Laboratory Activity 1: Free Falling Body and Projectile Trajectory Notes

Learning Objectives for Laboratory Activity 1: Free Falling Body and Projectile Trajectory

• Verify Mass and Composition Independence: Observe and verify that the physical characteristics of an object, specifically its mass, diameter, and composition, do not influence its downward acceleration or the total time of flight when dropped within a vacuum environment.
• Analyze Two-Dimensional Motion: Demonstrate that projectile motion is composed of two independent components: horizontal motion (which maintains a constant velocity) and vertical motion (which undergoes uniform acceleration due to the force of gravity).
• Evaluate Trajectory Variables: Determine how variables such as launch angle, initial velocity, and air resistance alter key trajectory markers, including the maximum height ($h_{max}$), total horizontal range ($R$), and the overall geometric shape of the flight path.

Introduction to Free Fall and Uniform Acceleration

• Definition of Free Fall: A free-falling body is an object that moves exclusively under the influence of gravity. In this state, the object experiences zero air resistance and zero friction.
• Galilean Principles: Historically demonstrated by Galileo Galilei, the principle states that all objects dropped near the surface of the Earth accelerate downward at the exact same rate. This occurs regardless of the object's mass, shape, or weight.
• Gravitational Acceleration ($g$): Gravitational acceleration is a constant vector pointing downward. Its value on Earth is approximately:     $g = 9.81\,m/s^2$
• Kinematic Equations for Vertical Motion: When an object is dropped from an initial height ($y_0$) with no initial vertical velocity ($v_{0y} = 0$), its motion over time ($t$) is described by:     $v_y = -gt$$y = y_0 - \frac{1}{2}gt^2$

Projectile Motion as Two-Dimensional Kinematics

• Definition of a Projectile: A projectile is any object launched into space that continues its motion governed by its own inertia and the force of gravity.
• Trajectory: The path followed by a projectile is called its trajectory, which takes the geometric form of a predictable parabola.
• Independence of Dimensions: The fundamental method for analyzing projectile motion is to separate it into two perpendicular and independent dimensions:
  • Horizontal Vector ($x$-axis): In the absence of air resistance, no horizontal forces act on the projectile. Consequently, the horizontal acceleration ($a_x$) is zero, and the horizontal velocity ($v_x$) remains constant throughout the flight.         $v_{0x} = v_0 \cos(\theta)$$x = v_{0x}t$
  • Vertical Vector ($y$-axis): The vertical component behaves identically to an object in free fall, experiencing a constant downward acceleration due to gravity ($a_y = -g$).         $v_{0y} = v_0 \sin(\theta)$$y = v_{0y}t - \frac{1}{2}gt^2$
  • Variables: In these equations, $v_0$ represents the initial launch speed and $\theta$ represents the launch angle relative to the horizontal plane.

Key Trajectory Milestones and Real-World Factors

• Maximum Height ($h_{max}$): This is the apex or highest point of the trajectory. At this specific instant, the vertical velocity component momentarily reaches zero ($v_y = 0$).
• Horizontal Range ($R$): This is the total linear horizontal distance covered by the projectile before it returns to the target plane. Assuming flat ground, the range is calculated as:     $R = \frac{v_0^2 \sin(2\theta)}{g}$
• Ideal Vacuum Conditions:
  • A launch angle of $45^\circ$ provides the absolute maximum horizontal range.
  • Complementary Angles: Angles that sum to $90^\circ$ (such as $30^\circ$ and $60^\circ$) will result in the exact same horizontal range, although the shapes of their flight paths (height and time) will differ significantly.
• Air Resistance (Drag Force): In real-world environments, air exerts an opposing drag force on moving objects. This force is determined by the object's velocity, cross-sectional area, shape, and mass. Air resistance warps the ideal mathematical parabola by:
  • Slowing the object during transit.
  • Truncating (shortening) the total horizontal range.
  • Creating a steeper angle of descent compared to the angle of launch.

Laboratory Equipment and Methodology

• Required Materials:
  • Computer, smartphone, or tablet with a web browser.
  • PhET Projectile Motion HTML5 Simulation.
• Simulation Screens: The lab utilizes the "Intro", "Vectors", and "Lab" sections of the PhET simulation dashboard.

Procedural Steps for Experimentation

Part 1: Free Fall and Independence of Mass

1. Access the "Intro" screen of the PhET simulation.
2. Set the pedestal height to $15\,m$.
3. Adjust the cannon muzzle angle to $90^\circ$ (pointing straight down).
4. Set Initial Speed to $0\,m/s$ to simulate a simple drop release.
5. Ensure "Air Resistance" is unchecked.
6. Test the following objects sequentially: Pumpkin, Cannonball, Human, and Piano.
7. Use the Time/Range/Height Tool to measure the time of flight for each object and record the results in Table 1.

Part 2: The Parabolic Trajectory and Launch Angle

1. Access the "Vectors" screen.
2. Set pedestal height to $0\,m$ (ground level) and Initial Speed to $15\,m/s$.
3. Check "Velocity Vectors" and select "Components" to view vector arrows.
4. Ensure "Air Resistance" is unchecked.
5. Launch the projectile at angles: $25^\circ$, $35^\circ$, $45^\circ$, $55^\circ$, $65^\circ$, and $75^\circ$.
6. For each angle, use the measurement tool to find Max Height, Total Range, and Total Time, recording them in Table 2.

Part 3: Real-World Effects of Air Resistance

1. Access the "Lab" screen.
2. Set pedestal to $0\,m$, angle to $45^\circ$, and Initial Speed to $18\,m/s$.
3. Trial 1 (Vacuum): Uncheck air resistance using a Cannonball. Record data.
4. Trial 2 (Real-World Drag): Check air resistance. Set Mass to $17.60\,kg$ and Diameter to $0.15\,m$. Record data.
5. Trial 3 (Mass Effects with Drag): Keep air resistance and diameter ($0.15\,m$) the same, but reduce Mass to $5.00\,kg$. Record data in Table 3.

Data Tables for Observation

Table 1: Free Fall Verification (Height = $15\,m$, Angle = $-90^\circ$, Speed = $0\,m/s$)

Table 2: Projectile Dynamics vs. Angle (Height = $0\,m$, Initial Speed = $15\,m/s$)

Table 3: Effects of Drag (Height = $0\,m$, Angle = $45^\circ$, Speed = $18\,m/s$)

Questions & Analysis

• Question 1: Did changing the object's physical mass or diameter alter the final drop time in a vacuum? Use the fundamental laws of gravity to explain why or why not.
• Question 2: Which specific launch angle resulted in the longest horizontal distance traveled? Compare the data points between the $35^\circ$ and $35^\circ$ trials—what unique mathematical pattern do you observe regarding complementary angles?
• Question 3: Review your visual observations regarding the green component velocity arrows. Why does the horizontal velocity arrow stay the exact same length throughout flight, while the vertical arrow shrinks, momentarily disappears, and flips orientation?
• Question 4: Based on Table 3, detail how fluid air resistance alters the flight path of a projectile. When dealing with drag forces, why does a lighter object (Trial 3) get impacted much more aggressively than a heavier object (Trial 2) despite being the exact same size?

Reference

University of Colorado Boulder. (n.d.). Projectile motion [Online simulation]. PhET Interactive Simulations. https://phet.colorado.edu/sims/html/projectile-motion/latest/projectile-motion_all.html