Learning Physics

Mirror Equation

The mirror formula relates the focal length of a spherical mirror to the object and image distances. Move the sliders to explore how convex and concave mirrors form real and virtual images.

1⁄_f  =  1⁄_u  +  1⁄_v

The canvas traces the principal axis, focal markers, and image position so the geometry stays tied to the algebra.

u 28.0

f -21.0

¹⁄_f = ¹⁄_u + ¹⁄_v, v ≈ −12, m ≈ 0.43

Pythagorean Theorem

The most fundamental relation in Euclidean geometry. The area of the square on the hypotenuse equals the sum of the areas on the two shorter sides.

a² + b² = c²

Adjust the two legs and the construction updates so the triangle and all three outward-facing squares remain visible at once.

a 15.0

b 15.0

c = √(a² + b²) ≈ 21.21

a² + b² = c² ≈ 225.00 + 225.00 = 450.00

Ideal Gas Law (PV = nRT)

Lock one variable and adjust the others to see how pressure, volume, amount, and temperature are related. The particle simulation shows molecules bouncing inside a cylinder whose height scales with volume.

PV = nRT

Keeping one quantity fixed makes the compensating changes in the remaining three easier to read as both a formula and a physical picture.

P 1.00

V 24.5

n 1.00

T 298.0

Locked n: move P, V, or T to solve for the amount of gas.

PV and nRT stay matched while the locked variable updates.

Binomial Square

Squaring a sum or difference is not just squaring each term separately. The cross term appears because the product picks up two matching mixed pieces, which is why 2ab sits in the middle.

(a + b)² = a² + 2ab + b² (a - b)² = a² - 2ab + b²

A geometric area model makes the identity easy to remember: one large square splits into two smaller squares and two congruent rectangles. The same pattern powers expansion, factoring, and completing the square.

Charles’ Law

Charles’ law says the volume of a gas is proportional to its absolute temperature when pressure and amount of gas stay fixed. Heating the gas makes the particles occupy more space, so the container must expand if pressure is to remain unchanged.

V ∝ T V₁⁄T₁ = V₂⁄T₂

The law is linear only when temperature is measured from absolute zero. That is why the formula uses kelvin rather than Celsius.

Circle Area

The area enclosed by a circle depends on the square of its radius. As the radius grows, each extra layer wraps all the way around, and the accumulation of those rings produces the familiar πr² formula.

A = πr²

Because the radius is squared, scaling is non-linear: a circle with twice the radius covers four times as much area.

Circle Equation

The standard equation of a circle describes all points that lie exactly r units from a fixed center. It is the coordinate version of the geometric definition of a circle.

(x - h)² + (y - k)² = r²

Writing the expression in completed-square form makes the center and radius immediately visible, while expanding it connects the graph back to quadratic algebra.

Compound Interest

Compound interest turns repeated percentage growth into exponential growth. Each compounding period earns interest on the original principal and on the interest that has already accumulated.

A = P(1 + r⁄n)^nt

The formula makes timing explicit: the same nominal rate leads to different totals depending on how often interest is added back into the balance.

Cone Surface Area

A cone’s total surface area combines two pieces: the base circle and the curved sheet that runs from the rim to the apex. That curved part depends on the slant height, not directly on the vertical height.

A = πr² + πrl = πr(r + l)

This split is useful in practice because many fabrication problems care about whether the base is included, while others only use the lateral surface.

Cone Volume

The volume of a cone is one third of the volume of a cylinder with the same base radius and height. The factor of one third reflects how each horizontal cross-section shrinks as it moves toward the tip.

V = ⅓πr²h

This comparison makes the formula easy to remember and links cone volume to the broader family of pyramid-like solids.

Coulomb’s Law

Coulomb’s law gives the electrostatic force between two point charges. The interaction acts along the line joining the charges, and its size depends on the product of the charge magnitudes.

F = k|q₁q₂|⁄r²

Distance has a strong effect: doubling the separation reduces the force to one quarter. The sign of the charges determines whether the force pulls the particles together or pushes them apart.

Cylinder Volume

The volume of a cylinder is the area of its circular base multiplied by its height. Because the cross-section stays constant from bottom to top, the formula is a direct application of prism volume.

V = πr²h

The squared radius matters: doubling r makes the base four times larger, so the whole cylinder holds four times as much volume if height stays fixed.

Degrees of Freedom

Degrees of freedom count the independent coordinates or energy modes needed to describe a system. In kinetic theory, they determine how thermal energy is distributed among translation, rotation, and sometimes vibration.

U = f⁄2 nRT ⟨E⟩ = f⁄2 k_BT

This is why monatomic and diatomic gases can have different heat capacities even at the same temperature: they do not have the same number of active modes.

Difference of Squares

The difference of two perfect squares factors into the product of a sum and a difference. It is one of the fastest ways to simplify expressions and solve equations without full expansion.

a² - b² = (a - b)(a + b)

The identity is closely tied to symmetry: the middle terms cancel when the conjugates are multiplied, leaving only a² - b².

Exponential Decay

Exponential decay models processes that shrink by a constant percentage over equal time intervals. Radioactive decay, capacitor discharge, and some cooling models all follow this pattern.

N(t) = N₀e^-λt

The key idea is that the quantity disappears faster when there is more of it, and slower as less remains. That proportionality is what creates the exponential curve.

Hooke’s Law

Hooke’s law describes the restoring force of an ideal spring for small displacements. Pull the spring farther from equilibrium and the opposing force grows in direct proportion.

F = -kx

The relationship is linear only within the elastic regime. Beyond that range, real materials stop behaving like ideal springs.

Kinetic Energy

Kinetic energy is the energy an object has because it is moving. The formula shows that speed matters more strongly than mass, since velocity is squared.

K = 1⁄2 mv²

This is why modest increases in speed can create large increases in impact or required stopping distance. The work-energy theorem ties that change directly to net work.

Lens Equation

The thin-lens equation links focal length, object distance, and image distance. It predicts where an image forms when light passes through a converging or diverging lens.

1⁄f = 1⁄d_o + 1⁄d_i

Like the mirror equation, it turns a geometric ray diagram into a compact algebraic rule. Changing sign conventions changes the bookkeeping, not the underlying optics.

Linear Equation

A linear equation is an equation whose variables appear only to the first power and are not multiplied together. In two variables, its graph is a straight line.

ax + by = c

Linear models matter because they are the simplest non-trivial relationships and because many more complicated models can be approximated linearly near a point.

Ohm’s Law

Ohm’s law connects voltage, current, and resistance in simple electrical circuits. It is one of the basic translation rules between how hard charges are pushed and how much current actually flows.

V = IR

The law is an idealized model for ohmic materials. Real devices such as diodes and lamps can depart from this linear behavior.

Period-Frequency Relation

Period and frequency describe the same repeating process from opposite viewpoints. One asks how long a single cycle lasts; the other asks how many cycles happen each second.

f = 1⁄_T,   T = 1⁄_f

Because they are reciprocals, increasing one necessarily decreases the other. This relation appears everywhere from waves and circuits to clocks and orbital motion.

Potential Energy

Potential energy is stored energy associated with position or configuration. In the near-Earth gravitational model, lifting an object higher increases its potential energy by an amount proportional to height.

U = mgh

This is the energy reserve that can later become kinetic energy as the object falls. Different systems use different potential formulas, but the idea of stored capacity for work is the same.

Slope-Intercept Form

Slope-intercept form writes a line in the easiest form for quick graphing. The constant term tells where the line crosses the vertical axis, and the slope tells how steeply it rises or falls.

y = mx + b

This form makes linear behavior easy to read at a glance and is the usual entry point for connecting algebra with coordinate geometry.

Surface Area of Sphere

The surface area of a sphere depends only on its radius and grows like r². The factor of four is one of the elegant surprises of classical geometry.

A = 4πr²

It says that the curved outer skin of the sphere equals four copies of the area of the largest circle that fits inside it.

Triangle Area

The area of a triangle is half the product of a chosen base and its perpendicular height. The factor of one half appears because two congruent copies make a full parallelogram.

A = 1⁄2 bh

This makes the formula flexible: the same triangle can be measured with different bases as long as the corresponding perpendicular height is used each time.

Trig Angle Sum Identity

Angle-sum identities explain how sine and cosine behave when two angles are added. They let complex rotations be decomposed into simpler pieces and sit underneath many standard trigonometric transformations.

sin(A + B) = sin A cos B + cos A sin B cos(A + B) = cos A cos B - sin A sin B

These formulas are especially useful in wave analysis, coordinate rotation, and derivations of double-angle and subtraction identities.

Quadratic Formula

The quadratic formula gives the exact solutions of any quadratic equation. The discriminant controls the geometry: it decides whether the parabola cuts the axis twice, touches it once, or misses it entirely.

x = (-b ± √(b² - 4ac))⁄2a

This is why the graph of a quadratic and the algebra of its roots are tightly linked. Moving the curve up or down changes the discriminant and the visible intersections at the same time.

Distance Formula

The distance formula is the Pythagorean theorem written in coordinate form. Horizontal and vertical changes become the two legs of a right triangle, and the segment between points becomes the hypotenuse.

d = √((x₂ - x₁)² + (y₂ - y₁)²)

This lets geometry and algebra speak the same language: every straight-line distance in the plane can be computed from coordinate differences.

Arc Length

Arc length measures how much of a circle’s boundary is covered by a given angle. In radians, the relationship is especially clean: length is just radius times angle.

s = rθ

That formula works because radians already measure angle by comparing arc length to radius. Double the radius or double the angle and the arc length doubles too.

Sector Area

A sector is a slice of a circle cut out by two radii and an arc. Its area is proportional to the central angle, so widening the angle increases the slice by the same fraction of the full disk.

A = 1⁄2 r²θ

This is the area counterpart to arc length. In radians, both formulas keep the geometry direct and compact.

Derivative as Slope

The derivative at a point is the instantaneous rate of change there. Geometrically, it is the slope of the tangent line that just touches the curve at that point.

f′(x) = lim_h→0 (f(x + h) - f(x))⁄h

As the point moves, the tangent rotates with the curve, making change visible instead of abstract. Steeper tangent means larger derivative magnitude.

Integral as Area Under Curve

A definite integral accumulates the signed area under a curve across an interval. It turns local values into a total quantity by adding infinitely many thin slices.

∫_a^b f(x) dx

This is why integrals appear in area, displacement, mass, charge, and probability. They gather small contributions into one whole answer.

Newton’s Second Law

Newton’s second law links force to acceleration through mass. Push harder and acceleration increases; keep the same force but increase mass and the acceleration drops.

F = ma

The equation explains both why light objects respond quickly and why heavy ones resist changes in motion. Force sets the rate at which velocity changes.

Wave Speed Relation

Wave speed connects how often crests pass by with how far apart they are. If the medium keeps the wave speed fixed, raising frequency forces wavelength to shrink.

v = fλ

This simple product ties together sound, light, strings, and water waves. It explains why frequency and wavelength trade off against each other.

Snell’s Law

Snell’s law describes how light bends when it passes between materials with different refractive indices. The change in angle reflects the change in propagation speed.

n₁ sin θ₁ = n₂ sin θ₂

When light enters a slower medium, it bends toward the normal. When it enters a faster medium, it bends away from it.

Torque

Torque measures how effectively a force causes rotation about a pivot. It depends on both how large the force is and how far from the pivot it acts, with angle built into the lever effect.

τ = rF sin θ

This is why pushing at the end of a wrench works better than pushing near the center. More lever arm means more turning effect.