Falling Objects: Free Fall

What is Free Fall?

An object is in free fall if and only if:
- Gravity is the only force acting on it.
- It is only moving up or down.
Scientific definition: Free fall is motion under gravity with no other forces (like air resistance) acting significantly on the object.

Acceleration due to Gravity

Near Earth’s surface, the acceleration is constant downward: g \approx 9.8 \,\mathrm{m\,s^{-2}}
In other units:
- 9.8 \,\mathrm{m\,s^{-1}} = 22 \,\mathrm{mph} = 35 \,\mathrm{kph}
Often simplified for simple calculations: use g \approx 10 \,\mathrm{m\,s^{-2}} when appropriate.

Velocity and Distance in Free Fall (with constant g)

Velocity as a function of time: v(t) = g \, t
Distance traveled as a function of time: d(t) = \tfrac{1}{2} \, g \, t^{2}
Example with g = 10:
- At t = 0 s: V = 0, d = 0
- At t = 1 s: V = 10 m/s, d = 5 m
- At t = 2 s: V = 20 m/s, d = 20 m
- At t = 3 s: V = 30 m/s, d = 45 m
- At t = 4 s: V = 40 m/s, d = 80 m
Summary: velocity increases linearly with time while distance increases with the square of time.

How does a dropped object behave? Mass and shape

Do two objects with different masses or different shapes fall differently?
- Intuition: a feather falls more slowly than a brick in normal air.
- The real reason: air resistance interacts with mass and shape, not just gravity alone.
In a vacuum (no air resistance):
- Feathers and bricks fall at the same rate; gravitational acceleration does not depend on weight.
Key takeaway: without air resistance, all objects accelerate downward at the same rate g.

Open for discussion: Elephant and Feather

Scenario A: with air resistance, both feather and elephant encounter drag; which hits first?
Scenario B: with air resistance eliminated (vacuum), which hits first?
Purpose: to distinguish gravity-driven acceleration from drag forces and to illustrate air resistance effects.

Factors influence air resistance

1) Smaller weight (mass) tends to make an object more susceptible to drag relative to its weight
2) Larger surface area increases drag force
3) Longer fall time allows drag to act longer, modifying the net acceleration
Consequences: heavier or more streamlined objects reach the ground sooner under air resistance; shape and surface area matter.

Galileo and Acceleration due to Gravity

Galileo demonstrated that gravitational acceleration is uniform (constant with time) by rolling objects down inclined planes, which slowed motion enough to reveal the constant acceleration.
Conclusions:
- Gravity produces a constant downward acceleration near Earth’s surface.
- The rate of fall is independent of the mass of the object (in the absence of significant air resistance).

Gravitational influence across distances (Earth and beyond)

Gravitational constant: G = 6.674 \times 10^{-11} \, \mathrm{m^{3}\,kg^{-1}\,s^{-2}}
Earth mass: M_{\text{Earth}} = 5.972 \times 10^{24} \, \mathrm{kg}
Distance from Earth to the Moon: r \approx 3.84 \times 10^{8} \ \mathrm{m}
Gravitational acceleration on the Moon due to Earth: a = \dfrac{G M_{\text{Earth}}}{r^{2}} \approx 0.003 \, \mathrm{m\,s^{-2}}
Lesson: gravitational influence falls with distance; gravity diminishes with increasing separation from the mass source.

Weightlessness and parabolic flight

Weightlessness can be experienced during parabolic flight maneuvers (G-FORCE ONE) where aircraft follow a parabolic trajectory.
Typical altitude range for such maneuvers is roughly between 24,000 and 34,000 feet.
Concept: even when in free fall, gravity is still present; weightlessness arises because apparent weight is not felt due to free-fall conditions.

Gravitational force on a 70-kg student at various locations (weight values)

Weight on Earth’s surface varies slightly by location due to gravity differences:
- Sea Level at Equator: ≈ 684.6 N
- Sea Level at the Poles: ≈ 688.3 N
- Chicago: ≈ 686.2 N
- Top of Mount Everest: ≈ 685.4 N
Other locations:
- Moon: ≈ 19 N
- Low-Earth Orbit: ≈ 697 N
- High-Earth Orbit (geosynchronous range): ≈ 18 N (approximate, as shown in the slide)
Note: These values illustrate how gravity varies slightly with location due to shape of the Earth and altitude.

Gravity on planets and the Sun

Gravity on a body is often expressed as a relative to Earth: g/g_{\text{Earth}}
The slide provides a table of relative gravities, masses, and radii (Earth-based reference = 1.00):
- Sun: g/g{\text{Earth}} \approx 27.95; Mass M{\text{Sun}} = 1.99 \times 10^{30} \, \mathrm{kg}; Radius R{\text{Sun}} = 6.96 \times 10^{8} \, \mathrm{m}; Surface gravity on the Sun: g{\text{Sun}} \approx 274.13 \, \mathrm{m\,s^{-2}}
- Mercury: g/g_{\text{Earth}} \approx 0.37; Mass ≈ 3.3 \times 10^{23} \, \mathrm{kg}; Radius ≈ 2.44 imes 10^{6} \, \mathrm{m}; g \approx 3.59 \, \mathrm{m\,s^{-2}}
- Venus: g/g_{\text{Earth}} \approx 0.90; Mass ≈ 4.88 \times 10^{24} \, \mathrm{kg}; Radius ≈ 6.06 imes 10^{6} \, \mathrm{m}; g \approx 8.87 \, \mathrm{m\,s^{-2}}
- Earth: g/g_{\text{Earth}} = 1.00; Mass ≈ 5.98 \times 10^{24} \, \mathrm{kg}; Radius ≈ 6.37 \times 10^{6} \, \mathrm{m}; g = 9.81 \, \mathrm{m\,s^{-2}}
- Moon: g/g_{\text{Earth}} \approx 0.17; Mass ≈ 7.36 \times 10^{22} \, \mathrm{kg}; Radius ≈ 1.74 \times 10^{6} \, \mathrm{m}; g \approx 1.62 \, \mathrm{m\,s^{-2}}
- Mars: g/g_{\text{Earth}} \approx 0.38; Mass ≈ 6.42 \times 10^{23} \, \mathrm{kg}; Radius ≈ 3.39 \times 10^{6} \, \mathrm{m}; g \approx 3.77 \, \mathrm{m\,s^{-2}}
- Jupiter: g/g_{\text{Earth}} \approx 2.65; Mass ≈ 1.90 \times 10^{27} \, \mathrm{kg}; Radius ≈ 6.99 \times 10^{7} \, \mathrm{m}; g \approx 25.95 \, \mathrm{m\,s^{-2}}
- Saturn: g/g_{\text{Earth}} \approx 1.13; Mass ≈ 5.68 \times 10^{26} \, \mathrm{kg}; Radius ≈ 5.85 \times 10^{7} \, \mathrm{m}; g \approx 11.08 \, \mathrm{m\,s^{-2}}
- Uranus: g/g_{\text{Earth}} \approx 1.09; Mass ≈ 8.68 \times 10^{25} \, \mathrm{kg}; Radius ≈ 2.33 \times 10^{7} \, \mathrm{m}; g \approx 10.67 \, \mathrm{m\,s^{-2}}
- Neptune: g/g_{\text{Earth}} \approx 1.43; Mass ≈ 1.03 \times 10^{26} \, \mathrm{kg}; Radius ≈ 2.21 \times 10^{7} \, \mathrm{m}; g \approx 14.07 \, \mathrm{m\,s^{-2}}
- Pluto: g/g_{\text{Earth}} \approx 0.04; Mass ≈ 1.40 \times 10^{22} \, \mathrm{kg}; Radius ≈ 1.50 \times 10^{6} \, \mathrm{m}; g \approx 0.42 \ \mathrm{m\,s^{-2}}
Note: These values illustrate how planetary mass and radius determine surface gravity relative to Earth.

Beyond Free Fall: Throwing a Ball Upward

If the ball is thrown upward:
- Gravitational acceleration is always downward (toward Earth’s center).
- Acceleration is in the opposite direction to the initial upward velocity while the ball goes up, slows, stops, and then accelerates downward.
Key idea: Direction of velocity changes, but the acceleration remains downward with magnitude g.

Ball’s acceleration at the top of its path

At the top (e.g., t = 2 s in the example path), velocity v = 0, but acceleration a = -g (downward).
Explanation:
- Gravity does not turn off at the top; the velocity is momentarily zero, but the gravitational acceleration is still acting downward and changing the velocity from upward to downward.
Correct interpretation in the multiple-choice context: v = 0 at the top, but a = -g (not zero).

Throwing a ball downward

If the ball is thrown downward with an initial velocity v0 > 0 (downward):
- It starts with a nonzero downward velocity.
- It will reach the ground more rapidly than if released from rest.
- Its velocity at impact will be larger than that of a dropped ball from the same height.
Equations adapt to initial velocity: for downward positive convention, v(t) = v0 + g t, \ d(t) = v0 t + \tfrac{1}{2} g t^{2}

Key formulas to remember

Free-fall with initial velocity zero: v(t) = g t, \ d(t) = \tfrac{1}{2} g t^{2}
General vertical motion with initial velocity v0: v(t) = v0 + g t, \ d(t) = v_0 t + \tfrac{1}{2} g t^{2}
Gravitational acceleration near Earth: g \approx 9.8 \, \mathrm{m\,s^{-2}}
Gravitational constant: G = 6.674 \times 10^{-11} \, \mathrm{m^{3}\,kg^{-1}\,s^{-2}}
Gravitational acceleration due to Earth at distance r: a = \dfrac{G M_{\text{Earth}}}{r^{2}}

Video and reference

Video reference for a strobe-like demonstration: https://techtv.mit.edu/videos/831-strobe-of-a-falling-ball
Practical takeaway: visualizes constant acceleration and distance growth under gravity.

Concrete takeaways for the exam

Gravity on Earth causes a near-constant downward acceleration, independent of mass (in absence of air resistance).
In air, heavier and more compact objects tend to reach the ground faster due to reduced air resistance relative to weight; shapes and cross-sectional areas influence drag.
In a vacuum, all objects fall at the same rate regardless of mass or shape.
Distinguish between velocity and acceleration behavior: velocity changes in time under constant acceleration, while acceleration remains constant (downward) even when velocity is zero at the top of a trajectory.
Galileo’s experiments with inclined planes demonstrated the uniform acceleration due to gravity and set the stage for mathematical descriptions of motion.
Gravitational force and acceleration vary with distance; on the Moon (due to Earth), the acceleration is tiny (~0.003 m/s²) at lunar distances; weightlessness can be achieved in parabolic flight without altering gravity.
The gravitational field strength of bodies other than Earth can be compared via g/g_Earth, and surface gravity can be calculated from mass and radius for planets and the Sun (as shown in the table).