Kinematics in Two Dimensions: Kinematics, Projectile Motion, and Angular Motion (Physics 130, Lecture 4)

Kinematics in Two Dimensions

Introduction to 2D Kinematics

Kinematics in two dimensions extends the study of motion to objects moving in a plane, requiring the analysis of both x and y components of displacement, velocity, and acceleration. This is essential for understanding real-world motion such as projectiles and circular paths.

• Displacement, velocity, and acceleration are all vector quantities and must be resolved into components.

• Motion can be decomposed into independent motions along perpendicular axes (usually x and y).

• Examples include hockey pucks on air tables, projectiles, and objects in circular motion.

Vector Decomposition and Motion Analysis

Decomposing Vectors

Any vector in two dimensions can be broken into x and y components using trigonometric relationships.

• Component form: For a vector v at angle θ:

  • $v_x = v \cos(\theta)$

  • $v_y = v \sin(\theta)$

  • $v = \sqrt{v_x^2 + v_y^2}$

  • $\tan(\theta) = \frac{v_y}{v_x}$

• Choosing coordinate axes aligned with the motion simplifies calculations.

• Example: A hockey puck with $v = 2.0\ \mathrm{m/s}$ at $30^\circ$ has $v_x = 2.0 \cos(30^\circ)$ and $v_y = 2.0 \sin(30^\circ)$.

Uniform Motion in 2D

When an object moves with constant velocity (no acceleration), its position as a function of time is given by:

• $x = v_{x} t + x_0$

• $y = v_{y} t + y_0$

• Displacement: $d = \sqrt{x^2 + y^2}$ or $d = vt$

• Example: If a puck starts at $(0,0)$ and moves for $5.0\ \mathrm{s}$, calculate $x$ and $y$ using the above equations.

Motion with Constant Acceleration

When acceleration is present, the kinematic equations must be applied to each component:

• $x = \frac{1}{2} a_x t^2 + v_{x0} t + x_0$

• $v_x = a_x t + v_{x0}$

• $v_x^2 = 2a_x(x - x_0) + v_{x0}^2$

• Similarly for y: $y = \frac{1}{2} a_y t^2 + v_{y0} t + y_0$, etc.

• Example: A puck with $\vec{a} = 10\ (\mathrm{m/s^2}) \hat{x} + 20\ (\mathrm{m/s^2}) \hat{y}$, initially at rest, after $0.10\ \mathrm{s}$: $v_x = 10 \times 0.10 = 1.0\ \mathrm{m/s}$, $v_y = 20 \times 0.10 = 2.0\ \mathrm{m/s}$.

Free Fall and Vertical Motion

Falling Objects: Downward and Upward

Objects in free fall experience constant acceleration due to gravity ($g = 9.8\ \mathrm{m/s^2}$ downward). The direction of velocity and acceleration determines the motion's characteristics.

• Falling Downward: Velocity increases in the negative direction; acceleration is constant and negative.

• Falling Upward: Velocity decreases (becomes less positive), reaches zero at the top, then increases in the negative direction as the object falls back down.

Position	Fall Time	Velocity	Acceleration
44.1 m	3 s	0 m/s	-9.8 m/s2
0 m	0 s	-29.4 m/s	-9.8 m/s2

Additional info: Table values inferred from the provided images and standard free-fall calculations.

Comparing Upward and Downward Throws

• For two balls thrown from a cliff (one up, one down) with the same initial speed, both will have the same speed upon reaching the ground (ignoring air resistance), but the ball thrown upward takes longer to fall.

• Key Concept: The final speed depends only on the initial height and speed, not the direction of the initial velocity.

Projectile Motion

Basic Principles

Projectile motion describes the motion of an object launched into the air, subject only to gravity (no air resistance). The path is a parabola.

• Horizontal motion: Uniform, $v_x = v_0 \cos(\theta)$, $a_x = 0$

• Vertical motion: Constant acceleration, $a_y = -g$

• Equations:

  • $x = v_0 \cos(\theta) t$

  • $y = v_0 \sin(\theta) t - \frac{1}{2} g t^2$

• Time of flight: Determined by vertical motion.

• Range: $R = \frac{v_0^2 \sin(2\theta)}{g}$ (for launch and landing at same height)

• Maximum height: $H = \frac{v_0^2 \sin^2(\theta)}{2g}$

• Example: A projectile launched at $30^\circ$ with $v_0 = 100\ \mathrm{m/s}$: calculate time in air, range, and maximum height using the above equations.

Additional info: The battleship example illustrates that the shell with the shorter range (lower trajectory) will hit first, as it spends less time in the air.

Circular Motion and Angular Kinematics

Describing Circular Motion

Objects moving in a circle experience acceleration even at constant speed, due to the continual change in direction of velocity.

• Arc length: $s = r \theta$ (θ in radians)

• Angular displacement: $\Delta \theta$ (radians)

• Angular velocity: $\omega = \frac{\Delta \theta}{\Delta t}$ (rad/s)

• Relationship to linear speed: $v = r \omega$

• Period (T): Time for one revolution, $T = \frac{1}{f}$

• Frequency (f): Revolutions per second (Hz)

Centripetal Acceleration

In uniform circular motion, the acceleration is always directed toward the center of the circle (centripetal acceleration).

• Magnitude: $a_c = \frac{v^2}{r} = r \omega^2$

• Direction: Always points toward the center of the circle.

• Example: A jet flying in a circle of radius $150\ \mathrm{m}$ at $80\ \mathrm{m/s}$: $a_c = \frac{(80)^2}{150} = 42.7\ \mathrm{m/s^2}$

Angular Acceleration

Angular acceleration is the rate of change of angular velocity.

• $\alpha = \frac{\Delta \omega}{\Delta t}$ (rad/s2)

• Analogous to linear acceleration.

• Direction is the same as the change in angular velocity.

• Example: A turntable accelerates from rest to $33\frac{1}{3}$ rpm in $5.00$ s: $\omega_f = 33\frac{1}{3} \times \frac{2\pi}{60} = 3.49$ rad/s, $\alpha = \frac{3.49}{5.00} = 0.698$ rad/s2

Rotational Kinematics Equations

The equations for rotational motion with constant angular acceleration mirror those for linear motion:

• $\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$

• $\omega = \omega_0 + \alpha t$

• $\omega^2 = \omega_0^2 + 2\alpha (\theta - \theta_0)$

Additional info: These equations are used for analyzing rotating objects such as merry-go-rounds, records, and wheels.

Summary Table: Linear vs. Rotational Kinematics

Linear	Rotational
Displacement: $x$	Angular displacement: $\theta$
Velocity: $v$	Angular velocity: $\omega$
Acceleration: $a$	Angular acceleration: $\alpha$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
$v = v_0 + a t$	$\omega = \omega_0 + \alpha t$
$v^2 = v_0^2 + 2a(x - x_0)$	$\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Key Concepts and Applications

• Always resolve vectors into components for 2D motion problems.

• Horizontal and vertical motions are independent in projectile motion.

• In circular motion, acceleration is always directed toward the center (centripetal).

• Rotational kinematics equations are analogous to linear kinematics.

• Applications include sports (hockey, baseball), engineering (rotating machinery), and natural phenomena (planetary orbits).