16 foundation readings sit behind this argument.
The question we were answering:
The idea we opened:
Explanation · Argument synopsis
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Reading a graph How does a curve show the relationship between two physical quantities?
The horizontal axis (abscissa) typically shows the independent variable, like position x or elapsed time t. The vertical axis (ordinate) shows the quantity that depends on it, like particle count or probability density p(x).
A flat horizontal line means the quantity does not change across the axis. A rising curve has positive slope (increasing rate), while a symmetric bell-shaped curve shows high concentration in the centre with tails tapering toward zero on both sides.
One worked example Plot particle positions −3, −1, +1, +3 along the horizontal axis with a vertical bar for particle count. The graph shows symmetric spread centered at zero. The average position is 0, but the points are visibly separated from the origin. A stopping point: Check axis labels, units, and whether the curve represents a count, a density, or an accumulated total.
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Fractions and ratios How does dividing into parts compare two quantities?
A fraction a/b compares quantity a to quantity b, or tells us how many parts of size 1/b make up a. When b is not zero, dividing a by b calculates how many units of b are contained within a.
In physical science, a ratio often carries units: dividing displacement in micrometres by time in seconds produces a rate in micrometres per second. A dimensionless ratio occurs when both quantities share the same units, such as the fraction of an ensemble that moves in a given direction.
One worked example If 1 particle out of 4 moves by 3 units, the proportion is 1/4 = 0.25. If the ensemble doubles to 8 particles and 2 move by 3 units, the proportion is 2/8 = 0.25. Multiplying both the count and the total by 2 preserves the underlying ratio. A stopping point: A ratio compares relative magnitudes; always state whether the physical units cancel or combine.
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A sign records direction How can movement add up to zero?
Put the starting point at zero on a ruler. A final mark three units right has displacement +3; one three units left has displacement −3. Adding the signed displacements gives zero. Adding the distances from the start gives six.
Displacement compares the final and initial positions. It is not the length of the path travelled between them.
One worked example Two walkers both finish one unit from their start, one at −1 and the other at +1. The signed average is zero. The average distance is one.
A stopping point: The sign names a direction relative to a chosen axis; it does not mean a negative distance.
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Squares and square roots Why does four times a mean square mean twice the RMS?
Both +3 and −3 have square 9. Squaring therefore removes the sign while giving greater weight to a larger magnitude. A square root returns the result to the original kind of unit: the square root of a square micrometre is a micrometre.
The square root of four q is twice the square root of q, for nonnegative q.
One worked example The mean square of −3, −1, +1, +3 is 5. Doubling each displacement gives −6, −2, +2, +6, whose mean square is 20. The RMS changes from approximately 2.236 to 4.472: twice as large, not four times as large.
A stopping point: Use the nonnegative square root when reporting a magnitude.
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Adding and averaging What does an average keep, and what does it lose?
Imagine four equally sized containers holding 3, 1, 1 and 3 units. Combining them gives 8 units. Sharing the total equally between the four containers gives 2 in each. Two is the average; the original containers were not all equal.
The number of observations and the sum of their values are different quantities. Doubling the number of repeated observations doubles the sum but leaves the average unchanged.
One worked example Add 3 + 1 + 1 + 3 to obtain 8. Count four values. Divide 8 by 4 to obtain 2. Repeating the list gives 16 divided by 8, still 2. A stopping point: An average is a total shared equally among the number of observations.
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Rates of change and derivatives What physical rate does a derivative measure?
When a quantity changes over an interval of time or space, the ratio of the change in output to the change in input gives an average rate. As the measurement interval shrinks toward zero, this average ratio approaches a definite limit: the instantaneous derivative.
Geometrically, the derivative is the slope of the tangent line to the function graph at a single point. Physically, it always carries the units of the output divided by the units of the input. For position over time, the derivative is velocity in metres per second; for concentration over position, it is a spatial gradient in particles per metre to the fourth power.
In 1905, Einstein used derivatives to relate microscopic flux to macroscopic concentration gradients, and to extract temperature and wavelength dependencies from radiation laws without guessing unmeasured intermediates.
One worked example The derivative of f with respect to x is the limit of the difference quotient as delta x approaches zero.
If position x(t) = c t^2 with c = 3 metres per second squared, the change between t and t + Δt is c(t+Δt)^2 - ct^2 = 2ctΔt + c(Δt)^2. Dividing by Δt gives 2ct + cΔt. In the limit Δt -> 0, the instantaneous velocity is exactly 2ct = 6t metres per second.
A stopping point: A derivative is an instantaneous rate bearing explicit units; it is not a fraction of two separate isolated zeros.
Interactive construction: local sensitivity and derivative units In section 8 of the light-quanta paper, Einstein predicts that when light liberates electrons from a cathode, increasing the light frequency ν increases the required stopping potential V linearly. The derivative dV/dν is the local sensitivity of stopping voltage to incident frequency.
Choose a frequency step Δν:
+1.0 × 10¹⁴ Hz (large step) +5.0 × 10¹³ Hz (medium step) +1.0 × 10¹³ Hz (small nudge) +2.0 × 10¹² Hz (fine nudge)
Observed sensitivity response Baseline frequency (ν₀): 6.00e+14 Hz Frequency nudge (Δν): +1.00e+13 Hz Potential change (ΔV): +4.135668e-2 V Sensitivity ratio (ΔV / Δν): 4.135667696e-15 V·s (or V/Hz) Universal ratio h/e: 4.135667696e-15 V·s Summary of frequency steps and sensitivity ratio Nudge size Δν (Hz) ΔV (V) Ratio ΔV / Δν (V·s) +1.0 × 10¹⁴ Hz 1.0e+14 4.1357e-1 4.135668e-15 +5.0 × 10¹³ Hz 5.0e+13 2.0678e-1 4.135668e-15 +1.0 × 10¹³ Hz 1.0e+13 4.1357e-2 4.135668e-15 +2.0 × 10¹² Hz 2.0e+12 8.2713e-3 4.135668e-15
Textual summary of the construction A derivative is not a dimensionless number; it has physical units determined by the ratio of output units to input units. Here, dividing volts by hertz yields volt-seconds. Regardless of how small the nudge step Δν is chosen, the ratio ΔV / Δν evaluates to the exact physical constant h/e ≈ 4.14 × 10⁻¹⁵ V·s, confirming that the sensitivity of stopping potential to frequency is universal and independent of the metal.
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Density is not probability What does the height of a probability curve mean?
Divide a ruler into bins. The fraction of observations in a bin estimates its probability. Dividing that fraction by the bin width estimates a density. Changing the ruler from metres to micrometres changes the numerical density but not the probability of the same physical interval.
The probability of the interval from a to b is the integral of the density over that interval.
A continuous distribution gives probability zero to any single exact position. At the starting time the ideal point source is instead an atom of probability one, not a finite curve with infinite height.
One worked example A uniform density of 0.25 per micrometre on a four-micrometre interval has total probability one. A one-micrometre subinterval has probability 0.25; a two-micrometre subinterval has probability 0.5.
A stopping point: Always distinguish curve height, interval area and an individual observation.
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Forces on charges, currents, and electromagnetic waves How do charges interact with electric and magnetic fields?
Electric charge is the intrinsic property of matter that produces and responds to electromagnetic fields. A charge q at rest in an electric field E experiences an electrostatic force F = q E. When moving with velocity v through a magnetic field B, the charge experiences an additional perpendicular magnetic force F = q (v x B). Together, these form the Lorentz force.
Moving a charge through an electric potential difference Delta V transfers potential energy Delta W = q Delta V. For a fundamental electron charge e = 1.602e-19 C accelerated across a potential of 1 V, the energy gained is defined as 1 electron-volt (1 eV = 1.602e-19 J).
In microscopic matter, an electron bound to an equilibrium position by a restoring force acts as a harmonic oscillator or resonator. When disturbed by incoming electromagnetic waves, it absorbs and reradiates energy at its natural resonant frequency nu_0. Einstein's 1905 light-quanta paper opened with Planck's model of such resonators in thermal equilibrium with radiant energy.
Maxwell's electrodynamics expresses four physical laws in words: electric charges act as sources of electric flux; magnetic field lines are closed loops with no isolated magnetic charges; a time-varying magnetic field induces a circulating electric field (Faraday induction); and electric currents alongside time-varying electric fields generate circulating magnetic fields (Maxwell-Ampere law). Combined, these equations govern self-propagating electromagnetic waves traveling at speed c in vacuum.
Einstein pointed out in 1905 that classical electrodynamics treated the relative motion of a magnet and a conductor with an artificial asymmetry: moving the magnet created an electric field in space that drove current, whereas moving the conductor created no electric field but rather a magnetic Lorentz force on electrons. Yet the physical current was identical in both descriptions.
One worked example Force equals charge times the sum of electric field and the cross product of velocity with magnetic field, and work equals charge times potential difference.
Accelerating an electron of charge e = 1.602e-19 Coulombs across an electric potential difference of Delta V = 1.0 Volt gives kinetic energy Delta W = (1.602e-19 C)(1.0 V) = 1.602e-19 Joules = 1 eV. An electron bound with effective spring constant k_s and mass m oscillates at natural frequency nu_0 = (1 / 2 pi) sqrt(k_s / m).
A stopping point: Charge is the property that makes electric forces; fields mediate force between separated charges without action-at-a-distance.
Functions and graphs Rates of change and derivatives Open this as a full reading page →
Fields, continuous waves, and harmonic functions What is a continuous wave field and how is its energy measured?
A field assigns a definite physical quantity to every location in space and time. A scalar field assigns a single number, such as temperature or particle density, while a vector field assigns a magnitude and direction, such as electric field strength E or magnetic field B.
Harmonic waves propagate oscillating field values through space according to the phase factor kx - omega t, where the wavenumber k = 2 pi / lambda relates to wavelength lambda and angular frequency omega = 2 pi nu relates to cyclic frequency nu. The wave crests travel at the phase speed c = lambda nu.
Because optical frequencies oscillate hundreds of trillions of times per second (for example, 600 THz corresponds to green-cyan light with wavelength lambda = c / nu = 499.65 nm), measuring instruments record the time-averaged energy flux rather than the instantaneous field oscillation. Over any complete period, the average of cos^2(omega t) is exactly 0.5.
When an isotropic source emits total power P into three dimensions, the energy spreads evenly over concentric spherical wavefronts of area 4 pi r^2. The intensity at distance r is given by I = P / (4 pi r^2). For a 1 W point source, the intensity is 0.0795775 W/m^2 at distance r = 1 m, and drops to one-quarter (0.0198944 W/m^2) at distance r = 2 m.
The Doppler shift in sound is an illustrative analogy for wave frequency shifts: sound waves travel through a material medium (air), which creates an asymmetry between a moving source and a moving listener. This analogy has a strict physical limit: light requires no material ether medium, and in 1905 Einstein demonstrated that electromagnetic wave transformation depends purely on relative velocity between observers.
One worked example Intensity equals total power divided by four pi r squared, and the time average of cosine squared over a period is one half.
At frequency nu = 600 THz (6.0e14 Hz) with light speed c = 299792458 m/s, the wavelength is lambda = c / nu = 499.65 nm. A 1 W source produces intensity I = 1 / (4 pi (1)^2) = 0.0795775 W/m^2 at 1 m, and I = 1 / (4 pi (2)^2) = 0.0198944 W/m^2 at 2 m.
A stopping point: A field is a value assigned to every place; observed optical intensity is a time average, not an instantaneous pulse.
Functions and graphs Rates of change and derivatives Open this as a full reading page →
Counting what crosses a boundary Why does a difference of flows change the density?
Take a short interval. If more tracers enter through its boundaries than leave, its count increases. Flux counts signed crossings per area per time; number density counts particles per volume. They are not the same quantity.
Density changes at the negative spatial derivative of the flux.
In the one-dimensional description, the same balance applies per unit transverse area. The minus sign says an increasing rightward flux removes more particles at the right boundary than arrive at the left.
One worked example If seven particles enter and five leave during a time interval, the count rises by two. If both flows are five, the count is unchanged even though particles keep crossing.
A stopping point: Conservation counts net crossings; it does not require each individual particle to stop.
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Events, reference frames, and the coordinate grid How do observers assign spatial and temporal coordinates to physical events?
An event is a physical occurrence localized at a single location in space and a single instant in time, such as a spark jumping, a particle collision, or the emission of a flash of light.
To measure events quantitatively, observers construct a reference frame: an imaginary rigid spatial lattice equipped with stationary clocks at every grid intersection. An observer does not rely on looking at a distant clock across space (which incurs light-travel delay); instead, the coordinate time of an event is recorded by the local clock located right where the event occurs.
Clock synchronization follows Einstein's operational convention: a light pulse sent from clock A at time t_A reaches clock B, reflects, and returns to A at time t'_A. Clock B is synchronized with clock A if its reading at reflection satisfies the midpoint assignment t_B = (t_A + t'_A) / 2.
One worked example The time at clock B equals one half of the sum of the emission time at clock A and the return time at clock A.
If a master clock A emits a light signal at t_A = 0 seconds toward stationary clock B, and the reflected signal returns to clock A at t'_A = 10 seconds, the synchronization convention assigns the time t_B = (0 + 10) / 2 = 5 seconds to the reflection event at clock B.
A stopping point: An event is something that happens at one place and one time; coordinate assignment is an operational measurement convention, not an absolute cosmic background.
Functions and graphs Open this as a full reading page →
Functions and graphs How does a mathematical function record relationships between physical quantities?
A physical function is an unambiguous rule that assigns to each value of an independent input (such as elapsed time t or coordinate position x) a specific value of a dependent quantity (such as particle displacement, local concentration, or field energy). It describes how one aspect of a physical system responds when another changes.
Plotting a function creates a curve on a coordinate plane. The horizontal axis represents the input, and the vertical axis represents the output. Every point on the curve represents a simultaneously paired measurement. Because both axes correspond to physical measurements, any slope or area derived from the graph carries the combined dimensions of those axes.
Distinguishing functional laws from empirical scatter is essential. A theoretical function predicts the ideal expectation of an ensemble, such as the mean square displacement growing linearly with time. Individual experimental trials fluctuate around this expectation, but the functional form governs the ensemble average.
One worked example Mean square coordinate displacement equals two times the diffusion coefficient times time.
For a diffusion coefficient D = 0.5 square micrometres per second, plotting mean square displacement against time t produces a straight line through the origin with slope 2D = 1.0 square micrometres per second. At t = 1 second the value is 1.0 square micrometre; at t = 4 seconds it is 4.0 square micrometres.
A stopping point: A graph is a record of paired quantities with physical units; a slope without units is not an explanation.
Interactive construction: from measurement table to curve In section 5 of the Brownian motion paper, Einstein shows that while the mean displacement is zero, the mean squared displacement grows linearly with time: ⟨x²⟩ = 2Dt. Consequently, the observable root-mean-square displacement grows as the square root of time: √⟨x²⟩ ∝ √t.
Plot next point (5/5) Plot all Reset Showing 5 of 5 points plotted. (0s, 0µm) (1s, 2µm) (4s, 4µm) (9s, 6µm) (16s, 8µm) Time t (seconds) RMS displacement (µm) Dashed line: continuous √t trajectory. Dots: discrete observations. Table of paired measurements (Brownian §5) Time t (s) ⟨x²⟩ (µm²) √⟨x²⟩ (µm) Status 0 0 0.0 Plotted 1 4 2.0 Plotted 4 16 4.0 Plotted 9 36 6.0 Plotted 16 64 8.0 Plotted
Textual summary of the construction Each observation pairs an elapsed time in seconds with an accumulated squared displacement in square micrometers. Plotting these pairs demonstrates that displacement does not scale proportionally with time (which would indicate constant drift velocity), but rather with the square root of time (the hallmark of diffusive random walks).
Reading a graph Open this as a full reading page →
Adding continuously What operation does an integral describe here?
Approximate the area under a curve by rectangles. Each contributes height times width. Add all contributions, then refine the widths. When those sums approach a limit, that limit is the integral.
For a probability density, a rectangle has units of inverse length times length, leaving a dimensionless probability. For total probability the whole area must be one.
Integration by parts comes from adding the product rule for differentiation over an interval: the integral of u times the derivative of v equals the endpoint product minus the integral of v times the derivative of u.
One worked example Integration by parts moves a derivative from one factor to another and retains the endpoint term.
A constant density of 0.25 per micrometre over 4 micrometres gives 0.25 times 4 = 1, regardless of how many equal rectangles we use.
A stopping point: The width and the endpoint term are part of the calculation, not decorations on the integral sign.
Density is not probability Open this as a full reading page →
Momentum, energy flux, and radiation pressure of light How does a beam of light carry energy and momentum?
In classical electromagnetic theory, light is a continuous wave that transports both energy and linear momentum through space. The Poynting vector (introduced by John Henry Poynting in 1884) describes the directional energy flux density in watts per square metre. Because electromagnetic waves travel at the speed of light c, the momentum p carried by a beam is directly proportional to its total energy E through the relation p = E / c.
When a beam of light shines on a target, the absorption of radiant power P delivers a continuous physical force F = P / c. For 1.0 Watt of absorbed light, the mechanical push is 1 / c = 3.33564e-9 Newtons per Watt. When the target acts as an ideal mirror, the light reverses direction and imparts twice its incident momentum, producing a force of 2P / c = 6.67128e-9 Newtons per Watt.
This radiation pressure was experimentally verified with high precision around the turn of the century by Pyotr Lebedev (1901) and Ernest Fox Nichols and Gordon Ferrie Hull (1901-1903), demonstrating that classical electromagnetic fields exert measurable macroscopic mechanical force.
In Einstein's mass-energy paper of September 1905, radiation momentum plays a crucial role: a body at rest that emits two equal pulses of light of energy L/2 in opposite directions imparts equal and opposite momenta (+L / (2c) and -L / (2c)), leaving the emitting body at rest with zero total transferred momentum.
One worked example Momentum equals energy divided by c, absorbed force equals power divided by c which is approximately 3.33564 times ten to the minus nine Newtons per Watt, and reflected force is twice that value.
A continuous laser beam absorbing 1.0 Watt of power at normal incidence delivers F = (1.0 W) / (299792458 m/s) = 3.33564e-9 N of force. If the surface reflects the entire 1.0 Watt, the force is 2(3.33564e-9 N) = 6.67128e-9 N.
A stopping point: Light's momentum is the push a beam gives when absorbed or reflected; it follows from Maxwell's electrodynamics without requiring quantum hypotheses.
Functions and graphs Rates of change and derivatives Open this as a full reading page →
Partial derivatives and held-fixed quantities What does a partial derivative mean when several variables change simultaneously?
Many physical quantities depend on more than one parameter: particle density depends on both position x and time t; gas entropy depends on both volume V and temperature T. When asking how such a quantity changes, one must specify which variable is moving and which variables are being held constant.
The partial derivative ∂f/∂x represents the rate of change of f with respect to x while holding time t strictly fixed. Conversely, ∂f/∂t represents the rate of accumulation at a fixed position x over time. In thermodynamics, holding temperature fixed produces an isothermal derivative, while holding entropy or volume fixed produces an adiabatic or isochoric derivative.
Einstein's derivation of the diffusion equation equates the time rate of accumulation at a fixed location to the divergence of spatial flux: ∂f/∂t = D ∂^2f/∂x^2. Both sides describe rates under different held-fixed constraints.
One worked example Spatial partial derivative of density with respect to x holding time t fixed.
Example 1 (Time fixed when moving through space): For diffusion profile f(x, t) = (4 pi D t)^(-1/2) exp(-x^2 / (4Dt)), the spatial gradient quantifies concentration variation along a channel. Here, elapsed time t is strictly fixed as the held-fixed parameter. A spatial snapshot taken at one fixed instant evaluates to ∂f/∂x = -x/(2Dt) f(x, t).
Time partial derivative of density with respect to time holding position x fixed.
Example 2 (Position fixed when tracking time): The time accumulation rate records concentration changes at a fixed location. Here, spatial position x is strictly fixed as the held-fixed parameter. A stationary probe at one coordinate records accumulation rate ∂f/∂t = (-1/(2t) + x^2/(4Dt^2)) f(x, t). Equating accumulation to spatial flux divergence yields ∂f/∂t = D ∂^2f/∂x^2.
Isothermal volume derivative with respect to pressure holding temperature T fixed versus adiabatic volume derivative holding entropy S fixed.
Example 3 (Thermodynamic derivatives: isothermal versus adiabatic): In gas thermodynamics, the volume response to pressure depends on thermal boundary conditions. The isothermal derivative (∂V/∂p)_T explicitly names temperature T as the held-fixed quantity while heat exchanges freely with a bath. Conversely, the adiabatic derivative (∂V/∂p)_S explicitly names entropy S as the held-fixed quantity under thermal insulation. Because isothermal compression permits heat release, the isothermal compressibility exceeds the adiabatic compressibility. Writing ∂V/∂p without identifying the held-fixed parameter is physically incomplete.
Monochromatic radiation entropy density derivative with respect to energy density holding volume V and frequency nu fixed.
Example 4 (Radiation entropy derivative in Light Quanta §3): For blackbody radiation at frequency nu, the entropy density s_nu depends on both spectral energy density u_nu and enclosure volume V. Einstein's derivation of Wien's displacement law takes the partial derivative (∂s_nu/∂u_nu)_(V, nu) = 1/T. Here both cavity volume V and radiation frequency nu are strictly held fixed as the held-fixed parameters while varying energy density u_nu.
Chain rule transformation of spatial partial derivative holding rest-frame coordinates fixed into moving-frame partial derivatives.
Example 5 (Transformed derivatives in Relativity §6): In coordinate transformations between a resting frame (x, y, z, t) and a moving frame (x', y', z', t'), the partial derivative (∂/∂x)_(y, z, t) explicitly holds resting coordinates y, z, and time t fixed. When expressed in moving coordinates via the relativistic chain rule, it becomes a combination of moving spatial derivative (∂/∂x')_(y', z', t') holding y', z', and t' fixed, and moving time derivative (∂/∂t')_(x', y', z') holding x', y', and z' fixed. Changing coordinate systems changes which physical quantities are held fixed.
A stopping point: A partial derivative is mathematically and physically undefined until every held-fixed parameter is explicitly identified.
Interactive construction: what is held fixed in a partial derivative In sections 3 and 4 of the Brownian motion paper, Einstein tracks the concentration of suspended particles c(x, t) as a function of both position x along a tube and elapsed time t. Because two independent variables can change, asking for “the rate of change of concentration” is ambiguous until you specify which coordinate is held fixed.
Choose which coordinate to hold fixed:
Hold time t fixed: Spatial derivative ∂c/∂x Hold position x fixed: Time derivative ∂c/∂t
Case A: Hold time t fixed (∂c / ∂x) Quantity held fixed: Time t (a single snapshot across the tube).
Meaning: Spatial concentration gradient. You inspect different positions along the tube at one frozen instant.
Physical units: particles / (µm³ · µm) = particles / µm⁴.
Role in Brownian motion: Fick’s first law of diffusion states that the particle flux is proportional to this spatial gradient: J = −D (∂c/∂x).
Thermodynamic examples: how the fixed constraint changes the derivative In thermodynamics, the same symbols have completely different numerical values depending on what is held fixed:
Thermodynamic partial derivatives and their held-fixed constraints Process Derivative Quantity held fixed Physical behavior Isothermal (∂p / ∂V)T Temperature T fixed Heat flows in or out to maintain constant temperature Adiabatic (∂p / ∂V)S Entropy S fixed (no heat exchange) Gas warms upon compression; stiffer response than isothermal Isochoric (∂p / ∂T)V Volume V fixed Rigid vessel; pressure rises directly with heating
Textual summary of the construction Writing ∂c/∂x asserts that t is held constant during differentiation. Writing ∂c/∂t asserts that x is held constant during differentiation. These two operations describe different physical phenomena and carry different physical dimensions. In thermodynamics, the subscript notation (∂p/∂V)T versus (∂p/∂V)S makes this essential distinction visible on the page.
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Energy of motion and inertia What can a change in energy of motion tell us when speed stays the same?
Work transfers energy when a force acts through a displacement. The energy associated with the motion of a body is called kinetic energy. It is not the same as the energy of its internal heating, chemistry, or other stored processes.
For the same body at ordinary slow speeds, doubling speed multiplies its energy of motion by four. At a fixed speed, doubling inertial mass doubles that energy. The word inertia describes resistance to a change of motion, not a measurement of gravitational weight.
In Newtonian mechanics, kinetic energy is one half times inertial mass times speed squared.
The formula is the low-speed rule. It cannot be assumed exact for a traveler moving at a substantial fraction of light speed. To identify inertia in a relativistic comparison, examine the coefficient as speed approaches zero.
One worked example Compare two bodies at 2 metres per second. In the Newtonian model a body of 3 kilograms has 6 joules of energy of motion; a body of 2 kilograms has 4 joules. The speed is unchanged, but the energy of motion differs by 2 joules.
At the same low speed, the kinetic-energy drop equals one half times the mass decrease times the squared speed.
This example assumes independently specified masses; it is not evidence for mass–energy equivalence. The mass-energy argument instead calculates an energy difference from emitted light and then identifies its low-speed coefficient.
A stopping point: Less energy of motion at the same low speed means a smaller inertial mass within the Newtonian approximation. The comparison alone does not supply an absolute internal energy.
Squares and square roots Fractions and ratios Open this as a full reading page →