Motion in a Plane

Introduction

Motion in a plane, or two-dimensional motion, is a fundamental topic in college physics. It involves analyzing the position, velocity, and acceleration of objects moving in two spatial dimensions, typically represented by the x- and y-axes. This topic is essential for understanding projectile motion, relative velocity, and the application of vector mathematics in physics.

Position, Velocity, and Acceleration Vectors in 2D

Position Vector

• Definition: The position vector \( \vec{r} \) locates a point P in the plane relative to the origin.

• In Cartesian coordinates: \( \vec{r} = x\hat{i} + y\hat{j} \), where x and y are the coordinates of point P.

• Magnitude of the position vector:

• The position vector can also be described by its magnitude and angle relative to the x-axis.

Velocity in a Plane

• Average velocity: The change in position vector divided by the time interval.

• Instantaneous velocity: The velocity at a specific instant, tangent to the path of motion.

• The velocity vector has both x and y components: \( v_x \) and \( v_y \).

• Direction: The instantaneous velocity vector is always tangent to the object's path.

Acceleration in a Plane

• Average acceleration: The change in velocity vector divided by the time interval.

• Instantaneous acceleration: The acceleration at a specific instant.

• Acceleration in two dimensions can result from changes in the magnitude and/or direction of velocity.

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

Vector Operations in Two Dimensions

Addition and Subtraction of Vectors

• Addition (Head-to-Tail Method): To add two vectors, place the tail of the second vector at the head of the first. The resultant vector is drawn from the tail of the first to the head of the second.

• Subtraction: Subtracting a vector is equivalent to adding its opposite (reverse direction).

• Vectors can be resolved into components along the x- and y-axes for easier calculation.

Projectile Motion

Characteristics of Projectile Motion

• A projectile is any object that moves in two dimensions under the influence of gravity alone (assuming air resistance is negligible).

• The path followed by a projectile is a parabola in the x-y plane.

• The motion can be analyzed by separating it into horizontal (x) and vertical (y) components.

Equations of Motion for Projectiles (Constant Acceleration)

• Horizontal motion (x-direction):

  • Acceleration:

  • Velocity: (constant)

  • Position:

• Vertical motion (y-direction):

  • Acceleration: (where downward)

  • Velocity:

  • Position:

• Initial velocity components:

Key Quantities in Projectile Motion

• Maximum height (H):

• Time of flight (T):

• Range (R):

Examples and Applications

• Model Car Example: Calculating average velocity using changes in x and y positions over a time interval.

• Paintball Gun Example: Determining the trajectory and range of a paintball fired at a given angle and speed.

• Home-Run Hit Example: Analyzing the flight of a baseball to determine if it clears a fence at a certain distance.

• Field Goal Example: Calculating the vertical position of a football at a given horizontal distance to see if it passes above the crossbar.

Summary Table: Key Equations for Projectile Motion

Additional info:

• Relative velocity in two dimensions and frames of reference are also important but not detailed in the provided slides. In general, relative velocity is calculated by vector addition or subtraction of velocities as observed from different frames.

• Projectile motion assumes negligible air resistance unless otherwise specified.