Study Notes: Gravity, Free Fall, and Terminal Velocity Review

Aristotle's Perspective (50 BC): * Aristotle posited that heavier objects fall faster than lighter objects. * He believed that this falling occurred at a constant speed.
Galileo's Perspective (1525 AD): * Galileo challenged the Aristotelian view, suggesting that all objects, regardless of their mass, fall at the same rate of uniform acceleration. * Experimental results using ticker tape analysis have historically supported Galileo's view of uniform acceleration over Aristotle's view of uniform motion (constant speed).
Comparison of Views: * Aristotle: Heavier objects = Faster fall; Motion = Constant speed. * Galileo: Different masses dropped from the same height reach the ground at the same time; Motion = Uniform acceleration.

Definition of Air Resistance (Drag Force): * All objects moving through a fluid, such as air, experience resistance. * Air possesses mass; when an object encounters enough air mass, its fall is slowed.
Relationship with Velocity: * The faster an object moves through a fluid, the more resistance or drag ( $F_D$ ) it experiences.
Impact of Surface Area and Pathing: * Objects with larger surface areas typically follow very indirect routes to the ground rather than a straight path. * Increasing an object's surface area generally increases the drag forces acting upon it, which in turn decreases the object's terminal velocity.
Aerodynamics and Fluid Density: * Improving an object's aerodynamics decreases its drag coefficient, thereby increasing its terminal velocity. * The density of the fluid the object is traveling through also influences its maximum velocity.

Mass and Attraction: * More mass results in a greater pulling force downward. * Relationship between mass ( $m$ ) and gravitational force ( $F_{grav}$ ): * For a mass of $1.0\,kg$ , $F_{grav} = 10\,N$ . * For a mass of $1000\,kg$ , $F_{grav} = 10000\,N$ .
Inertia: * Inertia is defined as the ability of an object to resist changes in its motion. * While heavier objects experience a greater force of gravity, they also possess more mass and thus more inertia, which is the resistance to being moved.

Mechanism of Acceleration Decrease: * As a person or object falls longer, they fall faster. * Increased speed results in an increased drag force ( $F_D$ ). * As the drag force increases, the rate of acceleration of the object decreases.
Definition of Terminal Velocity: * Terminal velocity is the maximum velocity an object reaches when its weight due to gravity ( $F_g$ ) is exactly equal to the drag force ( $F_D$ ). * At this equilibrium point ( $F_g = F_D$ ), the object stops accelerating and falls at a constant maximum velocity.
Example (Human Fall): * At approximately $t = 12.0\,s$ , a falling human reaches terminal velocity at roughly $200\,km/h$ .

Definition of Free Fall: * An object is in free fall when the only force acting upon it is gravity.
Falling in a Vacuum: * If air is removed to create a vacuum, there is no air resistance. * In a vacuum, all objects, regardless of mass, shape, or size, will fall at the exactly same rate.
Mathematical Proof of Mass Independence ( $g$ ): * The force of gravity ( $F_g$ ) equals the Universal Gravitational Force ( $F_G$ ). * Setting the equations equal: $m g = \frac{G M m}{d^2}$ . * The mass of the falling object ( $m$ ) appears on both sides of the equation and cancels out. * Result: $g = \frac{G M}{d^2}$ . * Conclusion: The acceleration due to gravity ( $g$ ) is independent of the mass of the falling object.

Dual Meaning of "g": * $g$ represents both the Gravitational Field Strength (measured in $N/kg$ , force per unit mass) and the Acceleration Due to Gravity (measured in $m/s^2$ ).
Variations by Location on Earth: * The value of $g$ varies based on geographic location: * Equator (sea level): $9.7805\,m/s^2$ * Mount Everest ( $8.8\,km$ peak): $9.7647\,m/s^2$ * Toronto: $9.8049\,m/s^2$ * North Pole (sea level): $9.8322\,m/s^2$
Standard Class Convention: * For the purposes of this course, for objects near Earth, $g = 9.81\,m/s^2\, \text{[Down]}$ . * Because gravity is constant near Earth's surface, the 5 uniform acceleration equations from kinematics are applicable.
Kinematics Calculation (Experiment Results): * To find acceleration ( $a$ ), use the formula: $\Delta d = v_i \Delta t + \frac{1}{2} a \Delta t^2$ . * Variables commonly used: $v_i = 0$ , $\Delta d = 4.90\,m$ , time ( $\Delta t$ ) based on average experimental measurement.