Kinematics and Particle Models: Study Notes for Introductory Physics

Kinematics and Particle Models

Particle Models

Particle models are simplified representations of objects in motion, treating them as point particles to analyze their movement without considering their size or shape.

• Constant motion: Represented by evenly spaced dots.

• Speeding up: Dots get farther apart as time progresses.

• Slowing down: Dots get closer together.

• Quadratic motion: Dots show increasing or decreasing spacing in a non-linear fashion.

• Measurement convention: The particle model does not consider the object's size; for example, measurement may start at the bottom of a rocket and end at the top.

Reference Frames and Arbitrary Choices

Directions and sign conventions in kinematics are chosen for convenience and can be arbitrary.

• Right: Often considered positive.

• Left: Often considered negative.

• Up: Often considered positive.

• Down: Often considered negative.

• Choices may be made differently to simplify calculations.

Motion and Kinematic Variables

Displacement and Position

Displacement is the change in position of an object, and is a vector quantity.

• Displacement ($\Delta x$): $\Delta x = x_F - x_i$

• Position ($x_i$, $x_F$): Location at specific moments.

Velocity and Speed

Velocity is the rate of change of position and is a vector; speed is the magnitude of velocity and is a scalar.

• Velocity ($v$): $v = \frac{\Delta x}{\Delta t}$

• Speed: $\text{Speed} = \frac{\text{distance traveled}}{\text{time}}$

Position and Velocity Graphs

Position and velocity graphs are related, but velocity graphs alone do not reveal the starting position.

• Position vs. Time: Shows how position changes over time.

• Velocity vs. Time: Shows how velocity changes over time.

• Key Point: You cannot determine the starting position just from velocity graphs.

Kinematic Variables and Units

Key Variables

• $x_i$, $x_F$: Position at specific moments

• $\Delta x$: Change in position over an interval

• $v_i$, $v_F$: Instantaneous velocities

• $\Delta v$: Change in velocity over an interval

• $v_{avg}$: Average velocity over an interval

• $t_i$, $t_F$: Moments in time

• $\Delta t$: Change in time

• $a$: Acceleration (assumed constant unless stated otherwise)

Units of Measurement

Variable	SI Units	Alternative Units
$x_i$, $x_F$, $\Delta x$	meters (m)	km, cm, inch, mile, ft
$t_i$, $t_F$, $\Delta t$	seconds (s)	min, hr, yr, etc.
$v_i$, $v_F$, $\Delta v$	m/s	km/hr, mph
$a$	m/s2	km/hr/s, G

Acceleration and Its Interpretation

Definition and Formula

Acceleration is the rate of change of velocity with respect to time.

• Formula: $a = \frac{\Delta v}{\Delta t} = \frac{v_F - v_i}{t_F - t_i}$

• Acceleration can be positive (speeding up) or negative (slowing down), depending on the direction of velocity and the sign convention.

Examples of Acceleration in Motion

• Forward, speeding up: Velocity increases above the x-axis ($+$ acceleration).

• Forward, slowing down: Velocity decreases above the x-axis ($-$ acceleration).

• Backward, speeding up: Velocity is negative and moving away from zero ($-$ acceleration).

• Backward, slowing down: Velocity is negative and moving toward zero ($+$ acceleration).

Kinematic Equations

Constant Velocity

When velocity is constant, displacement is given by:

• $v = \frac{\Delta x}{\Delta t}$

• $x_F = x_i + v \Delta t$

Constant Acceleration

If acceleration is constant, the following kinematic equations apply:

• $v_{avg} = \frac{1}{2}(v_i + v_F)$

• $x_F = x_i + v_{avg} \Delta t$

• $x_F = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$

• $v_F^2 = v_i^2 + 2a \Delta x$

Free Fall and Symmetrical Motion

Free Fall

Objects in free fall experience the same acceleration due to gravity, $g = 9.8\,\text{m/s}^2$, when no other forces act (e.g., air resistance is neglected).

• Free fall: Only gravity acts on the object.

• Graph: Velocity decreases linearly with time, slope $= -9.8\,\text{m/s}^2$.

Symmetrical Motion

• Distance up equals distance down.

• Launch speed equals landing speed.

• Time going up equals time going down.

• Velocity is zero at peak motion.

Comparing Motions: Numeric and Algebraic Methods

Numeric Comparisons

Solve for the motion of each object independently, then compare results.

Algebraic Comparisons

Final moment is defined by a relationship between two objects (e.g., when they meet).

Example: Helicopter and Box

A helicopter rises steadily at $2\,\text{m/s}$ while lifting a box hanging on a $2.5\,\text{m}$ rope. At $1.5$ seconds after the rope breaks, how far below the helicopter is the box?

Object	Initial Velocity	Acceleration	Equation Used	Result
Helicopter	$2\,\text{m/s}$	$0$	$x_F = x_i + v \Delta t$	$2.5 + 2 \times 1.5 = 5.5\,\text{m}$
Box	$2\,\text{m/s}$	$-9.8\,\text{m/s}^2$	$x_F = x_i + v_i \Delta t + \frac{1}{2} a (\Delta t)^2$	$2.5 + 2 \times 1.5 + \frac{1}{2}(-9.8)(1.5)^2 = -8.0\,\text{m}$

Distance below helicopter: $5.5 - (-8.0) = 13.5\,\text{m}$

Area Under Curves: Displacement from Velocity Graphs

Calculating Displacement

The area under a velocity vs. time graph gives the displacement.

• Rectangle: $\text{Area} = b \times h$

• Trapezoid: $\text{Area} = \frac{1}{2}(h_1 + h_2) \times b$

Examples

• For a velocity graph with constant velocity, use the rectangle formula.

• For a graph with changing velocity, use the trapezoid formula to find total displacement.

Example Calculation: If $v_i = 2\,\text{m/s}$, $v_F = 6\,\text{m/s}$, $\Delta t = 2\,\text{s}$, then $\Delta x = \frac{1}{2}(2 + 6) \times 2 = 8\,\text{m}$.

Additional info: These notes cover foundational concepts in kinematics, including graphical analysis, equations of motion, and problem-solving strategies for one-dimensional motion.