Motion in a Plane – Study Notes (College Physics, Chapter 3)

Motion in a Plane

Introduction to Two-Dimensional Motion

Motion in a plane refers to the study of objects moving in two dimensions, typically described using Cartesian coordinates (x and y axes). This topic is fundamental in physics as it extends the concepts of position, velocity, and acceleration from one dimension to two, allowing for the analysis of more complex motions such as projectile motion.

• Position Vector: The location of a particle in the plane is given by a position vector r, which points from the origin to the particle's location.

• Magnitude of Position Vector: The distance from the origin to point P is $r = |ar{r}| = \\sqrt{x^2 + y^2}$.

• Components: The position vector can be expressed as $\bar{r} = x\hat{i} + y\hat{j}$, where $x$ and $y$ are the coordinates of the point.

• Example: A ball moving in the x-y plane traces a path, and its position at any time is given by its coordinates $(x, y)$.

Velocity in a Plane

Velocity in two dimensions is a vector quantity that describes both the speed and direction of an object's motion. It can be analyzed as average or instantaneous velocity.

• Average Velocity: Defined over a displacement $\Delta \bar{r}$ and time interval $\Delta t$ as $\bar{v}_{av} = \frac{\Delta \bar{r}}{\Delta t}$.

• Instantaneous Velocity: The velocity at a specific point in time, tangent to the path: $\bar{v} = \lim_{\Delta t \to 0} \frac{\Delta \bar{r}}{\Delta t}$.

• Components: $v_x$ and $v_y$ are the x and y components of velocity, respectively.

• Example: In the motion of a model car, the average velocity vector points from the initial to the final position, and its components are calculated as $v_{av,x} = \frac{\Delta x}{\Delta t}$ and $v_{av,y} = \frac{\Delta y}{\Delta t}$.

Accelerations in a Plane

Acceleration in two dimensions involves changes in the magnitude and/or direction of velocity. Both average and instantaneous acceleration are important for describing motion.

• Average Acceleration: $\bar{a}_{av} = \frac{\Delta \bar{v}}{\Delta t}$, where $\Delta \bar{v}$ is the change in velocity over time $\Delta t$.

• Instantaneous Acceleration: $\bar{a} = \lim_{\Delta t \to 0} \frac{\Delta \bar{v}}{\Delta t}$, representing the acceleration at a specific instant.

• Direction: The acceleration vector always points toward the concave side of the curved path.

• Example: For a particle moving along a curved path, the instantaneous acceleration is found by taking the limit as the time interval approaches zero.

Vector Addition and Subtraction

Vectors in two dimensions can be added or subtracted using graphical or analytical methods.

• Head-to-Tail Method: Place the tail of the second vector at the head of the first; the resultant vector points from the tail of the first to the head of the second.

• Component Method: Add or subtract corresponding components: $\bar{A} + \bar{B} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$.

• Subtraction: Subtract by adding the opposite vector: $\bar{A} - \bar{B} = \bar{A} + (-\bar{B})$.

Projectile Motion

Projectile motion is a classic example of two-dimensional motion, where an object moves under the influence of gravity after being projected into the air.

• Trajectory: The path followed is a parabola, determined by the initial velocity and acceleration due to gravity.

• Independence of Motion: The horizontal and vertical motions are independent and can be analyzed separately.

• Equations of Motion:

  • Horizontal (x-direction): Position: $x(t) = x_0 + v_{0x} t$ Velocity: $v_x = v_{0x}$ Acceleration: $a_x = 0$

  • Vertical (y-direction): Position: $y(t) = y_0 + v_{0y} t - \frac{1}{2} g t^2$ Velocity: $v_y = v_{0y} - g t$ Acceleration: $a_y = -g$

• Initial Velocity Components:

  • $v_{0x} = v_0 \cos \theta$

  • $v_{0y} = v_0 \sin \theta$

• Range of Projectile: The horizontal distance traveled is $R = \frac{v_0^2 \sin(2\theta)}{g}$.

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

• Example: A paintball gun fires a ball at an angle; its trajectory and landing point can be calculated using the above equations.

Applications and Examples

Understanding motion in a plane is essential for analyzing real-world phenomena such as sports, vehicle motion, and projectile trajectories.

• Model Car Example: Calculating average velocity and displacement in two dimensions.

• Baseball Hit Example: Determining the flight path and range of a baseball hit toward a fence.

• Field Goal Example: Calculating whether a ball clears the goal post based on its trajectory.

Summary Table: Key Equations for Projectile Motion

Quantity	Equation	Description
Horizontal Position	$x(t) = x_0 + v_{0x} t$	Position in x-direction
Vertical Position	$y(t) = y_0 + v_{0y} t - \frac{1}{2} g t^2$	Position in y-direction
Horizontal Velocity	$v_x = v_{0x}$	Constant velocity (no horizontal force)
Vertical Velocity	$v_y = v_{0y} - g t$	Velocity affected by gravity
Range	$R = \frac{v_0^2 \sin(2\theta)}{g}$	Maximum horizontal distance
Maximum Height	$H = \frac{v_0^2 \sin^2 \theta}{2g}$	Maximum vertical height

Additional info: These notes expand upon the brief points and diagrams in the slides, providing full academic context and definitions for all key terms and equations relevant to two-dimensional motion and projectile motion in introductory college physics.