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.