The Physics
Hypertextbook
Opus in profectus

Gravitational Potential Energy

search icon

Practice

practice problem 1

Escape velocity
  1. Calculate the speed needed to escape the Earth from its surface.
  2. Calculate the speed needed to escape the sun from the Earth's orbit.
  3. Calculate the speed needed to escape the Milky Way from our solar system. (The sun is 25,000 light years from galactic center — halfway to the edge of the galaxy. Its orbit encloses a mass that is 78 billion times the mass of the sun.)

solution

A quick review of the origin of the escape velocity equation. Start with the law of conservation of energy.

E0 = E

The relevant energies are kinetic and gravitational potential.

K0 + U0 = K + U

Replace the energy symbols with their equations.

½m1v02 −  Gm1m2  = ½m1v2 −  Gm1m2
r0 r

Let r → ∞ and v → 0.

½m1v02 −  Gm1m2  + = 0 − 0
r0

Get rid of the subscripts and the minus sign.

½v2 =  Gm
r

Solve for v.

v = √ 2Gm
r

Substitute appropriate numbers and calculate the speed needed to…

  1. …escape the Earth.

    v = √ 2(6.67 × 10−11 N m2/kg2)(5.97 × 1024 kg)
    6.37 × 106 m

    v = 11,200 m/s

    Fast!

  2. …escape the sun.

    v = √ 2(6.67 × 10−11 N m2/kg2)(1.99 × 1030 kg)
    1.50 × 1011 m

    v = 42,100 m/s

    Damn fast!

  3. …escape the galaxy.

    v = √ 2(6.67 × 10−11 N m2/kg2)(78 × 109)(1.99 × 1030 kg)
    (25,000)(365.25×24×60×60)(3.00 × 108 m/s)

    v = 296,000 m/s

    Unfuckingbelievably fast!

practice problem 2

Event horizon
  1. If the sun were a black hole what would be the radius of its event horizon?
  2. If the Earth were a black hole what would be the radius of its event horizon?
  3. If you were a black hole what would be the radius of your event horizon?

solution

A quick review of the origin of the event horizon equation. Start with the law of conservation of energy.

E0 = E

The relevant energies are kinetic and gravitational potential.

K0 + U0 = K + U

Replace the energy symbols with their equations.

½m1v02 −  Gm1m2  = ½m1v2 −  Gm1m2
r0 r

Let v0 = c, r → ∞ and v → 0.

½m1c2 −  Gm1m2  + = 0 − 0
r0

Get rid of the subscripts and the minus sign.

½c2 =  Gm
r

Solve for r.

r =  2Gm
c2

Substitute appropriate numbers and calculate the radius of…

  1. the event horizon of the sun.

    r =  2(6.67 × 10−11 N m2/kg2)(1.99 × 1030 kg)
    (3.00 × 108 m/s2)2

    r = 2950 m

    I could easily walk this distance in under an hour — even if I doubled it to get the diameter. That's not far. Could the dying sun actually collapse to this size? Highly unlikely.

  2. the event horizon of the Earth.

    r =  2(6.67 × 10−11 N m2/kg2)(5.97 × 1024 kg)
    (3.00 × 108 m/s2)2

    r = 8.884 mm

    I love how close that is to 9 mm. It makes me see a 9 mm bullet in my mind. But this 9 mm event horizon is a radius and the 9 mm for bullets is a diameter. Doesn't really matter much since this is a hypothetical calculation. Crushing the Earth down to the point where it would fit into the barrel of any conventional weapon is totally absurd.

  3. the event horizon of you (or a person somewhat like you).

    r =  2(6.67 × 10−11 N m2/kg2)(70 kg)
    (3.00 × 108 m/s2)2

    r = 1 × 10−25 m

    What is the standard person? Most of the serious readers of this book (well, this website claiming to be a book) are probably young adults. Older adults don't have much reason to study physics. (How sad.) Let's give them a characteristic mass of 70 kg, because why not. (That's not your mass, but it's certainly within an order of magnitude of your's.)

    Why am I having this internal monolog with you? Your mass is 70 kg because I said so and your event horizon is ten orders of magnitude smaller than the radius of a proton (r ~ 10−15 m). Squeeze a human down to the size of a proton and they still won't become a black hole. You need to go ten billion times smaller.

practice problem 3

Cosmic expansion
When we look at galaxies and other objects outside our own Milky Way we see that they are generally moving away from us and that their recessional velocities are nearly directly proportional to their distance. This observation was first made in 1929 by the American astronomer Edwin Hubble (1889–1953) and is now known as Hubble's law. Mathematically, Hubble's law is written as…

v = Hr

where…

v =  the object's recessional velocity (usually stated in km/s)
r =  the object's distance from the Milky Way (usually stated in megaparsecs or Mpc)
H =  a constant of proportionality known as the Hubble constant (69.3 ± 0.8 km/s/Mpc).

Perform the following set of calculations.

  1. Given that one parsec equals 3.08568 × 1016 m…
    1. Convert the Hubble constant to SI fundamental units.
    2. By how much does a meter of space expand in ten years? About how big is this?
    3. How far away in light years is a distant quasar if it appears to be moving away from us at 90% of the speed of light?
    4. How far away in light years is the edge of the observable universe? (Your answer to this question can also be used to determine the age of the universe.)
  2. Will the universe continue expanding forever or will gravity eventually cause it all to collapse in a big crunch?
    1. Derive an expression for the critical density of the universe. (Hint: Use the formulas for escape velocity, the Hubble law, and the density of a uniform sphere.)
    2. Compute the critical density in terms of hydrogen atoms per cubic meter.
    3. Speculate on the fate of the universe given that the mean density of a galaxy is roughly one hydrogen atom per cubic centimeter while the mean density of the space between galaxies is about one hydrogen atom per cubic meter.
    4. Speculate on the fate of the universe given that the Hubble constant appears to be increasing.

solution

The solutions to this practice problem can be found in the discussion section of this topic.

practice problem 4

Rods from God
Signatories to the Outer Space Treaty of 1967 agreed that they "shall not place nuclear weapons or other weapons of mass destruction in orbit or on celestial bodies or station them in outer space in any other manner". That hasn't stopped them from doing feasibility studies, however, or looking for loopholes in the law. Is a massive object a weapon? What if there was a satellite orbiting the earth that was full of massive objects? Massive objects with dimensions similar to a telephone pole made of a dense material? Would that be a weapon? Drop one from space and tell me what happens.

Schemes like this have been in the works in the United States since 1964 (under the informal name of Project Thor) and are revived from time to time — for example, by the RAND Corporation in 2002 and by the US Air Force in 2003. They have been described as hypervelocity rod bundles, orbital telephone poles, and most poetically rods from God. They have met with legal, political, economic, and scientific skepticism. They were also used as a plot device in the 2013 action-adventure movie GI Joe: Retaliation.

A proposed rod bundle system
characteristic value
platform altitude 8000 km
rod diameter 40 cm
rod length 7 m
rod material tungsten carbide
rod density 15,630 kg/m3
Given the values in the table, determine the following quantities for a hypothetical, hypervelocity, orbiting, rod bundle system…

  1. the orbital speed of the platform
  2. the orbital period of the platform
  3. the mass of one rod
  4. the energy to de-orbit one rod (the energy needed to change its orbital speed to zero)
  5. the gravitational potential energy of one rod relative to the surface of the earth
    1. in joules
    2. in tons of TNT equivalent
  6. the impact velocity of a de-orbitted rod (disregarding aerodynamic drag and the rotational motion of the earth)

solution

  1. The orbital speed of the platform

    Fc = Fg

    mv2  = 
    Gm1m2
    r2
    r
    v = √ Gm
    r
    v = √ (6.67 × 10−11 N m2/kg2)(5.97 × 1024 kg)
    (6,670,000 m + 8,000,000 m)

    v = 5210 m/s

  2. The orbital period of the platform in hours and minutes

    t =  s  =  r    
    v v    
    t =  2π(6,670,000 m + 8,000,000 m)
    5,210 m/s
    t = 17,692 s = 4:55  
     
  3. The mass of one rod

    m = ρV = ρπrh
    m = (15,630 kg/m3)π(0.20 m)(7 m)
    m = 68,700 kg = 68.7 tonnes
  4. The energy to de-orbit one rod (the energy needed to change its orbital speed to zero)

    K = ½mv2
    K = ½(68,700 kg)(5,210 m/s)2
    K = 9.32 × 1011 J = 932 GJ
  5. The gravitational potential energy of one rod relative to the surface of the earth

    Ug = −  Gm1m2
    r
    Ug = −Gm1m2
    1  −  1
    r0 r
    Ug =  (6.67 × 10−11 N m2/kg2)(5.97 × 1024 kg)(68,700 kg)
    ×
    1  −  1
    (6,670,000 m) (6,670,000 m + 8,000,000 m)

    Ug = 2.24 × 1012 J = 2240 GJ = 2.24 TJ

    One ton of TNT is equivalent to 4.184 GJ by definition.

    Ug =  2240 GJ
    4.184 GJ/ton TNT

    Ug = 535 tons TNT

    I would call this half a ton of TNT to be less precise, but more meaningful.

  6. The impact velocity of a de-orbitted rod (disregarding aerodynamic drag and the rotational motion of the earth)

    v = √
    2∆Ug
    m
     
     
    v = √ 2(2.24 × 1012 J)
    68,700 kg
    v = 8075 m/s  
     

    Damn, that's fast.