General Physics 1: Uniformly Accelerated Motion and Free-fall

Module Information and Team

• Subject: General Physics 1
• Level: Grade 12, Quarter 1, Module 6
• Topic: Uniformly Accelerated Motion and Free-fall
• Publisher: Department of Education Division of Pasig City
• Development Team of the Self-Learning Module:
  • Writer: Aldeguer P. Realingo
  • Editor: Melvina S. Tarcena
  • Reviewer: Melvina S. Tarcena
  • Illustrator: Edison P. Clet
  • Layout Artist: Mark Kihm G. Lara
  • Management Team:
    • Ma. Evalou Concepcion A. Agustin (OIC-Schools Division Superintendent)
    • Aurelio G. Alfonso EdD (OIC-Assistant Schools Division Superintendent)
    • Victor M. Javeña EdD (Chief, SGOD and OIC-Chief, CID)
    • Librada L. Agon EdD (Education Program Supervisor - EPP/TLE/TVL/TVE)
    • Liza A. Alvarez (Education Program Supervisor - Science/STEM/SSP)
    • Bernard R. Balitao (Education Program Supervisor - AP/HUMSS)
    • Joselito E. Calios (Education Program Supervisor - English/SPFL/GAS)
    • Norlyn D. Conde EdD (Education Program Supervisor - MAPEH/SPA/SPS/HOPE/A&D/Sports)
    • Wilma Q. Del Rosario (Education Program Supervisor - LRMS/ADM)
    • Ma. Teresita E. Herrera EdD (Education Program Supervisor - Filipino/GAS/Piling Larang)
    • Perlita M. Ignacio PhD (Education Program Supervisor - EsP)
    • Dulce O. Santos PhD (Education Program Supervisor - Kindergarten/MTB-MLE)
    • Teresita P. Tagulao EdD (Education Program Supervisor - Mathematics/ABM)

Expectations and Learning Competencies

• The module aims to enable students to solve for unknown quantities in equations involving one-dimensional uniformly accelerated motion and free-fall motion.
• By the end of the module, students are expected to:
  1. Describe one-dimensional uniformly accelerated motion.
  2. Solve for unknown quantities in equations involving free-fall and uniformly accelerated motion.
  3. Appreciate the importance of understanding uniformly accelerated motion in daily human activities.

Review of Acceleration Concepts

• Definitional Terms:
  • Acceleration: The rate of change of velocity.
  • Formula: $a = \frac{V_2 - V_1}{t_2 - t_1}$
  • Deceleration: Negative acceleration (slowing down).
  • Average Acceleration: Total velocity per total elapsed time.
  • Formula: $a = \frac{V_i + V_f}{t_{total}}$
  • Instantaneous Acceleration: Acceleration at any specific instant of time.
  • Uniform Acceleration: A constant rate of change of velocity.

Uniformly Accelerated Motion (UAM)

• Description: Uniformly Accelerated Motion (UAM) occurs when an object has a constant acceleration. This means the velocity of the object changes by equal amounts in equal time intervals.
• Example Motion of a Car:
  • At $t = 0s$, Velocity = $0m/s$
  • At $t = 1s$, Velocity = $+5m/s$
  • At $t = 2s$, Velocity = $+10m/s$
  • At $t = 3s$, Velocity = $+15m/s$
  • At $t = 4s$, Velocity = $+20m/s$
  • Observation: The velocity increases by a constant amount of $+5m/s$ every second. The acceleration is constant.
• Graphical Representation: A velocity vs. time graph for UAM forms a straight line upward slant to the right.
• Kinematic Equations for UAM (Horizontal Line Motion):
  1. $d = \frac{V_i + V_f}{2} \times t$
  2. $V_f = V_i + at$
  3. $d = V_it + \frac{1}{2}at^2$
  4. $V_f^2 = V_i + 2ad$
• Variable Definitions:
  • $d = \text{position/distance}$
  • $V_i = \text{initial velocity}$
  • $V_f = \text{final velocity}$
  • $t = \text{time}$
  • $a = \text{acceleration}$

Free-fall Motion

• Nature of Free-fall: Free-fall is a specific type of uniformly accelerated motion where objects move solely under the influence of gravity in the absence of air resistance.
• Acceleration due to Gravity ($g$):
  • The constant for all free-falling bodies is $g = 9.8m/s^2$.
  • This acceleration is independent of the mass or weight of the falling objects.
• Scientific Contributions:
  • Galileo Galilei: Hypothesized that in the absence of air resistance and friction, objects fall at the same rate regardless of mass. He measured speeds using metal balls on inclined planes timed with a water clock.
  • Christian Huygens: Invented the pendulum clock (1656) and was the first to calculate $g$ using a pendulum's swing, a ruler, and a timepiece.
• Directional Effects:
  • $g$ decreases with increasing altitude.
  • Moving Upward: Velocity decreases at the rate of $9.8m/s$.
  • Moving Downward: Velocity increases at the rate of $-9.8m/s$ (the negative sign indicates the downward direction).
• Kinematic Equations for Free-fall ($a = g = -9.8m/s^2$):
  1. $y = \frac{V_i + V_f}{2} \times t$
  2. $V_f = V_i - gt$
  3. $y = V_it - \frac{1}{2}gt^2$
  4. $V_f^2 = V_i - 2gy$
• Variable Definitions:
  • $y = \text{vertical position}$
  • $V_i = \text{initial velocity}$
  • $V_f = \text{final velocity}$
  • $t = \text{time}$
  • $g = \text{acceleration due to gravity}$

Mathematical Problem Solving Examples

• Example 1: Horizontal Uniformly Accelerated Motion

  • Scenario: A car starting from rest undergoes UAM, reaching a velocity of $30.0m/s$ after traveling $80.0m$. Find the acceleration.
  • Given: $V_i = 0$, $V_f = 30.0m/s$, $d = 80.0m$
  • Formula Selection: $V_f^2 = V_i + 2ad$
  • Substitution: $(30m/s)^2 = (0)^2 + 2a(80.0m)$
  • Calculation: $900m^2/s^2 = 160m(a)$
  • Refining: $a = \frac{900m^2/s^2}{160m}$
  • Final Answer: $a = 5.6m/s^2$

• Example 2: Vertical Free-fall Motion

  • Scenario: A ball thrown vertically upward returns to its starting point in $4s$. Find its initial velocity.
  • Given: $t = 4s$, $a = -g = -9.8m/s^2$ (downward)
  • Formula Selection: $y = V_it - \frac{1}{2}gt^2$
  • Substitution (displacement $y = 0$ for a full trip): $0 = V_i(4s) - \frac{1}{2}(9.8m/s^2)(4s)^2$
  • Calculation: $0 = V_i(4s) - 78.4m$
  • Refining: $V_i(4s) = 78.4m$
  • Final Answer: $V_i = 19.6m/s$

Student Activities and Applications

• Practical Case Study: Food Relief during COVID-19

  • Scenario: Food relief dropped from a helicopter.
  • Comparison: Dropping from height $8m$ vs height $15m$.
  • Logic: At each second of fall, speed increases by approximately $9.8m/s$. An object from $15m$ height gains more speed because it is in the air longer. Therefore, it is safer to catch relief dropped from the lower level ($8m$).

• Activity 1: DIY Coin Toss Challenge

  • Task: Vertically toss a 5-peso coin.
  • Key Points for Labeling:
    1. Point A (lowest position)
    2. Point B (highest position)
    3. Final velocity ($V_f$)
    4. Velocity going upward ($v$)
    5. Velocity going downward ($v$)
    6. Acceleration at the highest point
    7. Acceleration before reaching the ground

• Activity 2: Horizontal & Vertical Motion Tracking

  • Horizontal (Concrete Deceleration):
    • Rate on dry concrete: $7.00m/s^2$
    • Rate on wet concrete: $5.00m/s^2$
    • Task: Find stopping distance from $30.0m/s$. Calculate time elapsed on wet concrete for $100m$.
  • Vertical (Ejected Ball):
    • Initial velocity: $4m/s$; Hits ground after $0.872s$.
    • Task: Find acceleration and velocity at the highest point, and velocity magnitude/direction upon ground impact.

• Activity 3: Advanced Problem Solving

  1. Travel time for a bus accelerating from $40km/hr$ to $80km/hr$ over $30km$.
  2. Acceleration of a car speeding from $5m/s$ to $9m/s$ in $20s$.
  3. Finding height of a branch from which a mango falls if its final velocity is $8m/s$.
  4. Launch time for a rocket accelerating at $20m/s^2$ to reach $400m/s$.
  5. Acceleration of a person starting from rest to catch a bicycle (constant $10m/s$) in $30s$.

• Valuing: Safety Protocols

  • The "tail-gating phenomenon" is a danger related to UAM.
  • Application: Following IATF social distancing (1-meter protocol) context for motorists and cyclists to prevent accidents.

Questions and Discussion

Pretest Questions:

• Q1: Best description of UAM? Answer: Object moving with constant acceleration.
• Q2: Distance traveled by a satellite released from rest after falling freely for $5s$? Answer: $122.5m$ ($d = 0.5 \times 9.8 \times 5^2$).
• Q3: Time to reach max height point B is $1.0s$, find vertical distance A to B? Answer: $4.9m$.
• Q4: Velocity when the ball falls and hits ground at point A? Answer: $9.8m/s$, downward.
• Q5: Activity where uniform acceleration is observed? Answer: Driving a car to accelerate at $1m/s$ when light turns green, then braking to stop.

Posttest Questions:

• Q1: Statement describing uniformly accelerated? Answer: Free-fall is a UAM in which only gravity affects motion.
• Q2: Rock dropped for $1.0s$, velocity? Answer: $-9.8m/s$ (downward).
• Q3: Coin tossed up at $+5m/s$, highest point reached? Answer: $1.28m$.
• Q4: Time for tennis ball to reach highest point (using $5m/s$ initial)? Answer: $0.50s$ (approx. $5 / 9.8$).
• Q5: Importance of understanding UAM? Answer: Includes avoiding tail-gating, analyzing body pursuit, and rocket launching (the exception would be simple car racing competition if it is not specifically about analyze/safe metrics).

References

• Bernido, C. C. and Bernido, M. V. C. (2008). Physics Essentials Portfolio.
• De Luna, M. J. M. et al. (2012). Exploring Science and Technology: Physics.
• Hewitt, Paul G. (2002). Conceptual Physics. 9th edition.
• Santos, G. N. C. and Ocampo, J. P. (2003). e-Physics: The Next Generation.
• Silverio, A. A. (2017). Exploring Life Through Science Series: General Physics 1.
• Padua, A. L. and Crisostomo, R. M. Exploring Practical and Explorational Physics: Modular Approach. 2nd edition.
• University Physics (OpenStax): Motion with Constant Acceleration.
• Norwell Schools: Free-fall motion documents.