Physics: Scalars, Vectors, and Motion Analysis

Classification of Physical Quantities

Physical quantities are fundamentally divided into two categories: scalars and vectors.

• Scalars: These are physical quantities that possess magnitude only and do not have any specific direction.

  • Examples: Length ($l$), Mass ($m$), Time ($t$ or $T$), Speed ($v$), Volume, Electric Current, and Amount of Substance.
  • Contextual Example: When a car travels at a speed of $1\,km/h$, it indicates how much distance is covered in a specific timeframe but does not specify the heading of the vehicle.

• Vectors: These quantities possess both a magnitude and a clearly defined direction.

  • Examples: Displacement, Velocity, Acceleration, and Force ($F$).

Representation and Symbols of Vectors

Vector quantities are distinguished from scalar quantities through specific notations in writing and graphical illustrations:

• Symbolic Representation:

  • Vectors can be represented by boldface letters (e.g., a or F).
  • Alternatively, they are represented by a letter with an arrowhead drawn directly above it (e.g., $\vec{F}$, $\vec{a}$, $\vec{v}$).
  • The use of a cap symbol (e.g., $\hat{a}$) is specifically used to denote the direction of a vector. For example, if a force is applied along the x-axis, it is denoted using $\hat{x}$. If it is along the y-axis, it is denoted using $\hat{y}$.

• Graphical Representation:

  • Vectors are drawn as arrows.
  • The length of the arrow corresponds to the magnitude of the quantity. A longer arrow indicates a greater force or higher velocity.
  • The arrowhead indicates the direction of the vector.

• Magnitude vs. Vector Notation: When discussing only the magnitude of a vector, the arrowhead is omitted. For instance, if a force of $2\,N$ is applied north, the magnitude can be written simply as $F = 2\,N$. This implies the scalar size of the vector quantity without needing to represent direction in calculations.

Distance versus Displacement

Understanding the distinction between these two terms is essential for defining motion:

• Distance: This is the total path length traveled from an initial position to a final position. It is a scalar quantity because it describes the magnitude of the path without considering direction.

  • Scenario: Moving from point A to point B through various turns or an arbitrary path. If the total curved path measures $65\,miles$, that total length is the distance.

• Displacement: This is defined as the shortest distance between two points in a specific direction. It is a vector quantity.

  • Scenario: A straight-line path from point A to point B. For example, a straight line pointing from A to B measuring $50\,miles$ in the direction of Northeast is a displacement.

• Comparison: If a vehicle travels in a perfectly straight line for $50\,meters$, the magnitude of the distance and the displacement will be identical ($50\,meters$). However, if the path is curved, the distance will be greater than the magnitude of the displacement. Displacement is described with direction, such as "$50\,meters$ in the direction of Mach."

Speed and Velocity

Movement quantities are derived by dividing the previous terms by time:

• Average Speed ($\bar{v}$): This is calculated by dividing the total distance traveled by the total time taken.     $\bar{v} = \frac{\text{Total Distance}}{\text{Total Time}}$

  • Average speed is a scalar quantity.
  • The standard unit is meters per second ($m/s$ or $m\,s^{-1}$).

• Instantaneous Speed: This is the speed of an object at a particular instant in time. The speedometers in cars or road signs showing speed limits refer to instantaneous speed.

• Velocity: This is a vector quantity defined as displacement divided by time. It includes both speed (magnitude) and direction.

  • Velocity can change if either the speed changes, the direction changes, or both change.
  • Instantaneous Velocity: At any single instant, the magnitude of the instantaneous velocity is equal to the instantaneous speed.

Acceleration and Deceleration

Acceleration ($\vec{a}$) describes how fast or slow the velocity of an object is changing. It is a vector quantity.

• Occurrence: Acceleration is produced whenever there is a change in the magnitude of velocity (speeding up or slowing down) or a change in the direction of motion (e.g., moving in a circle).

  • In circular motion at a constant speed of $20\,m/s$, acceleration is still present because the direction is constantly changing (always along the tangent of the circle).

• Formula: Acceleration is defined as the change in velocity ($V_{f} - V_{i}$) divided by the time ($t$ taken for that change.     $a = \frac{V_{f} - V_{i}}{t}$

  • The unit for acceleration is meters per second squared ($m/s^{2}$ or $m\,s^{-2}$).

• Uniform Acceleration: This occurs when the velocity changes by equal amounts in equal time intervals (e.g., velocity increasing by $20\,m/s$ every second: $20 \rightarrow 40 \rightarrow 60 \rightarrow 80$).

• Deceleration (Negative Acceleration): If a body is slowing down, such as when brakes are applied, the acceleration vector is in the opposite direction of the velocity vector. This is referred to as deceleration.

Practical Application: Acceleration Calculation

Problem: A car starts from rest and accelerates uniformly along a straight track, reaching a speed of $90\,km/h$ in $7\,seconds$. Determine the magnitude of the acceleration.

• Initial Velocity ($V_{i}$): Starting from rest implies $V_{i} = 0$.
• Final Velocity ($V_{f}$): $90\,km/h$. This must be converted to the SI unit ($m/s$).
  • Conversion Step 1 (Kilometers to Meters): $90 \times 1000 = 90,000\,m$.
  • Conversion Step 2 (Hours to Seconds): $1\,hour = 60 \times 60 = 3,600\,s$.
  • Calculation: $\frac{90,000}{3,600} = 25\,m/s$.
• Acceleration Calculation:     $a = \frac{25\,m/s - 0\,m/s}{7\,s}$$a = \frac{25}{7} \approx 3.6\,m/s^{2}$

Questions & Review

General Discussion and Class Review

• Calculation Practice:
  • Calculating $\frac{7.1 \times 10^{-25}}{9.3}$ yields approximately $7.6 \times 10^{-33}$.
• Height Conversion:
  • Question: A student's height is $168\,cm$. Determine the height in inches given $1\,inch = 2.54\,cm$.
  • Answer: Divide the height by the conversion factor: $\frac{168}{2.54} \approx 66.14\,inches$.
• Physical Properties:
  • An object's mass remains the same on Earth and on the Moon, but its weight will be different due to gravity.
  • The measurement of space in any direction is called length.
  • The instrument used to measure density is a hydrometer.
  • A liter ($L$) has a volume of $1,000\,cm^{3}$ (cubic centimeters).
• Significant Figures:
  • Question: In the multiplication $6.704 \times 1.20$, how many significant figures should be in the result?
  • Answer: Three. The rule is to use the least number of significant figures in the calculation ($1.20$ has three significant figures).
  • Note: Zeros appearing after a decimal point at the end of a number are significant (e.g., the zero in $1.20$). Zeros such as those in $0.01$ are not significant as they are placeholders unless preceded by a non-zero digit.