Motion in Two or Three Dimensions

Introduction

Motion in two or three dimensions extends the concepts of kinematics beyond straight-line motion, allowing us to analyze the position, velocity, and acceleration of objects moving along curved paths. This chapter covers the use of vectors to describe motion, projectile motion, circular motion, and the concept of relative velocity.

• Key Questions: Where does a batted baseball land? Is a cyclist accelerating when moving at constant speed around a curve? How do different observers describe the motion of a particle?

• Goal: To understand and mathematically describe motion in two and three dimensions using vectors.

Vectors in Kinematics

Position Vector

The position vector locates a particle in space relative to the origin. In three dimensions, it is given by:

• Definition: , where , , and are the coordinates of the particle.

• Displacement: The change in position is .

Motion in 3D

Motion in three dimensions is described by the position vector as a function of time:

• Each coordinate (, , ) is a function of time.

Velocity

Average Velocity

The average velocity between two points is the displacement divided by the time interval:

• Direction is the same as the displacement vector.

• Component form:

Instantaneous Velocity

The instantaneous velocity is the rate of change of position with respect to time:

• Component form: , ,

• Always tangent to the path of the particle.

• Magnitude (speed):

Acceleration

Average Acceleration

The average acceleration is the change in velocity divided by the time interval:

• Component form:

• Depends on the vector difference between velocities, not positions.

Instantaneous Acceleration

The instantaneous acceleration is the rate of change of velocity with respect to time:

• Component form: , ,

• Direction depends on whether speed is constant, increasing, or decreasing.

• Acceleration can be nonzero even if speed is constant (e.g., changing direction).

Problem Solving in Two and Three Dimensions

• Distinguish between average and instantaneous quantities.

• Distinguish between position, velocity, and acceleration vectors.

• points in the direction of change of ; points in the direction of change of $\vec{v}$.

• Motion in , , and directions can often be treated independently.

Motion in a Plane

Independence of Motion

Motion in the direction is independent from motion in the direction. The same kinematic equations apply to each direction separately:

• Position and velocity equations for each direction can be solved independently.

Projectile Motion

Basic Principles

Projectile motion is a special case of two-dimensional motion under constant acceleration due to gravity.

• Neglect air resistance and Earth's rotation.

• Acceleration due to gravity: downward.

• Horizontal velocity is constant: .

• Vertical motion: .

Equations of Motion

• Horizontal:

• Vertical:

• Initial velocity components: ,

Range and Maximum Height

• Range: (for same initial and final elevation)

• Maximum range at

• Maximum height:

• Time of flight:

Trajectory Equation

The path of a projectile is a parabola:

Effects of Air Resistance

• Air resistance complicates calculations; acceleration is no longer constant.

• Maximum height and range decrease.

• Trajectory is no longer a perfect parabola.

Uniform Circular Motion

Basic Concepts

Uniform circular motion occurs when an object moves in a circle at constant speed. The direction of velocity changes continuously, resulting in acceleration.

• Position can be described using polar coordinates: , .

• Velocity is always tangent to the circle.

• Acceleration is directed toward the center (centripetal acceleration).

Centripetal Acceleration

• Magnitude: , where is the radius of the circle.

• Period:

• Direction is always toward the center of the circle.

Nonuniform Circular Motion

• If speed varies, there is also a tangential acceleration component () parallel to the velocity.

• Total acceleration is the vector sum of radial and tangential components.

Relative Velocity

Frames of Reference

The velocity of an object depends on the observer's frame of reference. Relative velocity is the velocity of an object as seen by a particular observer.

• In one dimension:

• In two or three dimensions: use vector addition to combine velocities.

• Example: A passenger walking inside a moving train has a velocity relative to the train and a velocity relative to the ground.

Summary Table: Key Equations for Two- and Three-Dimensional Motion

Examples and Applications

• Projectile Motion: Calculating the range, maximum height, and time of flight for a baseball or paintball.

• Circular Motion: Determining the acceleration of a carnival ride moving in a circle.

• Relative Velocity: Analyzing the effect of wind on an airplane's path.

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