Physics, oriented

Start with what kind of quantity you have. Speed and distance are scalars; velocity, displacement and force are vectors, so they carry a direction and add by components. A round trip has a distance but zero displacement, and most trap answers swap one for the other. With constant acceleration, four equations link displacement, the two velocities, acceleration and time; choose the one that leaves out the quantity you neither know nor need. Horizontal and vertical motion run independently, so a projectile keeps its sideways speed while gravity changes only its vertical speed.

Newton's laws then explain why motion changes. With no net force, velocity stays the same, so a moving object needs no push to keep moving. The net force equals mass times acceleration, applied one axis at a time. Forces come in pairs, but the two act on different objects, which is why they never cancel. Weight is the pull of gravity, mg, while mass stays the same everywhere.

Most errors come from the free-body diagram. Draw one object, list only the forces acting on it, and never invent a force of motion. Centripetal force is a job some real force does, not an extra arrow. A scale reads the normal force, so it shows more than your weight in an elevator accelerating upward. Friction holds until the push exceeds its limit, then drops once sliding starts.

Rule: before you write F = ma, name every force on one object and its source; an arrow you cannot trace to anything is the error.

Work is force times displacement times the cosine of the angle between them, so a force at right angles to the motion does no work. The net work on an object equals its change in kinetic energy, one-half mv², which often gives a speed far faster than kinematics does. Because speed is squared, doubling it quadruples the energy, and with the same braking force the stopping distance quadruples too. Power is how fast the work happens, and a kilowatt-hour is 3.6 million joules.

If only gravity and ideal springs do work, kinetic plus potential energy stays constant. A sliding object's speed at the bottom then depends only on how far it has dropped, not on the shape of the slope. Friction is different: it turns mechanical energy into heat along the full path traveled, so include that term or your final speed comes out too high.

Momentum, mass times velocity, is the tool for collisions. With no outside net force the total momentum before equals the total after, even when a crash wrecks the objects. Kinetic energy survives only in elastic collisions; when objects stick together, the lost energy becomes heat and deformation. Impulse, force times time, equals the change in momentum, which is why airbags work: they stretch the stopping time and lower the peak force.

Rule: if a question gives heights and speeds, try energy; if it gives a collision, try momentum; and in both, ask what friction or a crumpling surface took away.

Every linear idea has a rotational twin. Torque is force times its lever arm, so the same push turns a door more easily far from the hinges. Moment of inertia depends on how far the mass sits from the axis, which is why a hoop rolls down a ramp more slowly than a solid ball. With no outside torque, angular momentum is conserved, so a spinning skater who pulls in their arms spins faster. An object stays upright while the line down from its center of gravity falls inside its base.

Gravity weakens with the square of distance. In orbit, astronauts float because they and their craft fall around Earth together, while gravity at the station's height is still about 90 percent of its surface value. Planets follow Kepler's three laws: ellipses, equal areas in equal times, and a period whose square grows with the cube of the orbit's size. The Moon pulls harder on Earth's near side than its far side, which raises two tidal bulges.

Fluid pressure grows with depth and pushes in every direction. A floating object displaces its own weight of fluid. Where a pipe narrows the flow speeds up, and where a fluid speeds up its pressure drops, which is Bernoulli's equation at work.

Rule: when an object spins, orbits or floats, find the conserved quantity or the balanced force first; the formula usually follows from it.

A mass on a spring or a pendulum swinging through a small angle moves in simple harmonic motion. Its period does not depend on how far you pull it, so a wider swing just moves faster. Push an oscillator at its natural frequency and you get resonance, with damping setting the limit.

A wave's speed equals its frequency times its wavelength, and the medium decides the speed. Raise the frequency in the same medium and the wavelength shortens; move into a new medium and the frequency stays fixed while speed and wavelength change. Sound in air at sea level travels about 761 miles per hour. Waves add when they overlap, which produces interference, standing waves on strings and beats between close notes. A moving source shifts the pitch you hear without changing the note it emits.

Light is an electromagnetic wave that needs no medium and travels at exactly 299,792,458 meters per second in a vacuum. It reflects at equal angles, and it bends toward the normal as it slows on entering glass or water. Past a critical angle it cannot escape, so it reflects completely, which is how optical fibers work. Two slits make bright and dark bands that only a wave can produce, and diffraction sets the finest detail any telescope can resolve. The sky is blue because air scatters short wavelengths far more than long ones.

Rule: decide what the medium fixes and what the source fixes before you change any variable; frequency belongs to the source, speed to the medium.

Two objects in contact trade heat until they share a temperature. Kelvin starts at absolute zero, and 0 K is about −273 °C, so use kelvins in every gas law and efficiency formula. Raising a temperature takes heat in proportion to the mass and the specific heat, but during melting or boiling the temperature stays flat while latent heat changes the phase. That is why steam burns worse than boiling water. Heat moves by conduction, convection and radiation, and a body radiates in proportion to the fourth power of its kelvin temperature.

The first law is energy bookkeeping: the change in internal energy equals the heat added minus the work the system does. An ideal gas's internal energy depends only on temperature, so in an isothermal process every joule of heat added leaves as work. A gas does work as it expands, equal to the area under its path on a pressure-volume graph, so the path matters.

The second law limits what engines can do. Heat flows on its own only from hot to cold, and no engine turns all of its heat into work. The best possible efficiency is the Carnot limit, one minus the cold temperature over the hot one, both in kelvins. Entropy, a measure of how spread out energy is, never decreases in an isolated system.

Rule: convert every temperature to kelvins before you divide by it, and name which way heat and work flow before you write a sign.

Like charges repel and unlike charges attract, with a force that weakens with the square of the distance. The electric field is force per unit charge, and voltage is energy per unit charge, so only differences in voltage matter. A capacitor stores charge in proportion to the voltage across it.

In a circuit, current is the rate of charge flow. Ohm's law, V = IR, holds for ordinary resistors but not for every component. Resistors in series share one current and their resistances add; in parallel they share one voltage, and the total resistance falls below the smallest branch. Kirchhoff's rules restate the conservation of charge at a junction and energy around a loop. A capacitor charging through a resistor reaches about 63 percent of its final voltage in one time constant.

A current makes a magnetic field, and a magnetic field pushes sideways on a moving charge, which is why that force never changes the charge's speed. A changing magnetic flux induces a voltage, and the induced current opposes the change that caused it. Generators, transformers and motors all run on that rule. Alternating current's effective value is about 0.707 of its peak for a sine wave.

At home, a fuse or breaker sits in the hot wire and opens before the wiring overheats. A ground-fault circuit interrupter cuts power when the current going out and coming back differ by more than about 5 milliamperes.

Rule: trace charge and energy through every junction and loop; a circuit answer that loses either one is wrong.

Special relativity starts from one fact: every observer measures the same speed of light. Moving clocks then run slow and moving lengths shrink, and Global Positioning System satellites must correct for these effects. Velocities never add past the speed of light, and mass is a form of energy, E = mc².

Quantum physics began when light was found to arrive in packets. In the photoelectric effect, light below a threshold frequency ejects no electrons however bright it is, because each photon's energy depends on its frequency alone. Electrons also behave as waves, and confining one allows only certain energies, which is why atoms emit sharp spectral lines.

A nucleus is held together by the strong force, and its mass is less than the mass of its parts. Splitting heavy nuclei or fusing light ones releases energy. In each half-life, half of a radioactive sample decays. Alpha particles stop in paper, beta particles in thin metal, and gamma rays need dense shielding. Absorbed dose is measured in grays and equivalent dose in sieverts.

The universe began about 13.8 billion years ago and has been expanding and cooling ever since. Light released about 380,000 years after that start still fills space as the microwave background. Most power plants turn heat into steam, the steam into turbine motion, and that motion into electricity, wasting part of the heat as the second law requires.

Rule: check which regime you are in before you choose your tools: near light speed, at atomic scale or inside a nucleus, everyday intuition stops being a guide.