Exercises on Satellites and Binding Energy

Helena Dedic

Beware: Many of the solutions to these exercises use $g=10m/s^{2}$ !

Exercise 1

Once every 74 years, Halley's comet visits the inner part of the Solar system. It moves at a speed of $5.5\times 10^{4}m/s$ when it is the closest to the Sun i.e.: when its distance from the Sun is $8.8\times 10^{10}m$ (for comparison, the Earth is $15\times 10^{10}m$ from the Sun). Halley's comet is believed to be a dirty snowball of density approximately 1 ${\tfrac {g}{cm^{3}}}$ and radius 20 km. Assume that the gravitational attraction by the Sun (mass of the Sun is $2\times 10^{30}kg$) is the only force acting on the comet and that the Sun is stationary in the Solar system. Note that the orbit is not circular.

(a) What is the mechanical energy of the Sun - Halley's comet system?

(b) What is the speed of Halley's comet when it is the farthest from the Sun at $5.3\times 10^{12}m$?

• SOLUTION EX 1
  
  

Exercise 2

Determine the binding energy of a satellite of mass 50 kg in an orbit of radius $6.67\times 10^{6}m$ around the Earth.

• SOLUTION EX 2
  
  

Exercise 3

An interstellar probe of mass 10 kg orbits the Sun at the distance equal to the distance between the Earth and the Sun. What is the minimum energy that the rocket must supply to the probe to send it into the interstellar space?

• SOLUTION EX 3
  
  

Exercise 4

A satellite (m = 50 kg) is initially placed in a circular orbit of radius $r_{i}=2R_{E}$ around the Earth. It loses 20% of its energy and descends to a new circular orbit of $r_{f}$.

(a) What is its initial mechanical energy?

(b) What is the radius of the new orbit?

(c) Did the speed increase or decrease because of the losses in the satellite's energy? Determine the percentage change in speed.

• SOLUTION EX 4
  
  

Exercise 5

An electron in a hydrogen atom has an energy of -13.6 eV.

(a) What will happen if the electron is supplied with an energy of 13.1 eV? Explain.

(b) What will happen if the electron is supplied with an energy of 15.0 eV? Explain.

• SOLUTION EX 5
  
  

Exercise 6

What is the escape velocity of a lunar module from the surface of the Moon?

• SOLUTION EX 6
  
  

Example 7

Electrons are supplied with energy of 8.3 eV. Electrons escaping from a sheet of lithium are observed to move with kinetic energy of 6 eV. What is the energy of the electrons in lithium?

• SOLUTION EX 7
  
  

Example 8

The dissociation energy of $O_{2}$ is 7.2 eV. What is the energy of an $O_{2}$ molecule? If we supply each molecule with 10 eV, what will be the total kinetic energy of the two oxygen atoms?

• SOLUTION EX 8
  
  

Example 9

A satellite of mass $m=500$ kg is in orbit of radius $r=1.5R_{E}$. What are its potential, kinetic, and total mechanical energies?

• SOLUTION EX 9
  
  

Example 10

What is the binding energy of the Earth - Sun system? What is the escape velocity of an object at the same distance as Earth from the Sun?

• SOLUTION EX 10
  
  

Example 11

A satellite of mass 10 kg is launched into orbit at an altitude of 50 km above the surface of Earth. The objective of this experiment is to investigate the state of the ozone layer. With what speed should the satellite be launched (assume that it has this speed at zero altitude).

• SOLUTION EX 11
  
  

Example 12

The dissociation energy of CO is 11 eV. How much energy must be supplied to a molecule of CO so that the total kinetic energy of the carbon and oxygen atoms is 20 eV after the dissociation?

• SOLUTION EX 12
  
  

Example 13

A shuttle of mass $m=10^{4}$ kg is sent into orbit to repair an ailing telecommunication satellite in a geosynchronous orbit. It "lands" in an orbit of radius $r$ which is 5% smaller than the radius of the geosynchronous orbit $r_{g}$. The pilot of the shuttle uses its boosters to get it into the orbit of radius $r_{g}$. How much energy must the boosters provide?

• SOLUTION EX 13
  
  

Example 14

The energy of an electron in a helium ion $He^{+}$ is -54.4 eV. An electron is supplied with 100 eV of energy. What is the total kinetic energy of the electron-$He^{++}$ ion after it has escaped?

• SOLUTION EX 14