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≈9.8 m s−2g \approx 9.8 \,\mathrm{m\,s^{-2}}
  • In other units:
    • 9.8 m s−1=22 mph=35 kph9.8 \,\mathrm{m\,s^{-1}} = 22 \,\mathrm{mph} = 35 \,\mathrm{kph}
  • Often simplified for simple calculations: use g≈10 m s−2g \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 tv(t) = g \, t
  • Distance traveled as a function of time: d(t)=12 g t2d(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 gg.

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=6.674×10−11 m3 kg−1 s−2G = 6.674 \times 10^{-11} \, \mathrm{m^{3}\,kg^{-1}\,s^{-2}}
  • Earth mass: MEarth=5.972×1024 kgM_{\text{Earth}} = 5.972 \times 10^{24} \, \mathrm{kg}
  • Distance from Earth to the Moon: r≈3.84×108 mr \approx 3.84 \times 10^{8} \ \mathrm{m}
  • Gravitational acceleration on the Moon due to Earth: a=GMEarthr2≈0.003 m s−2a = \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/gEarthg/g_{\text{Earth}}
  • The slide provides a table of relative gravities, masses, and radii (Earth-based reference = 1.00):
    • Sun: g/g<em>Earth≈27.95g/g<em>{\text{Earth}} \approx 27.95; Mass M</em>Sun=1.99×1030 kgM</em>{\text{Sun}} = 1.99 \times 10^{30} \, \mathrm{kg}; Radius R<em>Sun=6.96×108 mR<em>{\text{Sun}} = 6.96 \times 10^{8} \, \mathrm{m}; Surface gravity on the Sun: g</em>Sun≈274.13 m s−2g</em>{\text{Sun}} \approx 274.13 \, \mathrm{m\,s^{-2}}
    • Mercury: g/gEarth≈0.37g/g_{\text{Earth}} \approx 0.37; Mass ≈ 3.3×1023 kg3.3 \times 10^{23} \, \mathrm{kg}; Radius ≈ 2.44imes106 m2.44 imes 10^{6} \, \mathrm{m}; g≈3.59 m s−2g \approx 3.59 \, \mathrm{m\,s^{-2}}
    • Venus: g/gEarth≈0.90g/g_{\text{Earth}} \approx 0.90; Mass ≈ 4.88×1024 kg4.88 \times 10^{24} \, \mathrm{kg}; Radius ≈ 6.06imes106 m6.06 imes 10^{6} \, \mathrm{m}; g≈8.87 m s−2g \approx 8.87 \, \mathrm{m\,s^{-2}}
    • Earth: g/gEarth=1.00g/g_{\text{Earth}} = 1.00; Mass ≈ 5.98×1024 kg5.98 \times 10^{24} \, \mathrm{kg}; Radius ≈ 6.37×106 m6.37 \times 10^{6} \, \mathrm{m}; g=9.81 m s−2g = 9.81 \, \mathrm{m\,s^{-2}}
    • Moon: g/gEarth≈0.17g/g_{\text{Earth}} \approx 0.17; Mass ≈ 7.36×1022 kg7.36 \times 10^{22} \, \mathrm{kg}; Radius ≈ 1.74×106 m1.74 \times 10^{6} \, \mathrm{m}; g≈1.62 m s−2g \approx 1.62 \, \mathrm{m\,s^{-2}}
    • Mars: g/gEarth≈0.38g/g_{\text{Earth}} \approx 0.38; Mass ≈ 6.42×1023 kg6.42 \times 10^{23} \, \mathrm{kg}; Radius ≈ 3.39×106 m3.39 \times 10^{6} \, \mathrm{m}; g≈3.77 m s−2g \approx 3.77 \, \mathrm{m\,s^{-2}}
    • Jupiter: g/gEarth≈2.65g/g_{\text{Earth}} \approx 2.65; Mass ≈ 1.90×1027 kg1.90 \times 10^{27} \, \mathrm{kg}; Radius ≈ 6.99×107 m6.99 \times 10^{7} \, \mathrm{m}; g≈25.95 m s−2g \approx 25.95 \, \mathrm{m\,s^{-2}}
    • Saturn: g/gEarth≈1.13g/g_{\text{Earth}} \approx 1.13; Mass ≈ 5.68×1026 kg5.68 \times 10^{26} \, \mathrm{kg}; Radius ≈ 5.85×107 m5.85 \times 10^{7} \, \mathrm{m}; g≈11.08 m s−2g \approx 11.08 \, \mathrm{m\,s^{-2}}
    • Uranus: g/gEarth≈1.09g/g_{\text{Earth}} \approx 1.09; Mass ≈ 8.68×1025 kg8.68 \times 10^{25} \, \mathrm{kg}; Radius ≈ 2.33×107 m2.33 \times 10^{7} \, \mathrm{m}; g≈10.67 m s−2g \approx 10.67 \, \mathrm{m\,s^{-2}}
    • Neptune: g/gEarth≈1.43g/g_{\text{Earth}} \approx 1.43; Mass ≈ 1.03×1026 kg1.03 \times 10^{26} \, \mathrm{kg}; Radius ≈ 2.21×107 m2.21 \times 10^{7} \, \mathrm{m}; g≈14.07 m s−2g \approx 14.07 \, \mathrm{m\,s^{-2}}
    • Pluto: g/gEarth≈0.04g/g_{\text{Earth}} \approx 0.04; Mass ≈ 1.40×1022 kg1.40 \times 10^{22} \, \mathrm{kg}; Radius ≈ 1.50×106 m1.50 \times 10^{6} \, \mathrm{m}; g≈0.42 m s−2g \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)=v<em>0+gt, d(t)=v</em>0t+12gt2v(t) = v<em>0 + g t, \ d(t) = v</em>0 t + \tfrac{1}{2} g t^{2}

Key formulas to remember

  • Free-fall with initial velocity zero: v(t)=gt, d(t)=12gt2v(t) = g t, \ d(t) = \tfrac{1}{2} g t^{2}
  • General vertical motion with initial velocity v<em>0v<em>0: v(t)=v</em>0+gt, d(t)=v0t+12gt2v(t) = v</em>0 + g t, \ d(t) = v_0 t + \tfrac{1}{2} g t^{2}
  • Gravitational acceleration near Earth: g≈9.8 m s−2g \approx 9.8 \, \mathrm{m\,s^{-2}}
  • Gravitational constant: G=6.674×10−11 m3 kg−1 s−2G = 6.674 \times 10^{-11} \, \mathrm{m^{3}\,kg^{-1}\,s^{-2}}
  • Gravitational acceleration due to Earth at distance r: a=GMEarthr2a = \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).