Lernzettel: One-Dimensional Motion Fundamentals

Chapter 2: Motion in One Dimension — Revision Sheet

1. 📌 Essentials

• Motion along a straight line described by position, velocity, and acceleration.
• Displacement ($\Delta x$): change in position, a vector quantity.
• Distance: total path length traveled, scalar.
• Average velocity: $\displaystyle v_{avg} = \frac{\Delta x}{\Delta t}$.
• Instantaneous velocity: $\displaystyle v(t) = \frac{dx}{dt}$.
• Average acceleration: $\displaystyle a_{avg} = \frac{\Delta v}{\Delta t}$.
• Instantaneous acceleration: $\displaystyle a(t) = \frac{dv}{dt}$.
• Equations of motion (constant acceleration):
  • $\displaystyle v = v_i + a t$
  • $\displaystyle x = x_i + v_i t + \frac{1}{2} a t^2$
  • $\displaystyle v^2 = v_i^2 + 2 a (x - x_i)$
• Free fall: acceleration due to gravity, $a = -g = -9.80\, m/s^2$.
• At maximum height, vertical velocity is zero.
• Symmetry in projectile motion: ascent time = descent time.

2. 🧩 Key Structures & Components

• Position ($x(t)$): Location along the line at time $t$.
• Displacement ($\Delta x$): Final minus initial position.
• Velocity ($v(t)$): Rate of change of position.
• Acceleration ($a(t)$): Rate of change of velocity.
• Equations of motion: Derived for uniform acceleration.
• Gravity ($g$): Constant downward acceleration in free fall.
• Projectile components:
  • Vertical: affected by gravity.
  • Horizontal: constant if no air resistance.
• Graphs:
  • $x(t)$: position over time.
  • $v(t)$: velocity over time.
  • $a(t)$: acceleration over time.

3. 🔬 Functions, Mechanisms & Relationships

• Position ($x(t)$): Fully determines motion; known function or data.
• Displacement ($\Delta x$): $x_f - x_i$, includes direction.
• Velocity ($v(t)$): Slope of $x(t)$; instantaneous rate.
• Acceleration ($a(t)$): Slope of $v(t)$; change in velocity.
• Constant acceleration equations: Link initial and final states over time.
• Gravity: Constant downward acceleration; affects vertical motion.
• Projectile motion: Vertical and horizontal components act independently.
• Symmetry: Time to max height equals time descending.
• Velocity at max height: Zero in vertical component.
• Impact velocity: Same magnitude as initial if thrown upward, but opposite in direction.

4. 📊 Comparative Table

Item	Key Features	Notes / Differences
Displacement ($\Delta x$)	Change in position, vector	Depends on initial and final points
Distance	Total path length traveled	Scalar, may differ from $\Delta x$
Average velocity	$\displaystyle v_{avg} = \frac{\Delta x}{\Delta t}$	Vector, depends on start/end points
Instantaneous velocity	$\displaystyle v(t) = \frac{dx}{dt}$	Slope of $x(t)$ graph
Average acceleration	$\displaystyle a_{avg} = \frac{\Delta v}{\Delta t}$	Change in velocity over time
Instantaneous acceleration	$\displaystyle a(t) = \frac{dv}{dt}$	Slope of $v(t)$ graph
Equations of motion	$v = v_i + a t$, $x = x_i + v_i t + \frac{1}{2} a t^2$, $v^2 = v_i^2 + 2 a (x - x_i)$	For constant $a$
Free fall acceleration	$a = -g = -9.80\, m/s^2$	Downward, constant
Max height	$y_{max} = y_i + \frac{v_{i,y}^2}{2g}$	Vertical component only
Time to max height	$t_{max} = \frac{v_{i,y}}{g}$	Vertical component only

5. 🗂️ Hierarchical Diagram

Motion in One Dimension
 ├─ Position ($x(t)$)
 │    ├─ Known function or data
 ├─ Displacement ($\Delta x$)
 │    └─ $x_f - x_i$
 ├─ Velocity ($v(t)$)
 │    ├─ dx/dt
 │    └─ Slope of $x(t)$ graph
 ├─ Acceleration ($a(t)$)
 │    ├─ dv/dt
 │    └─ Slope of $v(t)$ graph
 ├─ Equations of motion (constant $a$)
 │    ├─ $v = v_i + a t$
 │    ├─ $x = x_i + v_i t + \frac{1}{2} a t^2$
 │    └─ $v^2 = v_i^2 + 2 a (x - x_i)$
 └─ Free fall
      └─ $a = -g = -9.80\, m/s^2$

6. ⚠️ High-Yield Pitfalls & Confusions

• Confusing displacement ($\Delta x$) with total distance traveled.
• Assuming velocity is zero at any point without checking the context.
• Misapplying equations outside constant acceleration conditions.
• Forgetting that gravity acts downward ($a = -g$).
• Overlooking the symmetry in projectile motion when calculating times.
• Mixing up average and instantaneous quantities.
• Ignoring the vector nature of displacement and velocity.
• Assuming horizontal motion is affected by gravity (it's not).
• Mistaking the maximum height formula for horizontal motion.
• Neglecting initial velocity components in projectile motion.
• Confusing free fall with other types of uniformly accelerated motion.

7. ✅ Final Exam Checklist

• Define and distinguish position, displacement, distance.
• Calculate average velocity and acceleration.
• Derive and apply equations of motion for constant acceleration.
• Understand the difference between average and instantaneous quantities.
• Recognize the significance of gravity in free fall and projectile motion.
• Find maximum height and time to reach it in vertical motion.
• Analyze motion graphs: $x(t)$, $v(t)$, $a(t)$.
• Solve problems involving initial velocity, acceleration, and displacement.
• Identify symmetry in projectile motion.
• Know that horizontal velocity remains constant in projectile motion (no air resistance).
• Apply the equations correctly within their valid conditions.
• Visualize motion with hierarchical diagrams.
• Avoid common pitfalls related to vector quantities and assumptions.

End of Revision Sheet