Two-Dimensional Motion: Quick Notes

Two-Dimensional Motion: Independence of x and y

  • Base equations and relationships in 2D/3D are unchanged; x and y components act independently.
  • Treat as two separate 1D problems (one for x, one for y) and combine results vectorially when needed.

Decomposing Motion into Components

  • Break any 2D problem into its x and y parts.
  • Solve each part with its own initial velocity and acceleration, then recombine.

Vector Form: Magnitude and Direction

  • Final velocity components: vx,\; vy
  • Magnitude: v = \sqrt{vx^2 + vy^2}
  • Direction: \theta = \tan^{-1}\left(\frac{vy}{vx}\right)

Projectile Motion Basics (gravity only)

  • ax = 0,\quad vx(t) = v_{0x}
  • ay = -g,\quad vy(t) = v_{0y} - g t
  • y(t) = v_{0y} t - \tfrac{1}{2} g t^2
  • g \approx 9.8\ \mathrm{m/s^2}

Example Problem: Separate x and y (7 s)

  • Given: v{0x}=22\ \mathrm{m/s},\; ax=24\ \mathrm{m/s^2};\quad v{0y}=14\ \mathrm{m/s},\; ay=12\ \mathrm{m/s^2};\; t=7\ \mathrm{s}
  • Displacements:
    • x = v{0x} t + \tfrac{1}{2} ax t^2
    • y = v{0y} t + \tfrac{1}{2} ay t^2
  • Final components:
    • vx = v{0x} + a_x t = 190\ \mathrm{m/s}
    • vy = v{0y} + a_y t = 98\ \mathrm{m/s}
  • Final speed: v = \sqrt{vx^2 + vy^2} \approx 214\ \mathrm{m/s}
  • Direction: \theta = \tan^{-1}\left(\frac{vy}{vx}\right) \approx 27.5^{\circ}

Final Velocity Vector (components)

  • Velocity components: vx = 190\ \mathrm{m/s},\quad vy = 98\ \mathrm{m/s}
  • Vector form can give magnitude and angle as above

Time of Flight and Range: Horizontal drop example

  • Scenario: plane speed v_{0x}=115\ \mathrm{m/s}, height H=1050\ \mathrm{m}
  • Time in air: t = \sqrt{\dfrac{2H}{g}} \approx 14.6\ \mathrm{s}
  • Final velocities: vx = 115\ \mathrm{m/s},\quad vy = -g t \approx -143\ \mathrm{m/s}
  • Final speed: v = \sqrt{115^2 + 143^2} \approx 184\ \mathrm{m/s}
  • Flight angle: \theta = \tan^{-1}\left(\dfrac{vy}{vx}\right) \approx -51.2^{\circ}
  • Horizontal range: R = v_x t \approx 1.68 \times 10^3\ \mathrm{m}

Strategy and Tips for 2D Motion Problems

  • Draw a diagram; set positive directions clearly.
  • Treat x and y as separate problems; verify at least three kinetic values per component.
  • Break any initial velocity given as magnitude + angle into components: v{0x} = v0\cos\theta,\; v{0y} = v0\sin\theta
  • If motion occurs in segments (varying accelerations), solve per segment and chain results.
  • Combine x and y results to obtain the final vector (magnitude and direction).
  • Note: air resistance is neglected in these baseline problems; real motion may differ.

Quick takeaway

  • Any 2D motion can be analyzed by independently solving x(t) and y(t) using their own initial conditions and accelerations, then recombining to get the overall motion.