Free Fall Motion – Study Notes

Objectives

• Describe the concept of free-fall motion.
• Appreciate the pervasive role of gravity in everyday life and cultivate curiosity about natural free-fall phenomena.
• Apply the standard kinematic equations to solve quantitative free-fall problems.

Drill: “Gravity Alone?” (Concept Check)

• 1. Rock dropped from a cliff → YES (solely gravity).
• 2. Bird flying in the sky → NO (wing‐generated lift & drag).
• 3. Feather in a vacuum → YES (air resistance absent).
• 4. Book falling from a table → YES (air resistance negligible if ignored).
• 5. Paper airplane → NO (aerodynamic lift & drag).

Core Concept: Free-Fall Motion

• An object is in free fall when gravity is the only force acting on it.
  • Air drag, wind, or any propulsion must be ignored/absent.
• On Earth the (near-surface) gravitational acceleration is
  $g = -9.8\;\text{m/s}^2$
  (negative sign ⇒ downward direction in a standard upward-positive coordinate system).
• All objects, regardless of mass or composition, share the same acceleration in free fall (Galileo’s insight; validated by Apollo 15 hammer–feather experiment on the Moon).

Everyday & Philosophical Relevance

• Explains why raindrops, falling apples, thrown balls, stones from cliffs, etc. accelerate downward.
• Underpins safety equipment design (airbags, helmets) and architectural codes (e.g., calculating impact speeds from rooftops).
• Sparks philosophical reflection on universality of natural laws and equality of inertial & gravitational mass (foundation of Einstein’s equivalence principle).

Kinematic Framework (for 1-D Vertical Motion)

• We reuse the constant-acceleration formulas, substituting $a$ with $g$.
• Sign convention: “Up” is positive; hence $a=g=-9.8\;\text{m/s}^2$.

Distance Fallen (Time Known)

• Formula: $d = v_i t + \frac{1}{2} a t^2$
  • Useful when release velocity $v_i$ and elapsed time $t$ are given.
  • If object is dropped (from rest): $v_i = 0$ → $d = \tfrac{1}{2} a t^2$.

Distance (Time Unknown)

• When final velocity is known instead of time: $v<em>f^2 = v</em>i^2 + 2 a d$
  • Rearrange to solve for $d$: $d = \dfrac{v<em>f^2 - v</em>i^2}{2a}$.

Time of Fall (Velocity Route)

• If initial & final velocities are known:
  $v<em>f = v</em>i + a t$ → $t = \dfrac{v<em>f - v</em>i}{a}$.

Distance in Terms of Average Velocity

• Combining definitions: $d = \frac{v<em>i + v</em>f}{2} \, t$
  • Highlights that displacement equals average velocity × time under constant acceleration.

Worked Examples

Example 1 – Vase Thrown Upward

• Given: $v<em>i = 26.2\,\text{m/s},\; v</em>f = 0,\; a = -9.8\,\text{m/s}^2$.
• Target: Maximum height.
• Use $v<em>f^2 = v</em>i^2 + 2 a d$ ⇒
  $0 = (26.2)^2 + 2(-9.8)d$ ⇒
  $d = \frac{- (26.2)^2}{2(-9.8)} = 35.02\,\text{m} \approx 35\,\text{m}.$
• Physical insight: upward motion slows until velocity zero, then reverses.

Example 2 – Shingles Dropped from Roof

• Given: $d = -8.52\,\text{m},\; v_i = 0,\; a = -9.8\,\text{m/s}^2$.
• Solve $d = \frac{1}{2} a t^2$ ⇒
  $t = \sqrt{\frac{2|d|}{|a|}} = \sqrt{\frac{2\times 8.52}{9.8}} = 1.32\,\text{s}.$
• Takeaway: drop time depends only on height (for constant $g$).

Example 3 – Downward Launch, Find $v_f$

• Given: $v_i = -2\,\text{m/s},\; d = -45\,\text{m},\; a = -9.8\,\text{m/s}^2$.
• Apply $v<em>f^2 = v</em>i^2 + 2 a d$:
  $v<em>f^2 = (-2)^2 + 2(-9.8)(-45) = 4 + 882 = 886,$$v</em>f = -\sqrt{886} \approx -29.7\,\text{m/s}.$
• Negative sign ⇒ downward direction.

Example 4 – Drop from 20 m

• Step A (Time): $t = \sqrt{\frac{2\times 20}{9.8}} = 2.02\,\text{s}.$
• Step B (Final Velocity): $v_f = 0 + (-9.8)(2.02) = -19.8\,\text{m/s}.$
• Interpretation: impact speed ≈ $20\,\text{m/s}$ (~72 km/h).

Practice Problems (Activity #2)

• 1. Ball thrown straight up with $v_i = 18\,\text{m/s}$. Question: Total time in the air?
  • Hint: Time up = Time down; top occurs when $v=0$.
• 2. Ball dropped from $20\,\text{m}$ building. Question: Impact speed?
  • Use Example 4 or $v_f^2 = 2 g d$.
• 3. Ball thrown upward $v<em>i = 18\,\text{m/s}$. a) Highest height? b) Time to reach that height? (use $v</em>f=0$ at peak).

Quick Reference: Key Equations

• Displacement: $d = v_i t + \frac{1}{2} a t^2$.
• Final velocity: $v<em>f = v</em>i + a t$.
• Velocity–displacement link: $v<em>f^2 = v</em>i^2 + 2 a d$.
• Average velocity form: $d = \frac{v<em>i + v</em>f}{2} t$.

Practical Safety Implications

• Engineering of guardrails, balcony heights, and sports safety nets depends on maximum potential energy $m g h$ & impact velocity $\sqrt{2 g h}$.
• Understanding fall times crucial for timing parachute deployment, drone drops, and fair-ground rides.

Conceptual Connections

• Builds on introductory constant-acceleration kinematics (lecture 1).
• Prepares ground for later topics: projectile motion (adds horizontal component), Newton’s laws (forces), and energy conservation (potential ↔ kinetic).

Summary

• Free fall = motion under gravity alone.
• All objects share $g$, hence motion is mass-independent.
• Four kinematic formulas suffice to analyze time, displacement, and velocity.
• Sign convention clarity prevents common sign-errors.
• Real-world design, safety, and scientific inquiry hinge on these principles.