Physics I: Week 1 – 1D Kinematics Study Notes

Physical Quantities and Measurement

Fundamental Quantities and SI Units

Physics relies on the measurement of fundamental quantities, each with standard units defined by the International System of Units (SI).

• Length – measured in meters (m)

• Mass – measured in kilograms (kg)

• Time – measured in seconds (s)

• Force – measured in newtons (N)

• Velocity – measured in meters per second (m/s)

These units form the basis for all physical measurements in classical mechanics.

Significant Figures and Scientific Notation

Accurate measurement and reporting require attention to significant figures and the use of scientific notation for very large or small numbers.

• Significant figures indicate the precision of a measurement.

• Scientific notation expresses numbers as a product of a coefficient and a power of ten, e.g., $2.2 \times 10^3$.

Unit Conversion

Converting Between Units

Unit conversion is essential for solving physics problems. It involves multiplying or dividing by conversion factors to change from one unit to another.

• Example 1: Convert 135 cm to meters: $135 \div 1000 = 0.135$ m

• Example 2: Convert 2.2 km to meters: $2.2 \times 10^3 = 2200$ m

• Example 3: Convert 1 hour to seconds: $1 \times 60 \times 60 = 3600$ s

• Example 4: Convert 200 seconds to hours: $200 \div 3600 = 0.0556$ h

Scalars and Vectors

Definitions and Examples

Physical quantities are classified as either scalars or vectors.

• Scalars have only magnitude (size). Examples: distance, mass, speed.

• Vectors have both magnitude and direction. Examples: displacement, velocity, acceleration.

Comparison Table

Quantity	Scalar	Vector
Distance vs Displacement	Distance	Displacement
Speed vs Velocity	Speed	Velocity

Vector Algebra

Vector Addition and Subtraction

Vectors are added using the parallelogram or triangle method. The sum of two vectors is called the resultant.

• Vector addition: $\vec{a} + \vec{b} = \vec{c}$

• Magnitude of resultant: $|\vec{c}| = \sqrt{a^2 + b^2}$ (for perpendicular vectors)

• Direction: $\tan \theta = \frac{a}{b}$

Reference Frames and Coordinate Systems

Defining Position

Motion is described relative to a reference frame, often using a coordinate system (e.g., x-axis).

• Reference frame: The perspective from which motion is measured.

• Coordinate system: Assigns numerical values to positions (e.g., $x$, $x'$).

1D Kinematics

Displacement

Displacement is the change in position of an object along a straight line.

• Formula: $\Delta x = x_f - x_i$

• Example: If $x_i = 10$ m and $x_f = 30$ m, $\Delta x = 20$ m

• Direction matters: Negative displacement indicates motion toward the origin.

Average Velocity

Average velocity is the rate of change of position over time.

• Formula: $\bar{v} = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}$

• Units: $[m/s]$

• Example: If a car travels 100 km in 2 h and 50 km in 1 h, average velocity: $\frac{150}{3} = 50$ km/h

Average Speed

Average speed is the total distance traveled divided by the total time taken.

• Formula: $\text{Average speed} = \frac{\text{Total distance}}{\text{Total time}}$

• Example: Swimming 50 m out and 50 m back in 48 s: $\text{Average speed} = \frac{100}{48} = 2.08$ m/s

Instantaneous Velocity

Instantaneous velocity is the velocity at a specific moment in time.

• Formula: $v = \frac{dx}{dt}$

Average Acceleration

Average acceleration is the rate of change of velocity over time.

• Formula: $\bar{a} = \frac{v_f - v_i}{t_f - t_i}$

• Units: $[m/s^2]$

• Example: If a car accelerates from 0 to 75 km/h in 5 s: $\bar{a} = \frac{75}{5} = 15$ km/h/s (convert units as needed)

Kinematic Equations for Constant Acceleration

Equations

For motion with constant acceleration, the following equations are used:

• $v = v_0 + a t$

• $x = x_0 + v_0 t + \frac{1}{2} a t^2$

• $v^2 = v_0^2 + 2 a (x - x_0)$

Graphical Analysis of Motion

Position, Velocity, and Acceleration Graphs

Graphs are useful for visualizing motion in one dimension.

• Position vs. Time: Slope gives velocity ($y = mx + n$, $m = \frac{\Delta x}{\Delta t}$)

• Velocity vs. Time: Slope gives acceleration; area under curve gives displacement

• Acceleration vs. Time: Area under curve gives change in velocity

Problem Solving in 1D Kinematics

Steps for Solving Problems

1. Identify knowns and unknowns

2. Choose appropriate kinematic equations

3. Solve algebraically, then substitute values

Example Problem

A car starts from rest and accelerates at $4.9$ m/s2 along a straight line. What distance does it travel before reaching a speed of $30$ m/s?

• Given: $a = 4.9$ m/s2, $v_0 = 0$, $v = 30$ m/s

• Use: $\Delta x = \frac{v^2 - v_0^2}{2a}$

• Solution: $\Delta x = \frac{30^2 - 0^2}{2 \times 4.9} = \frac{900}{9.8} = 91.8$ m

Summary Table: Key Kinematic Quantities

Quantity	Symbol	SI Unit	Vector/Scalar
Distance	d	m	Scalar
Displacement	$\Delta x$	m	Vector
Speed	v	m/s	Scalar
Velocity	$\vec{v}$	m/s	Vector
Acceleration	$\vec{a}$	m/s2	Vector

Additional info: These notes are based on handwritten lecture slides and class notes for introductory college physics, focusing on the foundational concepts of 1D kinematics, measurement, and problem solving.