Kinematics: Foundations and Objectives

Introduction to Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It focuses on quantities such as position, velocity, and acceleration, and analyzes their components in one and two dimensions.

• Key Objective: Understand and calculate the x and y components of an object's position, velocity, and acceleration.

• Applications: Motion of objects sliding down an incline, projectile motion, and free fall.

Recap of 1D Motion Equations

Equations of Motion in One Dimension

For motion along a straight line (horizontal or vertical), the following kinematic equations are used:

• Final velocity:

• Final position:

• Velocity-position relation:

Definitions:

• vf: Final velocity

• v0: Initial velocity

• a: Acceleration

• t: Time interval

• xf, x0: Final and initial positions

Example Problems – Simple 1D Motion

Solving 1D Kinematics Problems

These problems involve objects moving in a straight line, either accelerating or moving at constant velocity.

• Example 1: A car moving at 17 m/s accelerates at 5.3 m/s2 for 25 m. Find its speed after traveling 25 m.

• Example 2: A car accelerates from rest at 4 m/s2 for 10 s, then moves at steady speed for another 10 s. Find the total distance traveled.

• Example 3: A car traveling at 25 m/s sees a moose and brakes at 8.3 m/s2. With a reaction time of 0.60 s, find the total distance to stop.

Application: Use the kinematic equations above to solve for unknowns such as final velocity, time, or displacement.

Example Problems – 1D Free Fall

Free Fall Motion in One Dimension

Free fall refers to motion under the influence of gravity alone, typically in the vertical direction.

• Example 1: An object thrown upward at 30.0 m/s. Find the maximum height and time to reach it.

• Example 2: A feather dropped on the moon from 1.40 m. With lunar gravity m/s2, find the time to fall.

• Example 3: A person jumps to a height of 2.62 m. Find the takeoff speed.

Key Concept: For free fall near Earth's surface, m/s2 (downward).

1D vs 2D Motion

Comparing One-Dimensional and Two-Dimensional Motion

While 1D motion occurs along a straight line, 2D motion involves movement along a curved path, requiring analysis of both x and y components.

• 1D Motion: Use single-axis kinematic equations.

• 2D Motion: Objects move in a curved path; measurements are broken into x and y components for calculation.

• Method: Solve for each component separately, then combine results as needed.

Vectors in 2D Motion

Understanding Vectors and Their Components

Vectors are quantities with both magnitude and direction, such as displacement, velocity, and acceleration. Analyzing their components simplifies calculations in two dimensions.

• Vector Representation: A vector is specified by its magnitude and direction (angle).

• Component Form: The x-component and y-component are found using trigonometric functions:

• Application: Use components to add, subtract, or resolve vectors in physics problems.

Kinematics Case 1: Motion Up a Ramp

Analyzing Motion on an Inclined Plane

When an object moves up a ramp, its velocity can be resolved into horizontal and vertical components using trigonometry.

• Given: Jill moves up a hill at 3.0 m/s; horizontal component m/s.

• Find: The angle of the hill () and the vertical component ().

• Example: Climbing a 34° slope at 0.75 m/s for 1 minute. Vertical gain is calculated as:

Kinematics – Projectile Motion

Projectile Motion Fundamentals

Projectile motion involves objects launched into the air, experiencing both horizontal and vertical motion. The two components are analyzed separately due to differing accelerations.

• Vertical motion: Acceleration due to gravity ( m/s2).

• Horizontal motion: No acceleration (if air resistance is neglected); velocity remains constant.

Key Equations:

• Initial velocity components:

• Example: An air gun fires T-shirts at 28 m/s, 40° above horizontal. Find time to reach a fan 60 m away and the height at which the fan catches the T-shirt.

Application: Use the above equations to solve for time of flight, range, and maximum height in projectile motion problems.

Additional info: These notes are based on lecture slides and textbook images from "College Physics: A Strategic Approach" (Knight/Jones/Field), 4th Edition, Pearson.