The Physics
Hypertextbook
Opus in profectus

# Gravitational Potential Energy

## Problems

### practice

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.)
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?
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 base 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.
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.

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)

### numerical

1. Crash landing
1. Determine the impact speed of an out of control satellite initially at rest falling from an altitude of 3.5 × 105 m above the Earth.
2. Repeat this problem for an out of control satellite at the same altitude above the moon.
3. Are the numbers you just calculated reasonable; that is, would spacecraft with these initial conditions really hit the surface at these speeds?
2. A neutron star is the collapsed core of a star that has gone supernova. A typical neutron star has the about the same mass as our sun, but a diameter of only 20 km or so. Determine the release height that would give a 5 g marshmallow the same kinetic energy as a small, one kiloton nuclear bomb (4.2 × 1012 J) when it struck the surface.
3. The 1969, Apollo 11 mission to the moon was staged. Or more precisely, the flight vehicle used to transport three astronauts from the Earth to the moon and back was staged. What looks like one rocket on the launch pad was actually three — each with its own engines and propellant tanks, stacked one top of another. The actual astronaut-containing, spacecraft-that-went-to-the-moon was the last little bit perched on the top. (This was further divided up into modules, but that fact isn't relevant to this problem.) After each stage completed burning its fuel it was discarded. Since they serve no purpose, why bother carrying them around? The three stages were given the awkward code designations of S-IC-6, S-II-6, and S-IVB-6N, which I will just call stage one, stage two, and stage three. The final burn of the motors on the third stage gave the spacecraft the speed it needed to enter a free-return, translunar trajectory — essentially, it coasted all the way to the moon and back. (A little bit of fuel was needed to do other stuff, like enter orbit and land on the surface, but that fact isn't relevant to this problem.) Some relevant data from that final burn are highlighted in the table below.
1. What was the escape velocity from the Earth at the altitude of the final burn of the third stage of Apollo 11?
2. How does the actual speed of the spacecraft at the end of the burn compare to the escape velocity? (Hint: it's less.) Explain the significance of this comparison for the flight of Apollo 11. (Hint: they came back to Earth when the mission was over.)

### investigative

1. The twin Voyager space probes are the first human made objects to leave the influence of the solar wind and cross over into interplanetary space. Visit the Voyager mission status webpage for their most up to date speed and distance data.
1. Calculate the escape velocity of the Sun at the current locations of Voyager 1 and Voyager 2.
2. How do their current speeds compare to the escape speeds you calculated? (Hint: they should be greater.)
2. Determine your escape velocity. For the sake of simplicity, assume you are a sphere with a uniform density equal to that of water and a mass equal to your mass today. (Me telling you to "assume you are a sphere" is kind of a joke that physicists tell, by the way. Laugh at your earliest convenience.)