Projectile motion

88 topics across 6 chapters

Chapter 1

Math and coordinate foundations for projectile motion

Vectors and components (i, j; magnitude; direction)

2 subtopics

Practice: convert between polar and Cartesian vector forms

Practice: resolve a velocity vector into x/y components

Trigonometry for decomposing velocity (sin, cos, tan)

2 subtopics

Practice: compute angles from components using arctan2 and signs

Practice: use sin/cos to find components from speed and angle

Reference frames, axes choice, and sign conventions

2 subtopics

Choose axes for an inclined plane vs horizontal ground problems

Practice: apply consistent sign conventions in 2D kinematics

Units, dimensional analysis, and significant figures

2 subtopics

Practice: unit conversions (m/s ↔ km/h; degrees ↔ radians)

Practice: dimensional analysis checks for range/time/height formulas

Chapter 2

Kinematics basics (motion in 1D and 2D)

Displacement, velocity, acceleration (conceptual + graphical)

2 subtopics

Interpret slope and area on position/velocity graphs

Practice: sketch x(t), y(t), vx(t), vy(t), ax(t), ay(t) for a projectile

Constant-acceleration equations in 1D

2 subtopics

Practice: solve 1D constant-acceleration problems with the 5 equations

Practice: pick the correct kinematics equation without over-solving

2D motion as two independent 1D motions

2 subtopics

Set up coupled x/y equations with shared time variable

Practice: eliminate time between x(t) and y(t) in simple cases

Gravity as (approximately) constant acceleration near Earth

2 subtopics

Know and use g ≈ 9.8 m/s² (and sign) near Earth

Practice: identify when constant-g approximation is acceptable

Chapter 3

Ideal projectile motion (no air resistance)

Model assumptions for ideal projectile motion

2 subtopics

List the idealizations: no drag, flat Earth, constant g, no lift

Practice: decide which idealizations are violated in real scenarios

Launch speed and launch angle decomposition (v0x, v0y)

3 subtopics

↗ Practice: resolve a velocity vector into x/y components (see Chapter 1)

↗ Practice: use sin/cos to find components from speed and angle (see Chapter 1)

Practice: compute v0x and v0y for multiple launch angles quickly

Time of flight, maximum height, and range (level ground)

4 subtopics

Derive time of flight for level-ground launch/landing

Derive maximum height from vy(t)=0 (or energy)

Derive range formula R = v0² sin(2θ)/g (and interpret)

Practice: solve for θ or v0 given range/time/height constraints

Trajectory equation y(x) and parabola properties

2 subtopics

Derive y(x) by eliminating time and identify parabola coefficients

Practice: find where the trajectory intersects a given horizontal line

Energy and momentum perspectives (when useful)

2 subtopics

When to use conservation of energy vs kinematics in projectile problems

Practice: combine energy (speed vs height) with kinematics (timing)

Chapter 4

Problem-solving workflows and common projectile problem types

Given-unknown diagrams and variable selection (what to solve for first)

2 subtopics

Practice: draw a labeled diagram and list givens/unknowns with units

Practice: choose the first equation to solve based on unknown count

Common scenarios: launched from height or landing at different height

3 subtopics

Solve y(t) with nonzero initial height (y0)

Practice: find time to hit the ground from a cliff (quadratic in t)

Practice: range when landing height differs from launch height

Targeting problems: hit a target at known (x,y)

2 subtopics

Set up equations for passing through a point (xT, yT)

Practice: solve for launch angle(s) that hit a target at same height

Relative motion extensions: moving launch/landing platforms

2 subtopics

Add/subtract platform velocity (Galilean transformation)

Practice: projectile launched from a moving cart/plane problem

Error checking: limiting cases, symmetry, and sanity checks

2 subtopics

Practice: use symmetry (rise time = fall time on level ground) to check

Practice: check extreme angles θ→0 and θ→90 for range/height behavior

Chapter 5

Beyond the ideal model: drag, wind, and rotation

Linear drag (proportional to v): qualitative behavior and equations

2 subtopics

Write the linear drag force model Fd = -b v and terminal speed idea

Practice: compare ideal vs linear-drag trajectories qualitatively

Quadratic drag (proportional to v²): modeling and numerical solution idea

2 subtopics

Write the quadratic drag model Fd = -(1/2)ρCdA v² v-hat

Practice: implement quadratic drag update step in a spreadsheet/program

Wind and non-uniform g (when the ideal model breaks)

2 subtopics

Model wind as relative airspeed (v - v_wind) in drag force

Practice: decide when wind can be ignored for a given scenario

Spin and the Magnus effect (sports trajectories)

2 subtopics

Concept: lift from spin (Magnus) changes curvature and range

Practice: explain a curveball/soccer bend using force diagrams

Chapter 6

Measurement, simulation, and verification

Video analysis of a projectile (phone camera + tracker)

2 subtopics

Collect video data with a known length scale and consistent camera angle

Practice: extract x(t), y(t) points and fit to the ideal model

Build a simple simulation (step-by-step integration)

3 subtopics

Euler vs semi-implicit Euler updates for velocity/position

Practice: choose timestep size and see numerical stability effects

Practice: simulate ideal projectile and verify range/time formulas

Compare model vs data: estimate g, v0, and drag parameters

2 subtopics

Estimate g from fit to y(t) or y(x) and interpret error sources

Practice: infer an effective drag parameter by matching range reduction

Communicate results: plots, uncertainty, and assumptions

2 subtopics

Make clear plots: trajectory, vx/vy vs time, residuals

Practice: write an assumptions/limitations section for a lab report