2D Problems and Projectile Motion Study Guide

Definition of a Projectile: In the simplest mechanical terms, an object is classified as a projectile only if the following two conditions are met simultaneously: * Sole Influence of Gravity: The only force acting upon the object is gravity. This state is technically defined as "free fall." * Dual Velocity Components: The object must possess both a horizontal velocity component ( $v_{x}$ ) and a vertical velocity component ( $v_{y}$ ).
Non-Examples: Objects that do not meet both criteria—such as those with an engine providing constant thrust or those moving strictly in a 1D vertical or horizontal path—are not considered projectiles in this context.

Vector Directionality: Vectors are defined by both magnitude and direction. Consequently, the specific direction in which a vector acts determines how it is applied to a physical system.
The Principle of Independence: Motion in the horizontal plane ( $x$ ) is independent of motion in the vertical plane ( $y$ ). * Gravity's Role: Gravity acts exclusively in the vertical plane (directed [down]). Therefore, gravity only affects the vertical component of an object's motion. * Horizontal Motion: Because gravity does not act horizontally, there is no acceleration in the horizontal direction (provided air resistance is neglected). The horizontal velocity ( $v_{x}$ ) remains constant. * The "Vegas" Rule: A common mnemonic for this principle is "What happens in X stays in X, and what happens in Y stays in Y."
Time as a Scalar Bridge: While velocity, displacement, and acceleration are vectors tied to specific directions, time ( $t$ ) is a scalar quantity. It remains the same value regardless of the direction an object is moving and serves as the mathematical link connecting the horizontal and vertical equations of motion.

Key observations regarding the behavior of projectiles include: * Vertical Acceleration: Projectiles undergo constant acceleration ( $a_{y} = -9.81\,m/s^2$ ) due to gravity, causing the vertical velocity to change over time. * Horizontal Uniformity: In the absence of external horizontal forces, the projectile maintains a uniform horizontal velocity ( $v_{x}$ ). * Path Shape: The combination of constant horizontal motion and accelerated vertical motion results in a parabolic trajectory.

Scenario Description: A driver in a convertible (top down) is entering an underground parking garage. Window washers on a platform above the entrance are moving upwards. A cleaner knocks a pail of water off the platform. The goal is to determine if the pail will land on the driver based on the given kinematics.
Given Data (Vertical Motion of the Pail): * Initial Vertical Velocity ( $v_{iy}$ ): $1.50\,m/s$ [up] (The pail shares the initial velocity of the moving platform). * Vertical Displacement ( $d_{y}$ ): $-9.00\,m$ (The distance down to the driver). * Vertical Acceleration ( $a_{y}$ ): $-9.81\,m/s^2$ (Acceleration due to gravity).
Given Data (Horizontal Motion of the Car): * Horizontal Velocity ( $v_{x}$ ): $24.0\,km/h$ (The speed limit of the car). To convert to standard units: $\frac{24.0\,km/h}{3.6} = 6.666...\,m/s$ . * Horizontal Distance ( $d_{x}$ ): $12.0\,m$ (Initial distance from the car to the platform at the moment the pail is dropped).
Problem-Solving Approach: * One must first calculate the time ( $t$ ) it takes for the pail to fall the vertical distance of $9.00\,m$ , accounting for its initial upward velocity. * Once time is determined, calculate how far the car travels horizontally in that same duration using the formula $d_{x} = v_{x} \times t$ . * If the car's traveled distance equals the initial distance of $12.0\,m$ , the pail will hit the driver.