Understanding Free Fall and Gravity

• Free fall is motion where gravity is the only force acting on an object.
• Objects in free fall experience constant acceleration due to gravity, denoted as 'g.'
• Near the Earth's surface, 'g' is approximately 9.8 m/s² or 32 ft/s².

Acceleration Due to Gravity

• Acceleration due to gravity (g) is the acceleration experienced by objects in free fall.
• 'g' functions as a vector quantity, possessing both magnitude and direction.
• The direction of 'g' is always downwards, towards the center of the Earth.
• The standard value of 'g' is 9.80665 m/s², but it's often approximated as 9.8 m/s² for calculations.

Factors Affecting 'g'

• Altitude: 'g' decreases with increasing altitude because the distance from the Earth's center increases.
• Latitude: 'g' varies slightly with latitude due to the Earth's rotation and non-spherical shape, being greater at the poles and smaller at the equator.
• Local Geological Features: Variations in ground density can cause local differences in 'g.'

Equations of Motion for Free Fall

• When air resistance is negligible, kinematic equations of motion describe free fall.
  • v = v₀ + gt (velocity as a function of time).
  • y = y₀ + v₀t + (1/2)gt² (position as a function of time).
  • v² = v₀² + 2g(y - y₀) (velocity as a function of position).
  • v = final velocity.
  • v₀ = initial velocity.
  • y = final position (vertical).
  • y₀ = initial position (vertical).
  • t = time.
  • g = acceleration due to gravity.

Sign Conventions

• Establish a sign convention when applying these equations.
• Upward can be positive and downward negative, or vice versa, maintaining consistency.
• If upward is positive, 'g' is typically -9.8 m/s², indicating downward acceleration.
• If downward is positive, 'g' is +9.8 m/s².

Examples of Free Fall

• Dropping an Object: An object released from rest (v₀ = 0) falls downwards with increasing speed due to 'g.'
• Throwing an Object Upwards: When an object is thrown upwards, its velocity decreases until it reaches its highest point, where its velocity is momentarily zero, then it accelerates downwards.
• Projectile Motion: Free fall applies to the vertical component of projectile motion, where an object moves both horizontally and vertically under the influence of gravity.

Air Resistance

• Air resistance (drag) affects the motion of falling objects.
• Air resistance opposes the motion and increases with the object's speed.
• Terminal Velocity: When the force of air resistance equals the force of gravity, the object reaches a constant velocity.

Implications and Applications

• Understanding free fall and 'g' is crucial in various fields.
  • Physics education and research.
  • Engineering (structural design, aerodynamics).
  • Space exploration (trajectory calculations).
  • Sports (analyzing projectile motion in ballistics).

Calculating Distance in Free Fall

• The distance an object falls in free fall can be calculated using the equation d = (1/2)gt².
  • d = distance fallen.
  • g = acceleration due to gravity (9.8 m/s²).
  • t = time in seconds.

Impact of Initial Velocity

• If an object is thrown downwards with an initial velocity, the distance it falls is greater than if it were simply dropped.
• The equation becomes d = v₀t + (1/2)gt².
  • v₀ = initial velocity.

Time to Reach the Ground

• To calculate the time it takes for an object to reach the ground when dropped from a certain height use t = √(2h/g).
  • h = height from which the object is dropped.
  • g = acceleration due to gravity (9.8 m/s²).

Free Fall in a Vacuum

• All objects fall with the same acceleration, regardless of their mass or shape, when in a vacuum that has no air resistance.
• Galileo Galilei demonstrated this by dropping objects of different masses from the Leaning Tower of Pisa.

Gravitational Constant

• g relates to the universal gravitational constant (G), the mass of the Earth (M), and the radius of the Earth (R).
• g = GM/R².
  • G ≈ 6.674 × 10⁻¹¹ N⋅m²/kg².
  • M ≈ 5.972 × 10²⁴ kg.
  • R ≈ 6.371 × 10⁶ m.

Non-Uniform Gravitational Fields

• The acceleration due to gravity is not perfectly uniform across the Earth's surface.
• Variations occur due to differences in altitude, latitude, and the density of underlying rocks.

Practical Examples

• A skydiver jumping out of an airplane experiences free fall until air resistance becomes significant, eventually reaching terminal velocity.
• A ball thrown into the air is in free fall, decelerating as it rises and accelerating as it falls, neglecting air resistance.

Potential Energy in Free Fall

• An object in free fall converts gravitational potential energy into kinetic energy.
• The potential energy (PE) is given by: PE = mgh.
  • m = mass of the object.
  • g = acceleration due to gravity.
  • h = height above a reference point.

Kinetic Energy in Free Fall

• The kinetic energy (KE) of an object in free fall increases as its velocity increases.
• The kinetic energy is given by: KE = (1/2)mv².
  • m = mass of the object.
  • v = velocity of the object.

Conservation of Energy

• The total mechanical energy (PE + KE) of an object in free fall remains constant, assuming no air resistance or other external forces.