03_Lecture

Chapter 3: Motion in Two or Three Dimensions

3.1 Goals for Chapter 3

• Study of position, velocity, and acceleration vectors.

• Application of concepts to projectile motion.

• Extension of investigations to circular motion (both uniform and nonuniform).

• Exploration of relative velocity.

3.2 Introduction

• Understanding acceleration during curved motion, such as a car rounding a curve.

• Comparison of the impact timing of a dropped bullet with one fired towards a target.

3.3 Position Vectors

• The position vector of a particle is defined as:( r = xi + yj + zk )

• Where:

  • ( x, y, z ) represent coordinates in three-dimensional space.

3.4 Velocity

3.4.1 Average and Instantaneous Velocity

• Average velocity direction aligns with displacement.

• Instantaneous velocity vector is tangent to the path at any point.

• Definition of instantaneous velocity vector ( v ) when moving in the xy-plane (z and ( v_z ) are zero).

3.4.2 Example: Rover on Mars

• Problem involving a robotic rover and calculations of its position, displacement, and average velocity vectors based on time intervals.

3.5 Acceleration

3.5.1 Definition of Acceleration Vector

• Average acceleration ( a_{av} ) between two points is the change in velocity over time.

• Instantaneous acceleration found using:( a = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} )

3.5.2 Example: Average and Instantaneous Acceleration

• Detailed calculations for average and instantaneous acceleration over specified intervals.

3.6 Components of Acceleration

• Differentiation between parallel and perpendicular components of acceleration.

• Parallel components correspond to changes in speed, while perpendicular components relate to changes in direction.

3.7 Projectile Motion

• Definition: Any body projected with initial velocity affected by gravity and air resistance.

• Important note: Initial resistance to be neglected.

3.7.1 Equations of Motion Under Constant Acceleration

• Analysis of projectile motion separating x and y components.

• Key equations governing motion under constant acceleration.

3.8 Wind Resistance

• Impact of air resistance on projectile motion: peak heights and distances decrease, trajectories deviate from parabolic paths.

3.9 Example Calculations

• Various examples including motorcycle stunt riders and batted baseballs, demonstrating key calculations in projectile motion including velocity and range.

3.10 Summary of Concepts

3.10.1 Position, Velocity, and Acceleration Vectors

• The position vector ( r ): defined as ( r = xi + yj + zk ).

• Average velocity vector defined as ( \overline{v} = \frac{\Delta r}{\Delta t} ).

• Instantaneous velocity ( v = \frac{dr}{dt} ) and instantaneous speed is the magnitude of ( v ).

3.10.2 Projectile Motion Basics

• In projectile motion without air resistance, accelerations in x (( a_x = 0 )) and y (( a_y = -g )) are key.

• The trajectory of any projectile will form a parabola.

3.11 Circular Motion

3.11.1 Uniform Circular Motion

• Uniform circular motion: Constant speed in a circular path; acceleration directed inward.

• Period ( T ): time for one complete revolution.

3.11.2 Nonuniform Circular Motion

• Nonuniform motion occurs when speed varies; radial and tangential components are analyzed.

3.12 Relative Velocity

• Discusses scenarios where objects move relative to one another, illustrating velocity in various reference frames.

• Examples related to vehicles in motion and calculations of relative velocity in two or three dimensions.