Free Fall Motion & Physics
Master the fundamental concepts of free fall, gravity, and motion under constant acceleration!
What is Free Fall and Why is it Important?
Free fall is the motion of an object under the influence of gravity alone, with no other forces acting on it (ignoring air resistance). It’s one of the most fundamental concepts in physics and forms the basis for understanding motion under constant acceleration.
Why free fall matters: Understanding free fall is crucial for engineering, safety calculations, space exploration, and everyday applications. It helps us predict how objects move under gravity and design systems that account for gravitational effects.
Real-World Free Fall Examples:
- Skydiving: Before parachute deployment, skydivers experience near free fall for about 60 seconds
- Dropped Objects: A coin dropped from a building follows free fall motion
- Elevator Safety: Emergency brakes must account for free fall scenarios
- Space Missions: Spacecraft and astronauts experience free fall in orbit
- Sports: Basketball shots, diving, and gymnastics involve free fall components
- Construction: Dropped tools and materials follow free fall physics
Key Characteristics of Free Fall:
- Constant Acceleration: All objects fall with acceleration g = 9.81 m/s²
- Independent of Mass: Heavy and light objects fall at the same rate
- Increasing Velocity: Speed increases linearly with time
- Quadratic Distance: Distance increases with the square of time
- No Air Resistance: Ideal free fall ignores air drag effects
The Fundamental Free Fall Equations
Free fall equations are derived from the basic kinematic equations for motion under constant acceleration. These elegant formulas connect time, distance, velocity, and acceleration in predictable ways.
The Complete Free Fall Formula Set:
Distance Formula:
h = ½gt²
Where h = height, g = gravity, t = time
Time Formula:
t = √(2h/g)
Time to fall from height h
Velocity Formulas:
v = gt (from time)
v = √(2gh) (from height)
With Initial Velocity:
h = v₀t + ½gt²
v = v₀ + gt
Step-by-Step Derivation:
- Start with acceleration: a = g (constant downward)
- Velocity equation: v = v₀ + at = v₀ + gt
- Position equation: h = v₀t + ½at² = v₀t + ½gt²
- For free fall from rest: v₀ = 0, so v = gt and h = ½gt²
- Eliminate time: t = v/g, substitute: h = ½g(v/g)² = v²/(2g)
- Solve for velocity: v = √(2gh)
- Solve for time: From h = ½gt², we get t = √(2h/g)
Galileo’s Discovery: All Objects Fall Equally
Galileo’s revolutionary discovery showed that all objects fall at the same rate regardless of their mass. This contradicted the ancient belief that heavier objects fall faster and laid the foundation for modern physics.
Fall Time vs Height Analysis:
| Height | Fall Time | Final Velocity | Real-World Example | Safety Considerations |
|---|---|---|---|---|
| 1 m | 0.45 s | 4.4 m/s | Table height | Minor injury risk |
| 5 m | 1.01 s | 9.9 m/s | Single story | Serious injury risk |
| 10 m | 1.43 s | 14.0 m/s | Three stories | Life-threatening |
| 50 m | 3.19 s | 31.3 m/s | 15-story building | Terminal velocity approach |
| 100 m | 4.52 s | 44.3 m/s | 30-story building | Near terminal velocity |
| 500 m | 10.1 s | 99.0 m/s | Skyscraper height | Air resistance significant |
The Famous Leaning Tower Experiment:
- Historical Context: Galileo allegedly dropped objects from the Leaning Tower of Pisa
- Key Discovery: Objects of different masses hit the ground simultaneously
- Scientific Method: Observation contradicted accepted theory
- Modern Understanding: Mass cancels out in the acceleration equation
- Mathematical Proof: F = ma and F = mg, so a = g (independent of m)
- Real-World Verification: Apollo 15 hammer and feather experiment on the Moon
Gravity Variations and Environmental Factors
Gravitational acceleration varies slightly depending on location, altitude, and celestial body. Understanding these variations is crucial for precise calculations in engineering and scientific applications.
Gravity Values in Different Locations:
| Location | Gravity (m/s²) | Relative to Earth | Fall Time (10m) | Applications |
|---|---|---|---|---|
| Earth (standard) | 9.81 | 1.00× | 1.43 s | Most calculations |
| Moon | 1.62 | 0.17× | 3.51 s | Lunar missions |
| Mars | 3.71 | 0.38× | 2.32 s | Mars exploration |
| Jupiter | 24.79 | 2.53× | 0.90 s | Space missions |
| Earth (equator) | 9.78 | 0.997× | 1.43 s | Equatorial regions |
| Earth (poles) | 9.83 | 1.002× | 1.42 s | Polar regions |
Factors Affecting Gravitational Acceleration:
- Latitude: Earth’s rotation causes slight variations (9.78-9.83 m/s²)
- Altitude: Gravity decreases with height above sea level
- Local Geology: Dense rock formations can increase local gravity
- Earth’s Shape: Oblate spheroid shape affects gravity distribution
- Tidal Forces: Moon and Sun cause tiny gravitational variations
Air Resistance and Terminal Velocity
Air resistance opposes free fall motion and eventually limits the maximum speed an object can achieve. This maximum speed is called terminal velocity, where air resistance equals gravitational force.
Terminal Velocity Concepts:
Terminal Velocity Formula:
v_t = √(2mg/ρAC_d)
Where m = mass, g = gravity, ρ = air density, A = cross-sectional area, C_d = drag coefficient
Terminal Velocity Examples:
- Skydiver (spread-eagle): ~120 mph (54 m/s)
- Skydiver (head-down): ~200 mph (89 m/s)
- Raindrop: ~20 mph (9 m/s)
- Feather: ~2 mph (0.9 m/s)
- Penny: ~25 mph (11 m/s)
- Golf Ball: ~90 mph (40 m/s)
When to Consider Air Resistance:
- High Speeds: Objects approaching terminal velocity
- Large Objects: High surface area to mass ratio
- Long Falls: Extended time allows air resistance to build up
- Precision Required: Engineering applications needing exact values
- Safety Calculations: Parachute and safety system design
Practice Problems and Worked Solutions
Problem 1: Basic Free Fall Time
Question: A ball is dropped from a height of 45 meters. How long does it take to hit the ground, and what is its final velocity?
Click to see detailed solution
Given: h = 45 m, g = 9.81 m/s², v₀ = 0
Find time: t = √(2h/g) = √(2×45/9.81) = √(9.17) = 3.03 s
Find velocity: v = gt = 9.81 × 3.03 = 29.7 m/s
Verification: v = √(2gh) = √(2×9.81×45) = √(882.9) = 29.7 m/s ✓
Answer: Fall time = 3.03 seconds, Final velocity = 29.7 m/s (107 km/h)
Problem 2: Finding Drop Height
Question: An object takes 2.5 seconds to fall to the ground. From what height was it dropped?
Click to see detailed solution
Given: t = 2.5 s, g = 9.81 m/s²
Formula: h = ½gt²
Calculation: h = ½ × 9.81 × 2.5² = ½ × 9.81 × 6.25 = 30.7 m
Final velocity: v = gt = 9.81 × 2.5 = 24.5 m/s
Verification: t = √(2h/g) = √(2×30.7/9.81) = √(6.25) = 2.5 s ✓
Answer: Drop height = 30.7 meters
Problem 3: Free Fall with Initial Velocity
Question: A ball is thrown downward from a 20m building with an initial velocity of 5 m/s. Find the total fall time and impact velocity.
Click to see detailed solution
Given: h = 20 m, v₀ = 5 m/s (downward), g = 9.81 m/s²
Equation: h = v₀t + ½gt²
Substitution: 20 = 5t + ½(9.81)t² = 5t + 4.905t²
Rearrange: 4.905t² + 5t – 20 = 0
Quadratic formula: t = (-5 + √(25 + 4×4.905×20))/(2×4.905) = 1.67 s
Final velocity: v = v₀ + gt = 5 + 9.81×1.67 = 21.4 m/s
Answer: Fall time = 1.67 seconds, Impact velocity = 21.4 m/s
Problem 4: Comparing Different Planets
Question: Compare the fall time for a 10-meter drop on Earth (g = 9.81 m/s²) versus the Moon (g = 1.62 m/s²).
Click to see detailed solution
Given: h = 10 m, g_Earth = 9.81 m/s², g_Moon = 1.62 m/s²
Earth calculation: t_E = √(2h/g_E) = √(2×10/9.81) = √(2.04) = 1.43 s
Moon calculation: t_M = √(2h/g_M) = √(2×10/1.62) = √(12.35) = 3.51 s
Time ratio: t_M/t_E = 3.51/1.43 = 2.45
Velocity on Earth: v_E = √(2g_E h) = √(196.2) = 14.0 m/s
Velocity on Moon: v_M = √(2g_M h) = √(32.4) = 5.7 m/s
Answer: Moon fall takes 2.45× longer, with 2.45× lower impact velocity
