Motion in Two or Three Dimensions

Learning Outcomes

This chapter explores the fundamental concepts of motion in two and three dimensions, focusing on the use of vectors to describe position, velocity, and acceleration. Students will learn to:

• Represent the position and velocity of a particle using vectors in two or three dimensions.

• Determine the vector acceleration of a particle and interpret its components parallel and perpendicular to the particle's path.

• Solve problems involving projectile motion and curved paths.

• Analyze circular motion with constant or varying speed.

• Relate velocities of moving objects as observed from different frames of reference.

Introduction to Multidimensional Motion

Understanding motion in more than one dimension is essential for describing real-world phenomena such as the flight of a baseball, the path of a roller coaster, or the movement of a circling hawk. Extending the study of kinematics to two and three dimensions allows for a comprehensive analysis of these complex motions.

Vectors in Kinematics

Position Vector

The position vector locates a particle in space relative to an origin. In three dimensions, the position vector r has components along the x, y, and z axes:

• Definition: The position vector from the origin to point P is given by:

• Components: x, y, and z are the coordinates of the particle at a given time.

• Example: If a particle is at (2, 3, 5), its position vector is .

Velocity

Velocity describes the rate of change of position. It can be average or instantaneous:

• Average Velocity: The displacement divided by the time interval:

• Instantaneous Velocity: The rate of change of position at a specific instant:

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

• Example: For a particle moving along a curve, its velocity at any point points in the direction of motion.

Acceleration

Acceleration measures how velocity changes with time. It can also be average or instantaneous:

• Average Acceleration: Change in velocity divided by the time interval:

• Instantaneous Acceleration: The rate of change of velocity at a specific instant:

• Direction: The acceleration vector does not have to be tangent to the path; for curved paths, it points toward the concave side.

• Example: A car slowing down while rounding a curve experiences acceleration both in magnitude and direction.

Components of Acceleration

Parallel and Perpendicular Components

For motion along a curved path, acceleration can be decomposed into components:

• Parallel Component: Along the direction of velocity; changes the speed.

• Perpendicular (Normal) Component: Perpendicular to velocity; changes the direction of motion.

• Special Cases:

  • Constant speed: Acceleration is normal to the path.

  • Increasing speed: Acceleration points ahead of the normal.

  • Decreasing speed: Acceleration points behind the normal.

Projectile Motion

Basic Principles

A projectile is any object given an initial velocity and then follows a path determined by gravity and air resistance. Neglecting air resistance and Earth's curvature simplifies analysis.

• Separable Motion: Horizontal and vertical motions are independent.

  • Horizontal motion: Constant velocity ()

  • Vertical motion: Constant acceleration ()

• Example: Dropping and throwing balls horizontally from the same height; both hit the ground simultaneously if air resistance is neglected.

Equations for Projectile Motion

Assuming initial position at the origin ():

• Horizontal position:

• Vertical position:

• Horizontal velocity:

• Vertical velocity:

Effects of Air Resistance

When air resistance is considered:

• Acceleration is not constant.

• Maximum height and range decrease.

• Trajectory is no longer a parabola.

Circular Motion

Uniform Circular Motion

In uniform circular motion, an object moves at constant speed along a circular path.

• Acceleration is always perpendicular to velocity (centripetal acceleration).

• No parallel component; speed remains constant.

• Direction of acceleration: Toward the center of the circle.

• Magnitude of centripetal acceleration:

• Period for one revolution:

Nonuniform Circular Motion

If speed varies, motion is nonuniform circular motion:

• Radial (centripetal) acceleration:

• Tangential acceleration: , parallel to velocity, changes speed.

Relative Velocity

Frames of Reference

The relative velocity of an object depends on the observer's frame of reference, which consists of a coordinate system and a time scale.

• One Dimension: If point P moves relative to frame B, and frame B moves relative to frame A:

• Two or Three Dimensions: Use vector addition:

• Example: A passenger walking inside a moving train; the velocity relative to the ground is the sum of the passenger's velocity relative to the train and the train's velocity relative to the ground.

Summary Table: Types of Motion and Key Equations

Additional info: These notes expand on the provided slides and text, adding definitions, equations, and examples for clarity and completeness.