Kinematics in Two Dimensions Notes

Kinematics in Two Dimensions

Overview

• Lecture introduces concepts with examples.
• Workshops for practice.
• Labs to test models.
• Tutorial problems before workshops.

Lecture Topics

• Position, velocity, and acceleration in 2D motion.
• Vectors.
• Projectile motion.
• Circular motion: tangential velocity and centripetal acceleration.
• Relative velocity.

Vectors and Scalars

• Scalars: Magnitude only (e.g., time, mass, energy, distance).
• Vectors: Magnitude and direction (e.g., velocity, acceleration, force, displacement).
• Displacement vector: change in object’s position relative to the starting position.

Vector Components

• $A_x = |A| \cos\theta$
• $A_y = |A| \sin\theta$
• Pythagoras theorem: $c^2 = a^2 + b^2$

Unit Vectors

• Unit vectors i, j, k in x, y, and z axes, respectively.
• Vector representation: $A = A<em>x𝐢 + A</em>y𝐣 + A_z𝐤$

Position and Displacement Vectors

• Position vector: $r = x𝐢 + y𝐣$
• Displacement vector: $Δr = r<em>2 - r</em>1 = 𝐢(𝑥<em>2 − 𝑥</em>1) + 𝐣(𝑦<em>2−𝑦</em>1)$

Velocity

• Average velocity: $v = (r<em>2 - r</em>1) / (t<em>2 - t</em>1)$
• Instantaneous velocity: $v=\lim_{\Delta t \to 0} \frac{\Delta r}{\Delta t} = \frac{dr}{dt}$
• $v = v<em>x𝐢 + v</em>y𝐣$

Acceleration

• Average acceleration: $a = (v<em>2 - v</em>1) / (t<em>2 - t</em>1)$
• Instantaneous acceleration: $a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

Projectile Motion

• Only force is gravity (neglecting air resistance).
• $a<em>x = 0, a</em>y = -g = -9.8 m/s^2$
• Trajectory is a parabola.
• Initial velocity components: $𝑣<em>0 = 𝑣</em>{0x}𝐢 + 𝑣_{0y}𝐣$
• Horizontal motion: constant velocity.
• Vertical motion: constant acceleration.

Constant Acceleration Equations

• $v = v_0 + at$
• $s = v_0 t + \frac{1}{2}at^2$
• $v^2 = v_0^2 + 2as$

Circular Motion

• Rotation angle $𝜃 = \frac{Δ𝑠}{r}$
• Radian conversion: $360^0 = 2\pi$ rad, $1 rad = 57.3^o$

Uniform Circular Motion

• Constant speed around a circle.
• Centripetal acceleration: $a_c = \frac{v^2}{r}$
• Period: $T = \frac{2πr}{v}$
• Frequency: $f = \frac{1}{T}$

Angular and Linear Velocities

• Angular velocity: $ω = \frac{Δ𝜃}{∆𝑡}$
• Linear velocity: $v = rω$

Angular Acceleration

• $\alpha = \frac{\Delta \omega}{\Delta t}$
• Tangential linear acceleration: $a_{tang} = αr$

Relative Velocity

• $𝒗<em>{P/A} = 𝒗</em>{P/B} + 𝒗_{B/A}$
• $𝒗<em>{plane \,wrt \, ground} = 𝒗</em>{plane \,wrt \, air} + 𝒗_{air \, wrt \, ground}$