Chapter 4: Kinematics in Two Dimensions Summary

Motion in Two Dimensions

• Instantaneous velocity vector $\vec{v}$ is tangent to the trajectory.
• Velocity vector $\vec{v}$ changes in:
  • Magnitude (speed change)
  • Direction (object changes direction)
• Acceleration vector can be decomposed into:
  • $\vec{a_∥}$, parallel to velocity (changes speed)
  • $\vec{a_\bot}$, perpendicular to velocity (changes direction)

Projectile Motion

• 2D motion under gravity only.
  • Vertical Direction
    • Constant acceleration $\vec{g}$ downwards.
  • Horizontal Direction
    • Constant speed.
• Equations of Motion:
  • Vertical motion (constant acceleration):
    • $v<em>{fy} = v</em>{iy} + a_y\Delta t$
    • $y<em>f = y</em>i + v<em>{iy}\Delta t + \frac{1}{2}a</em>y \Delta t^2$
    • $v<em>{fy}^2 = v</em>{yi}^2 + 2a_y\Delta y$
    • $a_y = -9.80 \,\text{m/s}^2$
  • Horizontal motion (constant velocity):
    • $v<em>{fx} = v</em>{ix} = constant$
    • $x<em>f = x</em>i + v_x\Delta t$
• Launch angle affects range and maximum height.
• Range equation: $range = \frac{v_0^2 \sin{2\theta}}{g}$

Relative Motion

• Reference frame: Coordinate system for position measurements.
• Velocity Transformation Equation:
  • $\vec{v}<em>{CB} = \vec{v}</em>{CA} + \vec{v}_{AB}$

Uniform Circular Motion

• Particle moves at constant speed around a circle.
• Speed: $v = \frac{2\pi r}{T}$
• Angular position: $\theta \text{ radians} \equiv \frac{s}{r}$
• Angular displacement: $\Delta \theta = \theta<em>f - \theta</em>i$
• Average angular velocity: $\omega_{avg} \equiv \frac{\Delta \theta}{\Delta t}$
• Instantaneous angular velocity: $\omega \equiv \lim_{\Delta t \to 0} \frac{\Delta \theta}{\Delta t} = \frac{d\theta}{dt}$
• Tangential velocity: $v_t = \omega r$

Centripetal Acceleration

• Acceleration points towards the center of the circle.
• Magnitude: $a = \frac{v_t^2}{r} = \omega^2 r$

Nonuniform Circular Motion

• Speed is changing.
• Angular acceleration: $\alpha \equiv \frac{d\omega}{dt}$
• Tangential acceleration: $a_t = \alpha r$