates $t\to qZ$: it instead attaches the neutral boson to the internal quark line and has three cubic vertices. These are different final states. A complete gauge-invariant amplitude requires all companion attachments; for $qZ\gamma$ these include photon radiation from charged lines as well as the quartic graph.

Both flavour sums have the formwhere the loop function also depends on the final-state kinematics. After cancellation of a mass-independent part by GIM mechanism, the bottom-quark mass gives the largest surviving contribution: $m_b^2\gg m_s^2,m_d^2$, while $|V_{tb}{V}_{cb}|\simeq 0.042$ is not unusually small compared with the strange combination. The larger $|V_{cb}|$ than $|V_{ub}|$ then favours final charm. Schematically,The same flavour preference applies to the separate two-body modes $t\to cZ$ and $t\to c\gamma$. This estimates a relative flavour preference, not an absolute branching ratio. Loop factors and mass-difference suppression remain, so these channels are fully compatible with GIM. A resolved extra photon also introduces another electromagnetic coupling and three-body phase space; an experimental $Z\gamma$ definition needs photon-energy and isolation cuts to distinguish hard radiation from the soft limit.

At leading order, neglect the transverse momentum of the incoming partons and of the produced on-shell boson. In the hadron centre-of-mass frame,Then $m_W^2=(p_{+}+p_{-}{)}^2=x_{+}{x}_{-}s$, andSolving these product and ratio equations givesOver $-2<y<2$, each momentum fraction ranges betweenThey are anticorrelated: increasing $y$ increases $x_{+}$ while decreasing $x_{-}$. Higher-order recoil changes the simple two-to-one kinematics.

The massless two-body kinematics gives $q_1\cdot q_2=M^2/2$. The scalar scattering amplitude therefore has constant magnitude $|{\mathcal{A}}_{aa}|=|{\lambda }_a|M^2/2$. Use $\int d\Omega =4\pi$ and the identical-particle factor from part d:For the photons, the two-photon vertex of a scalar field-strength interaction following from $\Phi F_{\mu \nu }{F}^{\mu \nu }$ is, up to an irrelevant overall phase,Vertex qualification: the source prints same-index polarization contractions in the second product, whereas the contractions must be crossed as above. The printed expression fails the Ward identity. For a concrete counterexample take $q_1=(E,0,0,E)$, $q_2=(E,0,0,-E)$ and ${ϵ}_2=(1,1,0,-1)$. This is a transverse unit spacelike photon polarization vector, since $q_2\cdot {ϵ}_2=0$ and ${ϵ}_2^2=-1$. Replacing ${ϵ}_1$ by $q_1$ in the printed expression leaves $4{\lambda }_{\gamma }{E}^3$ instead of zero. The corrected crossed expression vanishes under either such replacement.

Choose physical spatial transverse photon polarizations along $x$ and $y$ in the parent rest frame. Then both crossed products vanish; the two equal-polarization combinations have $|{ϵ}_1\cdot {ϵ}_2{|}^2=1$ and the two orthogonal combinations give zero. There is no initial-spin average for a spin-zero parent. HenceThe calculation in this spatial transverse gauge also yields the intended result from the misprinted expression, but only the corrected vertex makes its gauge invariance explicit. For equal-magnitude couplings, the photon decay width is twice the two-scalar decay width because of its two nonzero polarization combinations.

A robot arm is an articulated mechanism whose joints move an end effector, such as a gripper or tool, relative to a base. Forward kinematics maps joint coordinates to the tool's position and orientation; inverse kinematics asks which joint coordinates realize a desired tool configuration. The mechanism's degrees of freedom and joint limits constrain which configurations are reachable.^[ref]

For a planar arm with two rotating joints and link lengths $l_1,l_2$, let ${\theta }_1$ measure the first link's angle from the horizontal and ${\theta }_2$ the second link's angle relative to the first. Its tip satisfiesWith both lengths $1\mathrm{m}$, angles $(0,\pi /2)$ put the tip at $(1,1)\mathrm{m}$. Angles $(\pi /2,-\pi /2)$ reach the same position with a different arm shape and tool orientation. The example explains why inverse kinematics can have multiple solutions and why specifying position alone differs from specifying a full configuration.

In a physical experiment, joint encoders report angles while an external camera measures the tip. Comparing measured and predicted positions tests the geometric model; load-dependent deviations reveal flexibility or calibration errors that ideal kinematics omits. Feedback control then uses measured deviations to adjust the actuators.

A roboticist develops or studies robots by combining their mechanical design, sensing, computation and control. The work connects a desired physical task to a machine that can actually perform it. Kinematics describes permitted motion; robot motion planning selects motions subject to constraints; feedback control uses measurements to correct execution. An algorithm and the apparatus it controls must therefore be evaluated together.

For an original planar robot arm example, let two links have lengths $l_1,l_2$ and relative joint angles ${\theta }_1,{\theta }_2$. Its end effector position isWith both lengths $0.5\mathrm{m}$ and angles $(0,\pi /2)$, the endpoint is $(0.5,0.5)\mathrm{m}$. A small first-joint offset $\delta {\theta }_1=0.01$ radians changes the position to first order by $(-y,x)\delta {\theta }_1=(-0.005,0.005)\mathrm{m}$, an error magnitude about $7.07\mathrm{mm}$. Accurate commanded numbers do not imply equally accurate physical geometry.

A proposed investigation compares measured endpoint positions with the kinematics model over several configurations, then checks whether the discrepancies follow a joint offset, a link-length error or changing mechanical deflection. Those alternatives suggest different corrections. It also tests execution on the real robot, rather than treating successful numerical planning as sufficient evidence of task completion.

Kris Hauser's author-hosted Robotic Systems by Kris Hauser connects geometric configurations, inverse kinematics, motion selection and contact constraints in one technical account.^[ref] Its contact discussion makes another key distinction: deciding when to touch the environment changes the applicable physical constraints. A roboticist may consequently work with mechanisms, algorithms or experiments while still needing a common model of what is sensed, commanded and physically achieved.

Robotics studies the design, modelling and control of machines that interact with their surroundings through sensing and motion. A robot combines sensors, actuators, computation and a mechanical structure. A robot arm must relate joint motion to the position and orientation of its end effector; a mobile robot must also estimate where it is relative to its environment.^[ref]

Kinematics asks which motions a mechanism permits. Dynamics determines the forces and torques needed to produce them, while feedback control corrects deviations using measurements. A desired path is useful only when the mechanism can realize it within constraints such as motor effort, contact and obstacles. These connections make robotics a meeting point of mechanics, geometry, statistical inference and control theory.