Chapter 3: Kinematics in Two Dimensions

Overview

This chapter introduces the study of motion in two dimensions, expanding upon the concepts of one-dimensional kinematics. It covers position, displacement, velocity, acceleration, and projectile motion, providing the foundational equations and problem-solving strategies for analyzing objects moving in a plane.

Motion in 2-D

Introduction to 2-Dimensional Motion

• 1-D Motion: Previously, motion was constrained to a straight line (one dimension).

• 2-D Motion: Objects move in a plane, requiring analysis of both horizontal (x) and vertical (y) components.

• Independence of Motion: The horizontal and vertical motions are independent and always perpendicular (at 90 degrees).

Position in Two Dimensions

Position Vectors and Components

• In 1-D, position is described by a single number with a sign indicating direction relative to an origin.

• In 2-D, position is described by the position vector r, which has both magnitude and direction.

• The position vector can be expressed in terms of its scalar components: rx and ry.

• Recall the discussion of vectors and their components from Chapter 1.

The Displacement Vector

Definition and Calculation

• The initial position of an object is ri at time ti, and the final position is rf at time tf.

• The displacement vector Δr is given by:

• In component form:

Velocity in Two Dimensions

Average and Instantaneous Velocity

• Average velocity vector:

• The direction of average velocity is the same as the displacement vector.

• Average velocity is independent of the path between endpoints.

• Instantaneous velocity:

• Instantaneous velocity is tangent to the object's path at a given point.

Acceleration in Two Dimensions

Average and Instantaneous Acceleration

• Average acceleration:

• Instantaneous acceleration:

Acceleration and Velocity Relationship

• An object accelerates when:

  • The magnitude of the instantaneous velocity changes (speed changes).

  • The direction of velocity changes (even if speed is constant).

  • Both speed and direction change.

• The instantaneous velocity vector v points in the direction of motion at all times.

• The acceleration vector a often points in a direction other than the direction of motion.

Separate x and y Motions

Component Analysis

• Motion in two dimensions can be analyzed as separate motions in the x and y directions.

• Each component follows the same constant acceleration equations as in 1-D, with subscripts indicating the direction.

2-D Constant Acceleration Equations

General Kinematic Equations

Additional info: These equations are valid for constant acceleration in each direction.

Note: 2-D Motion Equations

General and Specific Equations

• The equations above are general for 2-D motion with constant acceleration.

• Specific conditions (e.g., initial velocity or acceleration is zero) allow simplification for particular problems.

• Start with general equations and adapt to the specifics of each problem.

Problem Solving Procedure

Steps for Analyzing 2-D Motion

• Draw and label a diagram of the situation, including a coordinate system.

• Write down knowns and unknowns.

• Separate the motion into x (horizontal) and y (vertical) components, usually connected by the time interval.

• Consider each part separately using the appropriate equations.

• Solve the equations and interpret the answers.

Projectile Motion

Definition and Assumptions

• A projectile is an object launched and allowed to follow a path determined only by gravity.

• Projectile motion is a special case of 2-D motion:

  • Acceleration due to gravity is downward.

  • No air resistance (idealized case).

  • Horizontal acceleration is zero.

  • Earth's rotation is neglected.

Worked Examples

Example 1: Air Hockey Puck

• A puck moves on an air hockey table. Given initial velocities and accelerations in x and y, find the magnitude and direction of velocity after a time interval.

• Apply kinematic equations separately for x and y components, then combine for total velocity.

Example 2: Skateboarder Ramp

• A skateboarder launches off a ramp at a given velocity and angle. Find:

  • Maximum height above ground

  • Time to reach highest point

  • Horizontal distance from ramp to highest point

• Use projectile motion equations, resolving initial velocity into x and y components.

Example 3: Golf Ball Maximum Distance

• A golf ball is hit for maximum range. Find:

  • Time in air

  • Maximum hole length (distance)

• Apply projectile motion equations for range and time of flight.

Example 4: Golf Ball to Elevated Green

• Golf ball is hit to a green elevated above the tee. Find:

  • Speed at impact

  • Time of flight

  • Horizontal distance traveled

• Use kinematic equations with initial velocity components and vertical displacement.

Example 5: Airplane Package Drop

• An airplane releases a package while climbing. Find:

  • Distance from release point to where package hits ground

  • Angle of velocity just before impact

• Resolve airplane's velocity into components, apply projectile motion equations.

Example 6: Marble Thrown from Building

• A marble is thrown horizontally from a building. Given final velocity angle, find initial height.

• Use projectile motion equations and trigonometric relationships.

Additional info: Each example demonstrates the application of 2-D kinematic equations and vector analysis to real-world problems.