Week 2 - Kinematics: Scalars and Vectors

Kinematics: Scalars and Vectors

Scalars and Vectors

  • Definition of Scalar Quantity: A scalar quantity is one that can be described by a single number.

    • **Examples: **
    • Temperature
    • Speed
    • Mass

  • Definition of Vector Quantity: A vector quantity deals inherently with both magnitude and direction.

    • **Examples: **
    • Velocity
    • Force
    • Displacement

Representation of Vectors

  • Vector Arrow: By convention, the length of a vector arrow is proportional to the magnitude of the vector.
    • Example:
    • 8 lb vector
    • 4 lb vector
  • Direction of the Arrow: The direction of the arrow gives the direction of the vector.

Graphical Representation of Vectors

  • **Displacement Vector Example: **
    • Initial Position: Start
    • Magnitude: 2 km
    • Angle: 30.0°
    • Final Position: Finish
    • Note: The arrow represents the displacement vector.

Vector Addition and Subtraction

  • Purpose: Often it is necessary to add one vector to another.
  • Colinear Vectors: When vectors point in the same direction, they are called colinear.
  • Resultant Vector: The result of vector addition.

Graphical Addition of Vectors

  • **Example of Vector Addition: **
    • Vectors with Magnitudes:
    • 5 m
    • 3 m
    • Resulting Vector: 8 m
  • Method: Tail-to-head method used for graphical addition.

Moving to 2-Dimensional Vectors

  • **Representation of Vectors on a Plane: **
    • Vectors can be represented based on any two-dimensional coordinate system using cardinal directions (North, South, East, West).

Vector Analysis

  • Angle Representation:
    • Diagram demonstrating vectors maintained at a right angle (90°) is crucial in defining vector multiples and resolving components.

Mathematical Operations on Vectors

  • Vector Addition Example:
    • Example Calculation:
    • (R=(2.00extm)2+(6.00extm)2)1/2(R = (2.00 ext{ m})^2 + (6.00 ext{ m})^2)^{1/2}
    • Result: 6.32 m
  • Angle Calculation:
    • Angle determined based on vector components:
    • ( an heta = rac{2.00}{6.00})
    • Result: $ heta ext{ is } 18.4^ ext{o}$

Vector Components

  • Definition: The vector components of any vector A can be defined along the x and y axes.
  • Scalar Components: Results in vector components being expressed in terms of unit vectors $ ext{x} ext{ and } ext{y} $ with respective magnitudes.
  • Mathematical Representation:
    • A=A<em>x+A</em>yA = A<em>x + A</em>y
  • Unit Vector Definition:
    • xextˆx ext{ˆ} and yextˆy ext{ˆ} are unit vectors with magnitudes of 1.

Example of Resolving Vector Components

  • Example Calculation:
    • Given a displacement vector with magnitude 175 m at an angle of 50.0° relative to the x-axis:
    • yr=(175extm)(extsin50.0exto)=134extmy_r = (175 ext{ m})( ext{sin } 50.0^ ext{o}) = 134 ext{ m}
    • xr=(175extm)(extcos50.0exto)=112extmx_r = (175 ext{ m})( ext{cos } 50.0^ ext{o}) = 112 ext{ m}
    • Final vector representation:
    • r=(112extm)extxextˆ+(134extm)extyextˆr = (112 ext{ m}) ext{x} ext{ˆ} + (134 ext{ m}) ext{y} ext{ˆ}

Addition of Vectors by Components

  • Component Method:
    • To resolve vectors into their components:
    • C=A+BC = A + B
    • Each component can be added separately:
      • C<em>x=A</em>x+BxC<em>x = A</em>x + B_x
      • C<em>y=A</em>y+ByC<em>y = A</em>y + B_y

Real World Application: Problems

  • Example Problem (New Clark Bridge):

    • Task involves calculating the lengths of cables based on bridge dimensions.
    • Measurements to consider:
    • Tower's height above the road deck = 176 ft
    • Size of the cables based on total lengths BEC and AED.
    • Given measurements:
    • BEC = ??
    • AED = ??

  • Frank's Route Problem:

    • Distance calculation based on Manhattan map data:
    • Involves angle of path, shortest route estimation.

  • Cable Distance Problem:

    • Calculate the distance across a river based on angles and height:
    • Angles: 37.0°; Length: 18.0 m; Result: Distance across river (d) = ?

  • Hiking Route Issue:

    • Bill's total hiking distance and endpoint calculation after multiple directional movements (3.50 mi North, 1.00 mi Northeast, and 1.50 mi South).

  • Airplane Navigation Task:

    • Calculate resultant distance traveled by airplane given cruising altitude and directional changes:
    • Path: continue North for 500 km, then turn to 45.0° East of North for 200 km; Result: Total distance traveled?