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Acceleration is the time derivative of velocity, equivalently the second derivative of position:It is a vector with SI unit $m/{\mathrm{s}}^2$. A change in direction produces acceleration even if speed stays constant. For example, a car rounding a bend at steady speed needs an inward resultant force by Newton's second law. Thus acceleration does not mean only speeding up. When $v>0$, the dot product gives $dv/dt=\mathbf{v}\cdot \mathbf{a}/v$: a perpendicular acceleration changes direction without changing speed at that instant.

Take counterclockwise circular motion about the origin with radius $R=2\mathrm{m}$ and constant angular speed $\omega =0.5rad/s$. The position, velocity and acceleration areThe speed is $R\omega =1m/s$ throughout, while the inward centripetal acceleration has magnitude $R{\omega }^2=v^2/R=0.5m/{\mathrm{s}}^2$. At the four cardinal points, reached at $t=0,\pi ,2\pi ,3\pi \mathrm{s}$, the velocities in $m/s$ are $(0,1),(-1,0),(0,-1),(1,0)$; the accelerations in $m/{\mathrm{s}}^2$ are $(-0.5,0),(0,-0.5),(0.5,0),(0,0.5)$. The vectors turn together, remaining perpendicular. ^[ref]

The proof outline is just two applications of the chain rule to $\mathbf{r}(t)$; ${\sin }^2(\omega t)+{\cos }^2(\omega t)=1$ then fixes both magnitudes. The usefulness is geometric: turning requires a force even when a speed reading remains steady, and doubling speed at fixed radius quadruples the required centripetal acceleration. The same mechanism appears in a circular orbit.

Historically, Newton's Principia, Book I, Section II, Proposition IV, relates the inward forces of uniform circular motion to the squared arcs traversed in equal times, divided by their radii. In modern notation, its dependence on speed and radius is $v^2/R$; Newton presents a geometric argument rather than the coordinate derivatives above. ^[ref]

Two uniform discs of mass m and radius a stick at first edge contact. One initially translates at speed v and the other rests, with neither initially spinning. If the incoming velocity makes angle $\theta$ to the line of centres, the composite axial moment of inertia about its centre of mass is $3m{a}^2$. Conserved angular-momentum magnitude is $mav|\sin \theta |$, so the angular velocity magnitude is $v|\sin \theta |/(3a)$. The centre-of-mass speed is v/2. Splitting kinetic energy into translation and rotation gives $K_f=m{v}^2(1/4+{\sin }^2\theta /6)$. The difference from $m{v}^2/2$ is dissipated internally; it cannot be removed from angular-momentum conservation.

The drag force exerted by air on a body moving relative to it. Neglecting it is an approximation whose accuracy depends on the body, its speed and the ambient gas.

An actual teaching experiment in Arbor Scientific's The Level Air Track uses one glider, an air track, two photogates, and a ruler. Turn on the blower and place the glider near the middle. Adjust the feet until it does not drift when released. Put the photogates at least half a track length apart and align them with the glider's flag. Measure the length $L$ that interrupts the beams. Give a gentle push, let go, and record each beam-blocking time $\Delta t_i$. The readout is the pair of speeds $s_i=L/\Delta t_i$. The worksheet asks how those speeds compare, which forces remain after the push, and which interaction the air track minimizes.^[ref]

The motivating contrast is between a brief push and the subsequent glide. The air cushion reduces contact friction; it does not remove the downward gravitational weight or the upward support force. Similar successive speeds after release are consistent with a small horizontal net external force, rather than with a continuing push. The experiment approximates the zero net external force model over a finite track and observation interval.

To strengthen the documented procedure, use independent measurement calibration: check the flag length and each timer against a known length and time reference; measuring $L$ sets the length scale but does not calibrate the timer. Repeat launches in both directions without contact during the timed passage, and check for drift at several track locations. These are additional controls, not reported steps or measured results in the worksheet. A residual inclined plane produces a downhill component of gravitational weight with the same laboratory direction on both runs; friction and drag force oppose the direction of travel. Reversing the launch can therefore help distinguish these effects.

A photogate directly measures a beam-blocking time; $L/\Delta t_i$ is an average speed during that passage, not an exact instantaneous velocity. Assign the direction separately. Equal readings at two gates do not prove zero acceleration everywhere between them: opposing changes could cancel. Measurement uncertainty from timer resolution, an incorrectly measured flag length, residual slope, airflow disturbances, and remaining friction or drag force limit the inference. No measured glider speeds are assumed here.

The Alfvén velocity is the signed vector ${\mathbf{v}}_a=\mathbf{B}/\sqrt{{\mu }_0\rho }$, whose magnitude is the Alfvén speed. It follows the orientation of the magnetic field, unlike the nonnegative scalar speed.

In the isolated model, conservation of energy and angular momentum conservation reduce the exterior test particle's motion toHere $\epsilon$ is specific orbital energy and $h$ is specific angular momentum. The effective potential is ${\Phi }_{\mathrm{eff}}=h^2/(2r^2)-\mu /r$. A zero radial velocity is a radial turning point, and need not mean zero total speed: at the periapsis or apoapsis of a nonradial Kepler orbit, tangential velocity remains. Thus the vertical gap between $\epsilon$ and $-\mu /r$ represents total kinetic energy per unit mass, whereas the gap above ${\Phi }_{\mathrm{eff}}$ represents its radial part.

For $h>0$, the orbital eccentricity obeys $e^2=1+2\epsilon h^2/{\mu }^2$. Negative $\epsilon$ gives an elliptic orbit, including a circular orbit at its minimum allowed energy; zero $\epsilon$ gives a parabolic trajectory; positive $\epsilon$ gives a hyperbolic Kepler orbit. The effective potential tends to zero from below at infinity, so its outer region creates no extra positive escape barrier. This explains why escape velocity depends on energy even though angular momentum changes the path. The $h=0$ case is radial motion, rather than a nondegenerate parabola or hyperbola; inward radial motion can end in collision.

Angular speed is the nonnegative magnitude of angular velocity: $\omega =|\mathbf{\omega }|$. It measures the rate of rotation about an instantaneous axis without specifying its direction. For uniform circular motion of radius $R$, the ordinary speed is $v=R\omega$.

At radius $a$ in a ring rotating at rate $\omega$, a walker with signed tangential speed $u$ relative to the ring has inertial speed $a\omega +u$. The normal reaction per mass is $(a\omega +u{)}^2/a$; the curvature of the relative path matters as well as the centrifugal force and the Coriolis force.

The asthenosphere is a relatively weak region of the upper mantle below the mechanically strong lithosphere. A seismic low-velocity zone often occurs in this region, particularly beneath oceanic lithosphere, but mechanical weakness and seismic speed are distinct properties.

In Mass spectra and isotopes by Francis William Aston, printed pp. 10–12, Fig. 1, Francis William Aston describes successive electric and magnetic deflections in one plane, with opposite senses. This geometry focuses a range of ion speeds; it does not require the straight-through cancellation of a velocity selector.

The reason compensation is possible follows from the Lorentz force. In a short region of length ${\ell }_E$, a small electric deflection angle has magnitude ${\theta }_E\simeq qE{\ell }_E/(m{v}^2)$: the transverse acceleration acts for a time ${\ell }_E/v$, and the deflection angle is the acquired transverse velocity divided by $v$. A short magnetic region of length ${\ell }_B$ gives ${\theta }_B\simeq qB{\ell }_B/(mv)$, since the magnetic force already contains one factor of speed. Slower ions therefore receive a disproportionately larger electric deflection.

An original local approximation makes the focusing idea explicit. At fixed mass-to-charge ratio, replace each region by a small angular kick and include the drift distances in two positive constants $a$ and $b$. With $t=v/v_0$, write the detector coordinate asChoosing the geometry and fields so that $b=2a$ eliminates the first-order speed sensitivity. Indeed,A fractional speed error $\epsilon =t-1$ then contributes only at order ${\epsilon }^2$ in this approximation. This is a local thin-kick explanation, not Aston's exact pole geometry, focal-plane equation, or a prediction of its measured mass resolution. Finite slit widths, initial directions, and field boundaries still matter. Unlike a velocity selector, focusing can collect nearby speeds into a narrow image instead of discarding all but a narrow accepted range.

For an isotropic harmonic oscillator with inertial frequency $2\omega$, viewed in a frame rotating at $\omega$, initial rotating coordinates $(1,0)$ and zero relative velocity produce the displayed astroid trajectory. The complex coordinate satisfies $\ddot{\zeta }+2i\omega \dot{\zeta }+3{\omega }^2\zeta =0$, with solution $\zeta =\frac{3}{4}{e}^{i\omega t}+\frac{1}{4}{e}^{-3i\omega t}$. The relative speed is $(3|\omega |/2)|\sin 2\omega t|$, attaining its first maximum at $\pi /(4|\omega |)$ for nonzero $\omega$.

Astronomy studies celestial objects and phenomena, including stars, planets and galaxies. Astrometry measures their positions and motion; astrophysics uses physics to explain their properties and behaviour.

Much of astronomy begins with incoming light. An image tells us which direction the light arrived from, rather than directly providing the source's three-dimensional position. Two stars close together on the sky may be very far apart in space. The celestial sphere organizes these viewing directions; locating an object on it is a different task from measuring how far away that object is. Repeated observations turn this limitation into an opportunity: the changing direction of light contains evidence about both Euclidean distance and motion.

The particularly satisfying combination is stellar parallax with proper motion. The first supplies a geometric Euclidean distance, while the second measures how rapidly the viewing direction changes after accounting for the observer's annual motion. Together they give a transverse speed, connecting tiny angles on an image to kilometres travelled each second. This bridge from a flat sky map to moving stars is one reason astrometry matters to the study of the Milky Way.

A narrow fringe makes the center easier to resolve statistically; it does not certify an unbiased frequency. For example, with ideal $\pi /2$ pulses and a second-pulse oscillation phase offset ${\phi }_c$, the finite-pulse Ramsey transition probability approachesThis sign uses exactly the Rabi detuning convention and pulse Hamiltonian operator above, with the second pulse replacing ${\sigma }_x$ by $\cos {\phi }_c{\sigma }_x+\sin {\phi }_c{\sigma }_y$. The factor $2\pi$ converts angular frequency to ordinary frequency. For the original example, ${\phi }_c=0.010$ rad and $T=5\mathrm{ms}$ give a shift of about $-0.318\mathrm{Hz}$. For finite pulses or distributed flight times, use the full signal and its atomic clock frequency lock rather than inserting an unweighted mean $T$.

NIST-7 used optical pumping and fluorescence. Controls included field-sensitive transitions, independent beam-speed measurements, microwave power stabilization, and beam reversal for cavity quantum phase errors.^[ref]

The connection to special relativity is quantitative. A freely moving emitter's proper time advances more slowly than laboratory time by $\sqrt{1-v^2/c^2}$. With first-order propagation effects removed in the ideal comparison, its internal laboratory frequency is thereforeCorrecting an ensemble signal requires the effective velocity weights of preparation, interrogation and detection, not simply the oven's mean speed. Additional shifts from Zeeman splitting, unwanted light, collisions and cavity fields likewise require separate evaluation; feedback control locks to the observed line, including any uncorrected bias. Comparing clocks at different gravitational potentials additionally requires gravitational time dilation. The moving-clock experiments in twin paradox clock experiments compare calibrated clock readings along different world lines; the beam atoms' internal motion is part of the clock mechanism's own correction.

For unit mass, incoming speed $v>0$, impact parameter $b>0$, and potential energy $\mp \kappa /r$, put $q=\kappa /v^2>0$. At closest approach, conservation of energy and conservation of angular momentum give the displayed radii and tangential speeds $u_{\mp }=bv/p_{\mp }$. Consequently $p_{-}{p}_{+}=b^2$ and $u_{-}{u}_{+}=v^2$. These relations compare equal force magnitudes. The attractive radial case $b=0$ is a singular collision, so the speed-product statement does not extend to that endpoint.

In the flat-surface, constant-gravitational acceleration model, a coasting projectile conserves horizontal velocity. If it arrives at the top of its trajectory with horizontal speed $v_o$, its earlier velocity at angle $\theta$ above the horizontal has vertical component $v_o\tan \theta$. Its remaining height gain is therefore $v_o^2{\tan }^2\theta /(2g)$. The displayed burnout height follows. This is a local projectile motion approximation, not an exact Kepler orbit. Computing $v_o=\sqrt{g(R_E+h_o)}$ assumes constant gravitational magnitude also at the orbit; computing $v_o=\sqrt{g_0{R}_E^2/(R_E+h_o)}$ instead assumes an inverse-square law with surface value $g_0$. Those assumptions must be distinguished.

The nonnegative weight describing the rate and angular dependence of binary elastic collisions in the Boltzmann equation. In a scattering-angle representation with solid-angle measure $d\Omega$, it contains relative speed $g$ times differential scattering cross-section. A parametrization by collision normals has a different Jacobian. Any symmetry factor from integrating over a hemisphere instead of a sphere must be included consistently.

An infinite uniform line with proper charge per unit length ${\lambda }_0$ moving axially at speed $u$ has Lorentz-contracted density $\lambda =\gamma {\lambda }_0$ and current $I=\lambda u$. Transverse distance $r$ is unchanged by the boost. The rest electric field ${\lambda }_0\hat{\mathbf{r}}/(2\pi {ϵ}_{0}r)$ transforms into $\mathbf{E}=\lambda \hat{\mathbf{r}}/(2\pi {ϵ}_{0}r)$ and $\mathbf{B}=\mathbf{u}\times \mathbf{E}/c^2$. This equals the magnetic field ${\mu }_{0}I\hat{\mathbf{\phi }}/(2\pi r)$ from Ampère's law. The formulas concern a uniform stationary current pattern, not an accelerating finite filament's radiation.

Calculus connects local change with accumulated change. A derivative asks how fast a function changes near one input; an integral adds its contributions across an interval. The fundamental theorem of calculus explains why these operations fit together: accumulating a continuous rate produces a quantity whose derivative is that rate.

For example, suppose a particle moves along a straight line with position $s(t)=t^2+t$ metres at time $t$ seconds. Its velocity is $s^{\prime }(t)=2t+1$ metres per second. Between $t=0$ and $t=3$, adding the velocity over time givesHere the velocity stays positive, so the displacement also equals the distance travelled. If the particle reverses direction, the integral of its signed velocity still gives displacement, while the integral of its speed gives total distance.

This is useful because local rules are often easier to specify than whole trajectories. A differential equation can prescribe a particle's velocity or a population's growth rate; integration then reconstructs its evolution, subject to an initial condition. In the other direction, a derivative extracts local information from a known function, and optimization uses that information to locate or approach promising inputs. The same two operations therefore connect measurement, prediction and design.

The visual connection is developed in derivative, integral and fundamental theorem of calculus. The definitions depend on limits, which turn progressively finer finite measurements into exact mathematical statements. A graph with a corner need not have a derivative there, and the conditions permitting integration or interchange of limits matter; calculus is more than a collection of symbolic rules.

Historically, Isaac Newton described changing quantities and their rates in the language of fluents and fluxions; his manuscript History of the Method of Fluxions records that approach.^[ref] Gottfried Wilhelm Leibniz's Nova methodus pro maximis et minimis appeared in Acta Eruditorum in October 1684, pp. 467–473; the original article is preserved in the HKUST Library collection.^[ref] The modern limit definitions used here should not be read back into those seventeenth-century formulations.

The isotropic three-dimensional Cauchy velocity distribution $f_0={\rho }_{0}b/[{\pi }^2(v^2+b^2{)}^2]$, $b>0$, has density ${\rho }_0$ and one-dimensional marginal $F(u)={\rho }_{0}b/[\pi (u^2+b^2)]$. Its second velocity moment diverges, so $b$ is a scale rather than a finite velocity dispersion. At zero phase speed, $\int F^{\prime }(u)/udu=-{\rho }_0/b^2$, giving $k_J=\sqrt{4\pi G{\rho }_0}/b$. In the upper-half frequency plane, the marginal response is ${\rho }_0/(\omega /k+ib{)}^2$; the growing branch has $\omega =i(\sqrt{4\pi G{\rho }_0}-kb)$ for $0<k<k_J$.

Place an infinite sheet of surface charge density $\sigma$ at $z=0$ between grounded conducting planes $z=\pm d/2$, in vacuum. Reflection symmetry and Gauss's law give $E_z=\sigma \mathrm{sgn}(z)/(2{\epsilon }_0)$ in the two gaps. Integrating the electric field with zero electric potential at the conductors gives $V(z)=\sigma (d/2-|z|)/(2{\epsilon }_0)$. Each facing conducting surface carries induced surface charge density $-\sigma /2$. For $\sigma >0$, an electron released from rest at a plate reaches the sheet with kinetic energy $e\sigma d/(4{\epsilon }_0)$; its nonrelativistic speed follows from $m_e{v}^2/2$.