Physics from ZERO: Mechanics with Interactive Simulations

Explore how physics uses plus and minus signs to express direction on a coordinate axis, mapping cardinal directions to positive and negative values for mathematical clarity.

Set up the x-axis with rightward as positive and use meters. Compute displacement as final minus initial; from x=1 to x=7, delta x equals 6, showing direction and distance.

Define velocity as displacement per unit time with direction. Contrast it with speed; use delta x over delta t; rightward is positive, leftward negative; illustrate with examples.

Define acceleration as the rate of change of velocity, using a ball example to show delta v over delta t and uniformly accelerated motion.

Learn how the velocity-time graph links distance to the area under the curve and acceleration to the slope, illustrated for uniform and uniformly accelerated motion.

Use delta to define displacement as final minus initial. Express velocity as delta x over delta t and acceleration as delta v over delta t; v-t graph slope shows acceleration.

Analyze a velocity-time graph to determine acceleration and velocity at key times. Use area under the graph to compute distance and displacement, revealing a final -100 m displacement.

derive the formulas for uniformly accelerated motion. explore velocity, time, and displacement relationships and highlight signed quantities.

Explore uniformly accelerated linear motion along the x axis, solving for acceleration using v0=10 and v=4, locating the turning point and origin crossing, with v-t and x-t graphs.

Practice applying cosine and sine to find triangle side lengths for 30°, 45°, and 60° angles, including adjacent and non-adjacent sides, and relate hypotenuse to gravity mg.

Practice vector decomposition by projecting x and y components on axes with perpendicular lines, using cosine and sine for angles, and determining the signs of each component, including mg.

Explore vertical projectile motion through example problem 3, applying gravity g = 9.8 m/s² and coordinate axes, and use uniformly accelerated motion to reveal symmetry of motion.

Analyze horizontal projection with shadow visualization, showing horizontal uniform motion and vertical accelerated motion under gravity, while axes remain arbitrary. Solve example: time to floor, initial velocity, and impact speed.

Learn projectile motion by setting a coordinate system, decomposing velocity into x and y components, and deriving a downward-opening parabola to find maximum height and range.

Explore Newton's second law as the equation of motion, linking mass, forces, and acceleration through the resultant force. Learn about equilibrium, zero acceleration, and how balanced forces affect motion.

Explore equilibrium of forces in mechanics by balancing spring extensions with parallel springs and analyzing static friction on an incline, deriving friction limits and a 30-degree angle.

Explore motion on a smooth inclined plane, derive the equation of motion, and compute final speeds using constant acceleration and the v^2 formula, including friction in the second problem.

Draw force diagrams, set mass-specific positive directions, and form equations of motion to find accelerations and tensions in pulley systems with and without friction.

Focus on forces the object receives, like the normal force and gravity, and apply action and reaction only to moving interactions, not immovable objects.

Explore inertial force, a fictitious force opposite to acceleration, arising in accelerating frames like trains and cars, and how it balances tension to keep a hanging weight at rest.

Derive equations of motion for two objects in contact on a smooth floor, identify the normal force and action–reaction pair, and show solving as separate bodies and as one system.

Explore inertial forces on an inclined plane with a ball tethered to a string, using the platform perspective to find accelerations and tensions and to simplify equilibrium analysis.

Study a two-body problem with a mass on a board under kinetic friction, deriving accelerations, relative motion, the time to rest relative to the board, and the distance it slides.

Focus on forces to determine constancy, form an equation of motion, and apply formulas for acceleration; if forces vary, use work, energy, and momentum toward conservation of energy and momentum.

Define work as force times displacement and its sign rules; distinguish positive, negative, and zero work, and connect to energy as the capacity to do work.

Illustrate the work-energy relation by calculating forces on a moving object; gravity and normal do zero work, friction does negative work, so final kinetic energy equals initial energy plus work.

Explore how gravity does work on height differences and apply conservation of energy to relate kinetic and potential energy, including spring energy and friction on inclined planes.

Explore how work drives energy change, linking initial energy to final energy plus the work done across kinetic, gravitational potential, elastic, electrical, and electrostatic energies.

Explore why force and energy alone can't explain collisions and how momentum and impulse clarify such problems, including how sound energy affects conservation of energy.

Relate momentum to impulse by showing final momentum equals initial momentum plus impulse. Apply conservation of momentum when external impulses cancel, focusing on directions with no external force.

Explore the coefficient of restitution, linking relative velocities before and after collision to momentum conservation. Learn how the minus sign, relative velocity, and bounciness define the relation equation.

Using energy conservation, compute the pre-collision speed; apply momentum conservation and restitution to get post-collision velocities; conclude when A rebounds left (e > M/m) with h' = e^2 h.

Apply conservation of momentum and energy to problems with changing forces, linking initial and final states for cart and ball on a spring across horizontal and inclined surfaces.

Use equation of motion for constant forces, and conservation of energy and momentum for curved trajectories; circular motion and simple harmonic motion are exceptional cases where equations stay effective.

Explore circular motion by reviewing arc length and period, derive speed and angular velocity relations, and show acceleration toward the center as the force producing turning.

Analyze uniform circular motion of a mass on a string, deriving radius as h sin theta and the period from omega, and note the string break condition at theta.

Explore non-uniform circular motion in a vertical plane, using conservation of energy to relate speeds at A, B, and C, and analyze normal forces and gravity on a cylindrical surface.

Circular motion provides acceleration as r omega squared or v squared over r, letting us derive force from motion and apply energy conservation, including zero normal force or zero tension.

Discover universal gravitation: force is G M m / r^2 with r measured from the center, and potential energy uses zero at infinity with a negative sign.

Explore universal gravitation through problem solving: relate surface gravity g to GM/R^2, derive the second cosmic velocity via energy conservation, and compute circular-orbit speed and period for a satellite.

Explore simple harmonic motion as a back-and-forth motion with a restoring force toward a center, where velocity is zero at endpoints and maximum at center, described by F = -k(x−x0).

Explore how simple harmonic motion arises as the shadow of uniform circular motion, and derive its position, velocity, acceleration, and phase using sine, cosine, and omega.

Explore how a simple pendulum exhibits approximate harmonic motion at small amplitudes, derive its equation and period 2 pi sqrt(L/g), and learn the modeling approach mapping it to a spring.

Explore how simple harmonic motion is modeled by a pendulum under small-angle approximation and a horizontal spring, and apply energy conservation with k replaced by m g / l.

Explains problem 19 of horizontal SHM with two springs and a mass on a smooth floor; derives the equation of motion, reduces to a 3k spring, and finds the period.

Explore a small-amplitude pendulum on an inclined plane, using the simple pendulum formula with g replaced by g sin alpha to find the period and the length for half period.

Derives the net force for a vertical spring pendulum, showing gravity cancels with the elastic force to yield ma = -k x, revealing the simple harmonic motion model.

Model simple harmonic motion as a horizontal spring system using force diagrams. Relate the equation of motion to x versus t graphs, starting sine or cosine, and energy conservation.

Define torque as the rotational effect of force on a rigid body, driven by the arm and line of action. Balance torques and forces to stop rotation.

Identify the center of mass as the point where the gravity acts on a rigid body, enabling you to apply the center-of-mass formula to determine the balance point for torque.

Compute centers of mass for composite shapes using center of mass formula, coordinates, and mass ratios, including a rod with a two-mass ball and a disk with a displaced hole.

Explore a torque balance problem for a light rod with three masses leaning on a wall. Derive normal and static friction forces and assess stability with axis choices.