Chapter 3 Projectile Motion

Chapter 3: Motion in 2D

  • 3.1 Kinematics in Two Dimensions: An Introduction

  • 3.2 Vector Addition and Subtraction: Graphical Methods

  • 3.3 Vector Addition and Subtraction: Analytical Methods

  • 3.4 Projectile Motion

  • 3.5 Addition of Velocities

Overview

This chapter provides an in-depth exploration of two-dimensional motion, emphasizing the intricate concepts, methodologies, and mathematical tools that are essential for comprehending vector operations in various real-world applications. The topics covered include:

  • Two-Dimensional Motion

  • Vector Operations

  • Trigonometric Applications

  • Problem-Solving

Vector Addition Methods

  1. Tip-To-Tail

  2. Parrallellagram Method

Graphical Methods of Vector Addition

  • Tip to Tail Method:

    • In this illustration method, the tail of one vector is positioned at the tip of another vector. The resultant vector is illustrated by drawing a line from the tail of the first vector to the tip of the last vector added. This method visually depicts how vectors combine, making it easier to understand vector addition geometrically.

  • Parallelogram Method:

    • This technique requires aligning the tails of the vectors to be added. A parallelogram is drawn such that the two vectors represent adjacent sides. The diagonal of the parallelogram represents the resultant vector, which formalizes the mathematical determination of the resultant from two vectors while ensuring that the opposite sides align in length (congruence).

Understanding the Independence of Perpendicular Vector Quantities


Tip to Tail Method is our Typical Choice

To illustrate the addition of two vectors, such as displacements of 5 meters and 2 meters:

  • Procedure: Vectors are aligned in a tip to tail configuration to clearly display their orientations and resultant direction. The resultant vector is drawn from the tail of the first vector to the tip of the second vector, indicating total displacement in magnitude.

  • Example: For instance, when adding the distances of displacement from a starting point to reveal a cumulative distance traveled, this method enhances visual comprehension of motion.

Commutative Property of Vectors

  • The commutative property asserts that the order in which vectors are added does not influence the resultant's magnitude or direction. This can be expressed symbolically as: blue vector + red vector = red vector + blue vector, illustrating the flexibility in rearranging vectors while maintaining the same outcome.

Adding Multiple Vectors

  • Combining Three or More Vectors: When dealing with multiple vectors that share similar characteristics, such as velocity or displacement, the Tip to Tail method is applied methodically. Each vector must retain its direction, and care must be taken to ensure that all vectors are accurately represented with their respective angles and magnitudes, highlighting the vectors' relationships through systematic addition.

Parallelogram Method Explained

  • Explanation: The resultant vector can alternatively be calculated using the design properties of the parallelogram, taking the initial positions of the two vectors' tails and calculating their respective paths. The resultant extends from the common tail to the intersection of the lines extending from the heads of the vectors, demonstrating how vector addition can be visualized geometrically.

Opposite Vectors

Definition and Explanation

  • Considering a vector v measuring 17 m/s directed upwards and to the right, its opposite vector, denoted as -v, would measure the same magnitude (17 m/s) but directed downwards and to the left. This distinction showcases that while the magnitudes of opposing vectors are equal, their directional orientations are separated by 180 degrees, revealing crucial relationships in vector mathematics.

Independence of Vector Quantities

Key Concept

  • The concept of independence is illustrated through the resultant of perpendicular vector quantities. For example, analyzing a boat moving east at 8.0 m/s, while concurrently a river flows north at 5.0 m/s. Despite working independently, their combined effects result in a diagonal movement representing the overall resultant velocity experienced by the boat, exemplifying how perpendicular vectors can coexist and dramatically impact motion.

River Crossing Problem

Practical Scenario Explained

  • Consider a situation where an individual rows a kayak across a 200-meter-wide river with a rowing speed of 0.5 m/s against a current that flows at 0.3 m/s.


    • Solution: The total time required for crossing the river can be calculated by dividing the width of the river by the rowing speed: 200m / 0.5m/s = 400 seconds. As the river current affects the trajectory, the downstream distance can be calculated by multiplying the time elapsed by the current speed: 400 seconds * 0.3m/s = 120m. The overall resultant cross velocity can be derived using the Pythagorean theorem, leading to a resultant speed of approximately 0.58 m/s, depicted through the equation: (\sqrt{(0.5)^2 + (0.3)^2}).

Vector Operations Algebraically

Adding Vectors

  • Mathematical Approach: When vectors are positioned at right angles, the Pythagorean theorem can be directly applied to find resultant magnitudes. For vectors inclined at angles other than 90 degrees, it is advisable to decompose the vectors into their respective X and Y components, facilitating more straightforward addition.

Finding Resultant Magnitude

  • Right Angles: In cases where vectors are at right angles, the effective formula is expressed as (c^2 = a^2 + b^2) facilitating straightforward calculation of resultant vectors’ magnitudes.

  • Angle Vector Calculation: To determine the angle formed by the resultant vector, use the formula (θ = tan^{-1}(Y/X)), aiding in visualizing the vector’s orientation concerning the axes.

Basic Trigonometry Functions

Key Functions

  • The fundamental relationships of trigonometry are utilized to derive functionalities associated with vectors:

    • Resultant Magnitude: R = (\sqrt{x^2 + y^2})

    • For X component: x = R * cos(θ)

    • For Y component: y = R * sin(θ)

Finding Vector Components with Trigonometry

Detailed Procedure

  • To determine the vector components given the angle with respect to the horizontal, use sine and cosine functions to find the X and Y components promptly:

    • Example: Consider a velocity vector of 34 m/s directed 25° south of west. The:

      • Horizontal component (Vx) can be calculated as (34 * cos(25°) ≈ 30.8 m/s), denoting the westward component.

      • Vertical component (Vy) can be found through (34 * sin(25°) ≈ 14.4 m/s), representing the southward component of motion.

Pythagorean Theorem Application

Utilizing in Vector Problems

  • The Pythagorean theorem is employed in vector analysis to corroborate the interactions among components through the properties afforded by right-angled triangles. This relationship is essential when resolving intricate vector scenarios.

Example Problems

Projectile Motion and Real-World Applications

  • This section includes a thorough assortment of examples that tackle projectile motion, addressing a variety of angles and trajectories. The derivation of vector components and resultant vectors offers practical applications across diverse scenarios, reflecting real-world challenges involving 2D motion.

Practice and Review

Homework Assignments

  • Section Reviews

  • Physics Classroom Practice Problems