Vectors and Motion: A First Look

Introduction to Scalars and Vectors

• Physical Quantities
  • Physical quantities can often be categorized into scalars and vectors.
  • Examples of scalar quantities include:
  • Time
  • Temperature
  • Mass
  • Scalar quantities are fully described by a single number (with a unit).
  • Example: Mass can be $6 \text{ kg}$, Temperature can be $30^{\circ}\text{C}$.
  • Scalar quantities can take on positive, negative, or zero values.

Definition of Vector Quantities

• Vector Quantities
  • Many physical quantities have directional quality, which means they require more than just a number for complete representation.
  • Definition of a Vector Quantity: A vector quantity has both a size (magnitude) and a direction.
  • Questions to answer:
  • How far?
  • How fast?
  • Which way?
  • The magnitude of a vector represents its size or length, which can be positive or zero, but cannot be negative.
  • Graphical Representation of Vectors:
  • Represented graphically as arrows.
  • Arrow Characteristics:
    • Points in the direction of the vector quantity.
    • Length is proportional to the vector's magnitude.

Scalars vs. Vectors

• Scalar Examples:
  • Current Time
  • Outside Temperature
  • Your Mass
  • Only requires a single number for specification.
• Vector Examples:
  • The velocity of a race car (e.g., $120 \text{ mph}$, west).
  • The force exerted by an individual (e.g., pushing force in a certain direction).

Notation for Vectors

• When representing vector quantities with symbols, it is crucial to indicate that the symbol represents a vector:
  • Draw an arrow over the letter for vectors (e.g., $\vec{A}$)
  • Symbols without arrows (e.g., $A$) represent scalars.
  • In handwritten work, it is essential to consistently use arrows to denote vectors.
  • Important Note: The arrow over the symbol always points to the right, regardless of the vector's actual direction.

Displacement Vectors

• Displacement:
  • A quantity that specifies not only how far an object moves, but also the direction of that movement (left or right).
  • Since displacement involves both magnitude and direction, it is represented as a vector called the displacement vector.
  • Example Representation:
  • For Sam's travel: $\vec{d} = (100 \text{ ft}, \text{ east})$
  • The first value is the magnitude (size of displacement), and the second value specifies its direction.
  • Path Independence:
  • An object's displacement vector is defined by the initial and final positions, regardless of the specific path taken between these points.
  • Example involving Jane’s trip shows that displacement maintains the same vector representation even if the path varies.

Vector Addition

• Sam's Trip Example:
  • Sam walks east for $50 \text{ ft}$, then walks northeast for $100 \text{ ft}$.
  • His trip can be represented with two displacement vectors: $\vec{d}<em>{1}$ and $\vec{d}</em>{2}$.
  • The Net Displacement Vector: $\vec{d}<em>{\text{net}} = \vec{d}</em>{1} + \vec{d}_{2}$
  • Vector Addition Rules:
  • Vector addition follows different rules compared to scalar addition. Both the direction and magnitude must be accounted for.
  • Graphic Representation:
  • Vectors can be added by placing the tail of one vector at the tip of another (tail-to-tip method).

Trigonometry and Vectors

• In multi-dimensional vector addition, we often compute lengths and angles using trigonometry.
• Right Triangle Basics:
  • The longest side (hypotenuse) is opposite the right angle.
  • Other two sides are defined relative to one angle, denoted as $\theta$.
• Pythagorean Theorem:
  • $H = \sqrt{A^2 + O^2}$
• Trigonometric Ratios:
  • Sine: $\sin(\theta) = \frac{O}{H}$
  • Cosine: $\cos(\theta) = \frac{A}{H}$
  • Tangent: $\tan(\theta) = \frac{O}{A}$
  • Rearranging these ratios allows finding side lengths or angles based on known values.

Velocity Vectors

• Definition of Velocity:
  • Velocity is a vector quantity that describes the motion of an object.
  • Represents both the speed of the object and the direction of its motion.
  • The velocity vector points in the same direction as the displacement vector of motion.
• Motion Diagram Explanation:
  • A motion diagram can illustrate velocity vectors showing a car accelerating from rest.
  • The magnitude of the velocity vector corresponds to the car's speed, illustrating that longer velocity vectors indicate higher speeds.
• Average vs. Instantaneous Velocity:
  • Current examples show average velocity vectors, noting that as velocity increases, details regarding instantaneous velocity will be refined in future material.