Physics Lesson 3: Two-Dimensional and Projectile Motion Study Guide

Conceptual Foundations: Lesson three of the "Physics by Diana" series (also referred to as AP Physics 102) focuses on motion in two dimensions. This is commonly known as projectile motion or ballistic motion.
Central Principle: The fundamental concept is that horizontal motion (back and forth) and vertical motion (up and down) are independent of one another.
Key Tool for Analysis: Because these motions are independent, they can be analyzed separately using distinct horizontal and vertical components while remaining coordinated through a single parameter: time ( $t$ ).
Initial Example: Skateboarder on a Flatbed Truck: A skateboarder performing a trick (e.g., a kickflip) on the back of a moving truck appears counter-intuitive, but it is possible because the skateboarder and the board retain the horizontal velocity of the truck while performing a vertical jump.

The Scenario: In one hand, a gun is held perfectly level and fires a hair tie horizontally. At the exact same moment, an identical hair tie is dropped from the same height.
The Assumption: Air resistance is neglected for all calculations and scenarios in this lesson.
The Question: Which object hits the ground first?
The Answer: Both hair ties hit the ground at exactly the same time.
Explanation: Horizontal motion has no effect on the vertical acceleration due to gravity ( $g$ ). Since both objects share the same initial vertical velocity ( $v_{iy} = 0$ ) and the same downward acceleration ( $g = 9.81\,m/s^2$ ), their vertical displacement over time is identical.
Acoustic Verification: This can be demonstrated by placing one penny on the edge of a table and flicking a second penny from beside it; the sound of them hitting the floor simultaneously confirms they fall at the same rate.

Problem Description: A diver at a cliff named Shipwrecks must jump out sideways to clear rocks located at the base of the cliff.
Parameters:
- Cliff Height: $10\,m$ .
- Rock Extension: Rocks extend $1\,m$ out from the cliff base.
- Initial Horizontal Velocity ( $v_x$ ): $2\,m/s$ .
- Initial Vertical Velocity ( $v_y$ ): $0\,m/s$ (the diver jumps straight sideways).
- Acceleration due to Gravity ( $g$ ): Rounded to $10\,m/s^2$ for simplicity.
Vertical Analysis (Calculating Fall Time):
- Equation: $d = \frac{1}{2}at^2$
- Calculation: $10 = \frac{1}{2}(10)t^2 \rightarrow 10 = 5t^2 \rightarrow t^2 = 2 \rightarrow t = \sqrt{2} \approx 1.4\,s$
Horizontal Analysis (Calculating Landing Distance):
- Equation: $\Delta x = v_x \times t$
- Calculation: $\Delta x = 2\,m/s \times 1.4\,s = 2.8\,m$
Conclusion: Since the landing distance ( $2.8\,m$ ) is greater than the rock extension ( $1\,m$ ), the diver safely clears the rocks.

Drone Perspective (Front View): When viewed from far away in front, the sideways motion is not apparent, and the object appears to fall in a straight vertical line.
Overhead Perspective (Top View): When viewed from directly above, the downward fall is not apparent, and the object appears to move in a straight horizontal line at a constant velocity.
The Combined Path: The actual path of an object undergoing projectile motion is a parabola. This is observed in everyday life, such as pouring coffee out of a cup with initial horizontal velocity.
Wile E. Coyote Fallacy: The cartoon depiction where an object moves horizontally in a straight line and only starts falling after realizing there is no support is "fake news" (unphysical).

Definition of Projectile Motion: An object freely moving where the only influence is the downward acceleration of gravity ( $g = 10\,m/s^2$ ).
Problem Scenario: A player kicks a purple soccer ball off the ground.
Initial Conditions:
- Initial Velocity ( $v$ ): $20\,m/s$ .
- Launch Angle ( $\theta$ ): $37^{\circ}$ off the horizontal.
Vector Decomposition:
- Horizontal Component ( $v_x$ ): $v \cos(\theta) = 20 \cos(37^{\circ}) \approx 16\,m/s$ .
- Vertical Component ( $v_y$ ): $v \sin(\theta) = 20 \sin(37^{\circ}) \approx 12\,m/s$ .
Trajectory Characteristics:
- The horizontal velocity remains constant at $16\,m/s$ throughout the flight (neglecting air resistance).
- The vertical velocity starts at $12\,m/s$ , decreases to $0\,m/s$ at the peak, and then accelerates downward.
- The trajectory is perfectly symmetric.
Calculating Time and Distance:
- Time to reach peak ( $t_{peak}$ ): $v_y / g = 12 / 10 = 1.2\,s$ .
- Total Flight Time ( $t_{total}$ ): $1.2\,s \times 2 = 2.4\,s$ .
- Sideways Distance ( $d$ ): $d = v_x \times t_{total} = 16\,m/s \times 2.4\,s = 38.4\,m$ .
Calculating Peak Height:
- Average Vertical Velocity ( $v_{avg, y}$ ): $\frac{12 + 0}{2} = 6\,m/s$ .
- Height ( $h$ ): $v_{avg, y} \times t_{peak} = 6\,m/s \times 1.2\,s = 7.2\,m$ .

The Vomit Comet: A aircraft (zero-g plane) used to experience weightlessness.
Mechanism: The plane follows a parabolic trajectory, moving up and diving back down just like a thrown soccer ball. During this arc, passengers are in free fall relative to the plane, creating a weightless environment.
Pilot Technique: Pilots monitor a ball on a string in the cockpit. When the ball goes slack and begins to float, it indicates the plane is perfectly following the free-fall parabola.

Scenario Background: Reimagined as Ethel (daughter of a pirate) throwing eggs at Romeo's window. This is a non-symmetric projectile problem.
Variables:
- Distance to Wall: $4\,m$ .
- Initial Launch Height: $1\,m$ off the ground.
- Window Height: Bottom starts at $6\,m$ above ground.
- Initial Launch Velocity ( $v$ ): $30\,m/s$ (approximately $67.11\,mph$ ).
- Launch Angle: $60^{\circ}$ .
Decomposing Velocity:
- $v_x = 30 \cos(60^{\circ}) = 15\,m/s$ .
- $v_y = 30 \sin(60^{\circ}) \approx 26\,m/s$ .
Timeline to Impact:
- Time to hit the wall: $t = \frac{\Delta x}{v_x} = \frac{4}{15}\,s$ .
Determining Height of Impact:
- Equation of motion: $y = y_0 + v_{y}t + \frac{1}{2}at^2$
- Substitution: $y = 1 + (26)(\frac{4}{15}) + \frac{1}{2}(-10)(\frac{4}{15})^2$
- Result: $y \approx 7.5\,m$ .
Outcome: Since the egg hits at $7.5\,m$ and the window starts at $6\,m$ , the egg breaks the window (vandalism) provided the window is at least $1.5\,m$ tall.

Parametric Equations: Horizontal and vertical positions are linked by the parameter of time ( $t$ ).
- Horizontal position: $x = v_x t \rightarrow t = \frac{x}{v_x}$
- Vertical position: $y = v_y t + \frac{1}{2}at^2$
Substitution: Replace $t$ in the vertical equation with standard horizontal terms:
- $y = v_y (\frac{x}{v_x}) + \frac{1}{2}a(\frac{x}{v_x})^2$
- $y = (\frac{v_y}{v_x})x + (\frac{g}{2v_x^2})x^2$
Geometric Form: The equation takes the standard form of a parabola ( $y = bx + cx^2$ ). Since gravity ( $g$ ) is negative (pointing down), the result is an upside-down parabola.

Perpendicular Independence: Vertical acceleration does not affect perpendicular sideways motion.
Strategy for Complex Problems: When faced with non-symmetric or complex problems, identify the flight time ( $t$ ) first to link the horizontal and vertical components.