Game Math and Physics Basics in 2D

Explore one-dimensional motion in Unity by comparing displacement vectors to distances along the x axis, using position x changes and arrows to illustrate left and right movement.

Explore vertical displacements and distances using displacement vectors and arrows for up and down movement. Distinguish displacement from distance: vectors carry direction, while distance is a scalar.

Examine relative positions and distances in 2d space, showing there is no absolute origin and how displacement is described from each object's perspective.

Investigate relative positions along coordinate axes, noting that there is no absolute origin, describe displacement from each object's perspective, and measure the single distance between objects.

Explore world space versus object space in 2D game math by examining local coordinate systems, global origin, and how positions and relative measurements vary across objects.

Explore world space versus object space through a local coordinate system, translating positions from an alien's perspective and understanding relative coordinates and measurements in 2D.

Learn 2D vector components using unit vectors along the x and y axes, building any vector as a linear combination with clear horizontal and vertical components.

Relate the positions of the spaceship and Earth using a displacement vector and its x and y components in a 2D Unity context.

Explore how a tilted object affects displacement vectors, and use basic trig to resolve their earth-frame and ship-frame components in the ship's local coordinates.

Explore how translating a scene horizontally or vertically affects the displacement vector and its components, and see that the vector remains unchanged when the entire scene moves together.

Explore the z axis in Unity as perpendicular to the screen and learn how z values and rotation about the z axis determine object placement in the x y plane.

Rotate the Earth object about the z axis to illustrate 2d rotations; observe the local coordinate system shift and the rotation values in the inspector.

Explore rotating two objects about their midpoint and observe the joining vector’s length remains invariant while its direction changes; learn to restore original orientations.

Explore how displacement vectors and their components define relationships among three colored objects in 2D space, including position, overlap, and rotation transformations.

Compare distance and displacement vector length in 2d motion by applying the pythagorean theorem to the x and y components of the displacement, illustrating their difference.

Explore tiny-scale 2D motions, calculating distances with small x, y changes and the Pythagorean theorem for rigid shapes; learn to sum segment distances and the role of computers.

Explore how to relate objects using vectors and angles in unity, measuring angles from the positive x axis with a protractor and computing vector length with the Pythagorean theorem.

Describe a displacement on the unit circle using a magnitude and angle, then convert to x and y components. Relate cosine and sine to coordinates, e.g., 30° gives x=√3/2, y=1/2.

Explore how to express a doubled unit vector as a displacement with clear horizontal and vertical components, using 45-degree geometry, unit vectors, and the distributive property.

Learn how a 2d position vector uses magnitude and angle to derive its x and y components from the origin, e.g., x = sqrt(3)/2, y = 1/2 at 30 degrees.

Explore graphical vector addition of displacements in 2d by tracing individual motion vectors along paths to obtain the resultant displacement.

Explore vector addition by adding the x and y components to form the resultant. Link the graphical path to the underlying math and verify the x and y sums.

Master graphical vector subtraction by visualizing the vector from V1 to V2, then add it to V1 to reach V2 using the origin and Unity examples.

Explore how to subtract two 2D vectors by decomposing them into horizontal and vertical components, compute V1−V2 and V2−V1, and verify the results graphically.

Learn how to find the midpoint of a segment by averaging x and y coordinates, revealing the centroid for two points and extending to three objects in two-dimensional space.

Locate the centroid of three points by drawing medians to find their intersection, then average x and y coordinates to get the centroid, e.g., (8/3, 4/3) and (0, -1/3).

Explore how a weighted average explains balance by adding mass to one side and observing the shift of the balance point. Prepare for unity-based modeling next.

Compute the center of mass for two masses via a weighted average of mass and position, giving the balance point on a fulcrum. Extend to x and y coordinates.

Learn coordinate translation moves objects in 2d by applying x plus one for horizontal shifts and y plus two for vertical shifts, illustrated with points A, B, C, and D.

Illustrates diagonal translations by showing how moving along a diagonal changes coordinates: x+3,y+4, then x-6,y+7, aligning motion with final positions.

discover the balance point of a two-mass distribution in 2d and see how translating the masses along the x and y axes shifts the balance point.

Rotate the ship by 45 degrees, then translate to new coordinates, transforming from a local ship coordinate system to world space using the local x and y axes.

Learn to simplify square roots by factoring into a largest square and a factor, then distribute the root across factors; sqrt(24)=2 sqrt(6), sqrt(15) stays sqrt(15), sqrt(18)=3 sqrt(2).

Explore constant speed motion along diagonal paths in 2d game math, using the distance formula and pythagorean theorem to show speed equals total distance divided by total time.

This lecture defines constant velocity in 2d as displacement over time and shows it as a vector with x and y components in horizontal and vertical motion.

Explore constant velocity in diagonal motion by using displacement vectors and components; compute velocity as displacement divided by time, demonstrating 1 m/s in both x and y directions.

Explore the difference between speed and velocity, with speed as distance over time and velocity as displacement over time. Compute velocity components and the speed magnitude using the Pythagorean theorem.

Explore zero velocity, displacement vectors, and the distinction between velocity and speed through a 2d game physics example, showing how displacement can be zero while speed remains nonzero.

Learn how averaging replaces varying daily rates with a single uniform rate while preserving the cumulative result. Use calories per day as an example in game math and 2D physics.

Explore average speed in 2D motion across two horizontal segments, comparing speeds of one and three meters per second, and show how averaging yields a constant two meters per second.

Explain average speed for a two-segment journey, compare leg distances and speeds, and compute the overall rate by averaging leg speeds, illustrating constant speeds per leg.

Compute the average velocity in 2d by averaging the two leg velocity vectors, (v1 + v2)/2, with horizontal segments showing x components and zero y components.

Explore how to compute distance and displacement from speed and velocity vectors using units, time, and vector components, including negative directions and unit cancellation.

Learn how to compute average velocity for diagonal motion by averaging the two leg velocities to obtain a single constant velocity vector of 1.5 m/s horizontal and 3 m/s vertical.

Multiply velocity components by time to get the displacement vector: x = -4 m/s × 25 s = -100 m, y = 10 m/s × 25 s = 250 m.

Compute the displacement from point a to point b, find its magnitude with the Pythagorean theorem, form a unit vector, then scale by the desired speed to obtain velocity vector.

Explore constant acceleration and how speed grows or decelerates over time, starting from an initial speed and measured in meters per second and meters per second squared.

Visualize acceleration in 2-d motion by interpreting speed changes over equal time intervals, using meters per second, with the Pythagorean theorem for diagonal paths.

Apply the average acceleration formula from the change in speed over a time interval. Observe how speeds rise from 10 m/s to 40 m/s over 10 s, yielding 3 m/s².

Shows how two-dimensional acceleration vectors change velocity vectors by updating each component over time; uses an example with the horizontal (x) and vertical (y) components to illustrate one-second updates.

Visualize how the velocity vector changes under acceleration in 2d, with a constant horizontal component and a vertical component that increases or decreases, measured in meters per second per second.

Compute the final position by adding the initial position to velocity multiplied by time. Use consistent units, such as meters and seconds; for example, (1,2) + (2,3)×2 = (5,8) meters.

Compute the final velocity vector as v_final = v_initial + a t, treating velocity and acceleration as vectors. Use a time-based example to confirm unit consistency with meters per second.

Explore basic vertical acceleration by releasing an object and observing gravity causing downward acceleration and velocity. There is no horizontal acceleration, and velocity vectors point straight down as time passes.

Explore the force vector concept by observing how direction and magnitude affect motion, and learn to add and balance multiple forces to predict a resultant.

Balance forces by applying equal opposite horizontal and vertical components to keep an object stationary. The lecture shows vector addition and how opposite forces cancel to a zero net force.

Explore balancing forces in two-dimensional space by applying angled vectors and calculating their components to cancel each other, achieving a zero resultant force.

Visualize four force vectors and apply the graphical tip-to-tail method to sum them. The resultant is the zero vector, showing forces cancel in pairs.

presents graphical vector addition by moving forces without changing length or direction, showing that the cumulative result equals a single resultant vector and predicting motion.

Explore how summation notation adds force vectors by x and y components to produce the resultant force in newtons.

Visualize balancing two force vectors in 2D by placing them tip-to-tail and adding a third vector to return to the start, confirming a zero net force with components.

Find a missing force vector by solving for its x and y components; x = -3, y = 0, then verify via component sums in newtons.

Compute the displacement from A to B, normalize to a unit vector, and scale by the desired magnitude to obtain the force in that direction.

Explore the force-acceleration relationship F = ma, showing how mass and force determine acceleration and its vector form in newtons and meters per second squared.

Explore how force produces acceleration and shapes velocity in 2D, using F=ma and v = v0 + a t, then find speeds at t=1 s and t=5 s from components.

Explains how the position vector evolves from initial position, velocity, and acceleration using r(t)=r0+v t+a t^2, showing linear and quadratic contributions to the trajectory.

Explore how a kick sets an initial velocity and shapes a projectile's trajectory under gravity, with and without friction, using a position vector with a g t^2 term.

Shows a projectile motion example using 30 m/s horizontal, 40 m/s vertical, and 9.8 m/s^2 gravity; after 1 s, position is 35 m in x and −4.9 m in y.

Decompose projectile motion into independent x(t) and y(t) functions, showing how time governs horizontal and vertical components to define the trajectory.

Understand how x(t) describes an object's horizontal position over time, derive x(t)=30 m/s·t, and read off position from time using the slope of the x–t graph.

Explore the two-dimensional motion of a projectile by analyzing y(t) and x(t) as separate time-based components of the position vector, including max height and horizontal range.