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.

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 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 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.

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.

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.

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.

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.

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.

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.

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.

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.