Chapter 2: 1-D Motion & Kinematics – Study Notes

Chapter 2: 1-D Motion & Kinematics

Introduction to 1-D Motion

One-dimensional (1-D) motion describes the movement of objects along a single axis, typically the x-axis. This chapter introduces the foundational concepts and equations used to analyze such motion, including position, displacement, velocity, and acceleration.

Coordinate System and Position

• Coordinate System: Defines the reference axis (or axes) and the origin (reference point) for measuring motion.

• Position (x): The location of an object along the chosen axis, measured from the origin.

Displacement

• Displacement (Δx): The change in position of an object. Formula: $\Delta x = x_f - x_i$ where $x_f$ is the final position and $x_i$ is the initial position.

• Example: If a car moves from $x_i = 0$ m to $x_f = 50$ m, then $\Delta x = 50$ m.

Velocity

• Average Velocity (vav): The rate of change of displacement over time. Formula: $v_{av} = \frac{\Delta x}{\Delta t}$

• Instantaneous Velocity: The velocity at a specific instant, given by the slope of the position vs. time graph at a single point.

• Graphical Interpretation:

  • On a position vs. time plot, the slope between two points gives the average velocity.

  • The slope at a single point gives the instantaneous velocity.

Acceleration

• Average Acceleration (aav): The rate of change of velocity over time. Formula: $a_{av} = \frac{\Delta v}{\Delta t}$

• Instantaneous Acceleration: The acceleration at a specific instant, given by the slope of the velocity vs. time graph at a single point.

• Graphical Interpretation:

  • On a velocity vs. time plot, the slope between two points gives the average acceleration.

  • The slope at a single point gives the instantaneous acceleration.

Kinematic Equations for Constant Acceleration

When acceleration is constant, the following kinematic equations describe 1-D motion:

Equation	Missing Variable
$v = v_0 + at$	$\Delta x$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$v$
$v^2 = v_0^2 + 2a \Delta x$	$t$
$\Delta x = \frac{1}{2}(v_0 + v)t$	$a$
$x = x_0 + v t - \frac{1}{2} a t^2$	$v_0$

Variables: $x$ = position, $x_0$ = initial position, $v$ = final velocity, $v_0$ = initial velocity, $a$ = acceleration, $t$ = time.

Solving Kinematics Problems: Holt's Kinematic Algorithm (HoKA)

• Choose a coordinate system.

• Make a table of the variables: $x$, $x_0$, $v$, $v_0$, $a$, and $t$.

• Identify all known quantities; typically, there are at least two unknowns (one needed, one not needed).

• Use the equation that does not contain the unwanted variable to solve for the desired quantity.

Worked Examples

Example 1: Freefall

• Problem: Noodle the cat fell off his cat tower and landed with a speed of 24 m/s.

  • (a) How far up was he when he fell?

  • (b) For how long was he falling?

• Solution: Use kinematic equations with $v_0 = 0$, $v = 24$ m/s, $a = g = 9.8$ m/s2 (downward).

  • Distance: $x = \frac{v^2}{2g} = \frac{(24)^2}{2 \times 9.8} \approx 29.4$ m

  • Time: $t = \frac{v}{g} = \frac{24}{9.8} \approx 2.4$ s

Example 2: Maximum Height

• Problem: A small rock is thrown straight up. Neglect air resistance. When the rock is at its maximum height, its velocity is zero. What is its acceleration at this instant?

• Solution: The acceleration is $-g = -9.8$ m/s2 (downward), even at the instant when velocity is zero.

Example 3: Uniform Deceleration

• Problem: A truck covers 40.0 m in 8.50 s while uniformly slowing down to a final velocity of 2.80 m/s.

  • (a) Find the truck's original speed.

  • (b) Find its acceleration.

• Solution:

  • Original speed: $v_0 = 6.6$ m/s

  • Acceleration: $a = -0.45$ m/s2

Example 4: Two Cars on a Line

• Problem: A red car and a green car move toward each other along an x-axis. At time $t = 0$, the red car is at $x_r = 0$ and the green car is at $x_g = 220$ m. If the red car has a constant velocity of 20 km/h, the cars pass each other at $x = 44.5$ m. But if the red car has a constant velocity of 40 km/h, they pass each other at $x = 76.6$ m. What are (a) the initial velocity and (b) the constant acceleration of the green car?

  • Initial velocity of green car: $v_{gp} = -13.9$ m/s

  • Constant acceleration of green car: $a_g = -2.0$ m/s2

Summary Table: Kinematic Equations

Equation	Missing Variable
$v = v_0 + at$	$\Delta x$
$x = x_0 + v_0 t + \frac{1}{2} a t^2$	$v$
$v^2 = v_0^2 + 2a \Delta x$	$t$
$\Delta x = \frac{1}{2}(v_0 + v)t$	$a$
$x = x_0 + v t - \frac{1}{2} a t^2$	$v_0$

Additional info: The notes are based on lecture slides and include both conceptual explanations and worked examples, suitable for introductory college physics students studying kinematics in one dimension.