AgentSkillsCN

02 Kinematics 2d

分析周期性运动，包括简谐运动、弹簧振动与单摆运动。 当学生需要解决振荡、周期、频率、振幅，或分析谐振系统中的能量问题时，此方法尤为适用。

SKILL.md
--- frontmatter
id: kinematics-2d
subject: physics
display_name: Two-Dimensional Kinematics
description: |
  Analyzes motion in two dimensions including projectile motion and relative velocity.
  Use when students need to decompose vectors, solve projectile problems, or work with relative motion.

grade_band: 9-12
khan_tags: [physics, kinematics, projectile-motion, vectors]
standards:
  - NGSS.HS-PS2-1
  - CC.HSN-VM.A.1
  - CC.HSN-VM.B.4

objectives:
  - Decompose vectors into x and y components
  - Apply kinematic equations independently to horizontal and vertical motion
  - Solve projectile motion problems for range, max height, and time of flight
  - Analyze relative velocity problems in two dimensions

prerequisites:
  - kinematics-1d
  - algebra-linear-equations

estimated_time_minutes: 60

validator:
  type: numeric_solver
  config:
    unit_library: physics
    default_tolerance: 0.02
    require_units: true

sources:
  - name: OpenStax College Physics
    chapter: 3
    url: https://openstax.org/books/college-physics/pages/3-introduction-to-two-dimensional-kinematics
    license: CC-BY-4.0

Two-Dimensional Kinematics

Misconceptions

MisconceptionCorrection
"Horizontal and vertical motions affect each other"Horizontal and vertical motions are independent; a bullet fired horizontally hits ground same time as one dropped
"Objects moving horizontally don't fall"Gravity acts on all objects equally regardless of horizontal motion
"Projectiles slow down at the top of their arc"Horizontal velocity is constant; only vertical velocity becomes zero at peak
"Thrown objects have a force pushing them forward"After release, only gravity acts (ignoring air resistance)
"Range is maximum at 90°"Maximum range occurs at 45° launch angle

Key Concepts

Vector Components

Any vector v can be decomposed into perpendicular components:

  • •Horizontal: v_x = v × cos(θ)
  • •Vertical: v_y = v × sin(θ)
  • •Magnitude: v = √(v_x² + v_y²)
  • •Direction: θ = tan⁻¹(v_y / v_x)

Independence of Motion

Projectile motion can be analyzed as two separate 1D problems:

  • •Horizontal: constant velocity (a_x = 0)
  • •Vertical: constant acceleration (a_y = -g = -9.8 m/s²)

The same time t applies to both directions.

Projectile Motion Equations

Horizontal motion (a_x = 0):

code
x = x₀ + v₀ₓt
v_x = v₀ₓ = v₀ cos(θ)

Vertical motion (a_y = -g):

code
y = y₀ + v₀ᵧt - ½gt²
v_y = v₀ᵧ - gt
v_y² = v₀ᵧ² - 2g(y - y₀)

Key Projectile Formulas (launched from ground level)

code
[1] Time of flight: T = 2v₀sin(θ)/g
[2] Maximum height: H = v₀²sin²(θ)/(2g)
[3] Range: R = v₀²sin(2θ)/g
[4] At any time t: x = v₀cos(θ)t, y = v₀sin(θ)t - ½gt²

Relative Velocity

Velocity of object A relative to object B:

code
v_AB = v_A - v_B

For perpendicular velocities (boat crossing river):

  • •Resultant velocity: v = √(v_boat² + v_river²)
  • •Drift distance: d = v_river × t_crossing

Worked Examples

Example 1: Horizontal Projectile

Problem: A ball is thrown horizontally at 15 m/s from a cliff 45 m high. How far from the base does it land?

Solution:

  1. •

    Find time to fall (vertical motion):

    • •y = y₀ - ½gt²
    • •0 = 45 - ½(9.8)t²
    • •t = √(90/9.8) = 3.03 s
  2. •

    Find horizontal distance:

    • •x = v₀ₓ × t = 15 × 3.03 = 45.5 m

Example 2: Angled Launch

Problem: A soccer ball is kicked at 20 m/s at 30° above horizontal. Find the range.

Solution:

  1. •Use range formula: R = v₀²sin(2θ)/g
  2. •R = (20)²sin(60°)/9.8
  3. •R = 400 × 0.866/9.8 = 35.3 m

Example 3: River Crossing

Problem: A boat travels at 4 m/s in still water. The river flows at 3 m/s. If the boat aims straight across a 100 m wide river, find the drift downstream.

Solution:

  1. •Time to cross: t = width/v_boat = 100/4 = 25 s
  2. •Drift: d = v_river × t = 3 × 25 = 75 m

Explanation Patterns

  1. •Always start with a diagram showing coordinate axes, initial velocity vector, and trajectory
  2. •Decompose initial velocity into v₀ₓ and v₀ᵧ immediately
  3. •Write separate equations for x and y motion
  4. •Identify what connects them: usually time t
  5. •Solve for time from whichever equation has enough information
  6. •Substitute time into the other equation to find the unknown
  7. •Check units and reasonableness: distances in meters, times in seconds

Common Problem Types

  1. •Horizontal launch: v₀ᵧ = 0, find range or time
  2. •Angled launch from ground: find range, max height, or time of flight
  3. •Angled launch from height: combines both
  4. •River/wind problems: relative velocity with perpendicular components
  5. •Find launch angle: given range and initial speed