Unit 1 Study Guide: Representing Motion and Motion in One Dimension

Representing Motion

Introduction to Physics and Scientific Study

• Physics is the study of the fundamental laws governing the natural world. Success in physics requires regular, diligent practice, as it is a skill-based discipline.

• Scientific study is based on the expectation of rationality and order in the universe, reflecting the consistency of physical laws.

Mathematical Prerequisites

• Algebra: Solving equations for unknowns.

• Trigonometry: Pythagorean theorem, sine, cosine, tangent, and their inverses.

• Vectors: Breaking vectors into components and adding them.

Base Quantities and SI Units

Base Quantity	SI Unit
Length [L]	Meter [m]
Time [T]	Second [s]
Mass [m]	Kilogram [kg]

Base quantities are defined in terms of a standard.

SI Prefixes

Prefix	Abbreviation	Value
kilo	k	103
centi	c	10-2
milli	m	10-3
micro	μ	10-6
nano	n	10-9
mega	M	106

Note: Memorize the most common prefixes for quick conversions.

Units and Unit Conversions

• Units are essential in all answers. For example, speed should be reported as "60 miles per hour," not just "60."

• Only quantities with the same units can be added or subtracted.

• Unit conversions use conversion factors, e.g., 1 in. = 2.54 cm.

Example: To convert 21.5 inches to centimeters:

$21.5\ \text{inches} \times \frac{2.54\ \text{cm}}{1\ \text{in}} = 54.6\ \text{cm}$

Motion in One Dimension

Quantities in Motion

• Displacement ($\Delta x$): Change in position, $\Delta x = x_f - x_i$ (final minus initial position), units: meters (m).

• Distance traveled: The total length of the path taken, regardless of direction.

• Velocity ($v$): Rate of change of displacement. Can be average or instantaneous.

• Speed: Rate of change of distance traveled. Can be average or instantaneous.

• Acceleration ($a$): Rate of change of velocity. Can be average or instantaneous.

Scalars vs. Vectors

Quantity	Scalars	Vectors
Distance/Displacement	Distance traveled (always positive)	Displacement ($\Delta x = x_f - x_i$; can be positive or negative)
Speed/Velocity	Average speed = $\frac{\text{distance traveled}}{\text{time elapsed}}$	Average velocity = $\frac{\Delta x}{\Delta t}$
Instantaneous	Magnitude of instantaneous velocity	$v_{inst} = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} = \frac{dx}{dt}$
Acceleration		Average: $a = \frac{\Delta v}{\Delta t}$ Instantaneous: $a_{inst} = \lim_{\Delta t \to 0} \frac{\Delta v}{\Delta t} = \frac{dv}{dt}$

Displacement and Examples

• Displacement can be positive or negative depending on direction.

• Example: If a car moves from 30 m to 55 m, $\Delta x = 55 - 30 = +25$ m. If it moves from 55 m to 30 m, $\Delta x = 30 - 55 = -25$ m.

Distance vs. Displacement

• Distance is the total path length; displacement is the straight-line change in position.

• Example: Walking 40 m east, then 30 m west: distance = 70 m, displacement = 10 m east.

Average Speed vs. Average Velocity

• Average speed: $\frac{\text{distance traveled}}{\text{time elapsed}}$

• Average velocity: $\frac{\text{displacement}}{\text{time elapsed}}$

• Example: If a runner moves from $x_1 = 50.0$ m to $x_2 = 30.5$ m in 3.00 s, average velocity $= \frac{30.5 - 50.0}{3.00} = -6.5$ m/s.

Instantaneous Velocity

• Defined as the slope of the tangent to the position vs. time curve at a specific instant.

• $v_{inst} = \frac{dx}{dt}$

Instantaneous Acceleration

• Defined as the slope of the velocity vs. time curve at a specific instant.

• $a_{inst} = \frac{dv}{dt}$

Kinematic Equations (Constant Acceleration)

Assumption: Acceleration is constant.

Equation	Missing Variable
$v_f = v_i + a t$	$x$
$x = x_i + v_i t + \frac{1}{2} a t^2$	$v_f$
$v_f^2 = v_i^2 + 2 a (x - x_i)$	$t$
$x = x_i + \frac{1}{2}(v_i + v_f)t$	$a$

Summary Table: Scalars and Vectors in Kinematics

Units	Scalars	Vectors
meters	Distance traveled	Displacement ($\Delta x = x_f - x_i$)
meters/second	Average speed = $\frac{\text{distance traveled}}{\text{time elapsed}}$	Average velocity = $\frac{\Delta x}{\Delta t}$
meters/second	Instantaneous speed = magnitude of instantaneous velocity	$v_{inst} = \frac{dx}{dt}$
meters/second2		Average acceleration = $\frac{\Delta v}{\Delta t}$ Instantaneous acceleration = $\frac{dv}{dt}$

Worked Examples

• Displacement: For $x(t) = 2.10 t^2 + 2.80$, displacement from $t_1 = 3.00$ s to $t_2 = 5.00$ s is $x(5) - x(3) = 55.3 - 21.7 = 33.6$ m.

• Average velocity: $\frac{33.6\ \text{m}}{2\ \text{s}} = 16.8$ m/s.

• Instantaneous velocity: $v(t) = \frac{dx}{dt} = 4.20 t$, so at $t = 5.00$ s, $v = 21.0$ m/s.

• Average acceleration: If $v_1 = 15.0$ m/s, $v_2 = 5.0$ m/s, $\Delta t = 5.0$ s, $a = \frac{5.0 - 15.0}{5.0} = -2.0$ m/s2$.

Additional info:

• Graphs of position, velocity, and acceleration provide visual summaries of motion. The slope of the position-time graph gives velocity; the slope of the velocity-time graph gives acceleration.

• Understanding the difference between scalars and vectors is crucial for solving kinematics problems correctly.