Kinematics in Two Dimensions: Kinematics, Projectile Motion, and Angular Motion

Kinematics in Two Dimensions

Introduction to 2D Kinematics

Kinematics in two dimensions involves the study of motion in a plane, where both the magnitude and direction of vectors such as velocity and acceleration are important. This topic extends the concepts of one-dimensional motion to more complex scenarios, such as projectile motion and circular motion.

• Vector Quantities: Displacement, velocity, and acceleration are all vectors, meaning they have both magnitude and direction.

• Decomposition: Any vector in two dimensions can be decomposed into x and y components.

• Applications: Examples include the motion of a hockey puck on an air table, projectiles, and objects in circular paths.

Acceleration in 2D

Vector Nature of Acceleration

Acceleration in two dimensions can change the magnitude and/or direction of velocity. The average acceleration is defined as:

$\vec{a}_{\text{avg}} = \frac{\Delta \vec{v}}{\Delta t}$

• Perpendicular Component: Changes the direction of velocity (e.g., in circular motion).

• Parallel Component: Changes the magnitude of velocity (speed up or slow down).

• Uniform Circular Motion: Acceleration is always directed toward the center of the circle (centripetal acceleration).

Vectors and Decomposition

Decomposing Vectors into Components

To analyze motion in two dimensions, vectors are decomposed into x and y components using trigonometric functions:

• $v_x = v \cos(\theta)$

• $v_y = v \sin(\theta)$

• $v = \sqrt{v_x^2 + v_y^2}$

• $\tan(\theta) = \frac{v_y}{v_x}$

Example: A hockey puck has $v = 2.0$ m/s at an angle of $30^\circ$:

• $v_x = 2.0 \cos 30^\circ = 1.7$ m/s

• $v_y = 2.0 \sin 30^\circ = 1.0$ m/s

Constant Velocity Motion

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: $d = \sqrt{x^2 + y^2}$

• Distance traveled: $d = vt$

Example: A puck starts at $(0,0)$ with $v_x = 2$ m/s, $v_y = 1$ m/s. After $5$ s:

• $x = 2 \times 5 = 10$ m

• $y = 1 \times 5 = 5$ m

• Total distance: $\sqrt{10^2 + 5^2} = 11.2$ m

Motion with Constant Acceleration

Kinematic Equations in 2D

For motion with constant acceleration, the kinematic equations are applied separately to each component:

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

• $v_x = a_x t + v_{x0}$

• $v_x^2 = 2 a_x (x - x_0) + v_{x0}^2$

• $y = \frac{1}{2} a_y t^2 + v_{y0} t + y_0$

• $v_y = a_y t + v_{y0}$

• $v_y^2 = 2 a_y (y - y_0) + v_{y0}^2$

Example: A puck at rest receives $\vec{a} = 10 \frac{m}{s^2} \hat{x} + 20 \frac{m}{s^2} \hat{y}$ for $0.10$ s:

• $v_{fx} = 10 \times 0.1 = 1$ m/s

• $v_{fy} = 20 \times 0.1 = 2$ m/s

• $|\vec{v}_f| = \sqrt{1^2 + 2^2} = 2.24$ m/s

• Angle: $\theta = \tan^{-1}(2/1) = 63.4^\circ$

Free Fall: Upward and Downward Motion

Vertical Motion Under Gravity

Objects in free fall experience constant acceleration due to gravity ($g = 9.8$ m/s2 downward). The motion can be analyzed for both upward and downward trajectories.

Case	Position	Fall Time	Velocity	Acceleration
Falling Downward	Decreases	Increases	Becomes more negative	-9.8 m/s2
Falling Upward	Increases then decreases	Increases	Becomes less positive, then negative	-9.8 m/s2

• At maximum height, velocity is zero; acceleration remains $-9.8$ m/s2.

• Time to rise equals time to fall (if starting and ending at same altitude).

Projectile Motion

Motion Under Gravity

Projectile motion describes the path of an object launched into the air, subject only to gravity. The motion is a combination of uniform motion in the horizontal direction and uniformly accelerated motion in the vertical direction.

• Horizontal motion: $v_x = v_0 \cos \theta$ (constant)

• Vertical motion: $a_y = -g$, $v_y = v_0 \sin \theta - g t$

• Trajectory: Parabolic path

• Time of flight: Determined by vertical motion

• Range: $R = v_0 \cos \theta \cdot t_{\text{flight}}$

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

Example: A projectile is fired at $30^\circ$ with $v_0 = 100$ m/s. To find time in air:

• Vertical velocity: $v_{0y} = 100 \sin 30^\circ = 50$ m/s

• Time to top: $t_{\text{up}} = \frac{v_{0y}}{g} = \frac{50}{9.8} \approx 5.1$ s

• Total time: $t_{\text{flight}} = 2 t_{\text{up}} = 10.2$ s

Circular Motion and Angular Variables

Describing Circular Motion

Objects moving in a circle are described by angular variables such as angular displacement, angular velocity, and angular acceleration.

• Angle (radians): $\theta = \frac{s}{r}$, where $s$ is arc length, $r$ is radius

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

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

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

Units: 1 revolution = $2\pi$ radians = 360°

Period and Frequency

• Period (T): Time for one revolution (units: s)

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

• $f = \frac{1}{T}$

• Linear speed: $v = \frac{2\pi r}{T} = 2\pi r f$

Centripetal Acceleration

Uniform Circular Motion

In uniform circular motion, the speed is constant but the direction of velocity changes, resulting in an acceleration toward the center of the circle (centripetal acceleration).

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

• Always points toward the center of the circle

• Responsible for changing the direction of velocity, not its magnitude

Example: A jet pulls out of a dive at 80 m/s in a circle of radius 150 m:

• $a_c = \frac{(80)^2}{150} = 42.7$ m/s2

Angular Kinematics

Equations for Rotational Motion

Rotational kinematics with constant angular acceleration is analogous to linear kinematics:

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

• $\omega = \omega_0 + \alpha t$

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

Example: A record accelerates from rest to 33 1/3 rpm in 5.00 s:

• Convert rpm to rad/s: $33\frac{1}{3} \text{ rpm} = 3.49$ rad/s

• $\alpha = \frac{3.49 - 0}{5.00} = 0.698$ rad/s2

Summary Table: Linear vs. Angular Kinematics

Linear Kinematics	Angular Kinematics
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$\theta = \theta_0 + \omega_0 t + \frac{1}{2} \alpha t^2$
$v = v_0 + a t$	$\omega = \omega_0 + \alpha t$
$v^2 = v_0^2 + 2a(x - x_0)$	$\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$

Key Concepts and Applications

• Decompose vectors into components for 2D motion analysis.

• Use kinematic equations for each direction independently.

• Projectile motion is a combination of constant horizontal velocity and constant vertical acceleration.

• In circular motion, centripetal acceleration always points toward the center.

• Angular variables describe rotational motion; their equations mirror those of linear kinematics.

Additional info: These notes provide a foundation for understanding more advanced topics in dynamics, such as energy and momentum in two dimensions, and rotational dynamics.