RANDY L. PHELPS

This is the "Newton's Cannon" Exercise

The following Applet was written and provided by Michael Fowler.  The applications were written by Randy Phelps.

Purpose:   The velocity an object needs to attain orbit is referred to as the orbital velocity. The actual value is determined by where, relative to the Earth's center, the projectile is "launched".  In this simulation, the projectile is launched from a ficticious tall mountain.  It is not hard to imagine that, with very little velocity, a cannon ball for example, will not travel far, and will crash to the Earth after it is fired.  It is also not hard to image that, with just enough velocity, a cannon ball would clear the Earth, and go into orbit.  In effect, the cannon ball "falls" toward the Earth, but never quite hits it.  This is a good way to think of orbits!  If just the right velocity (the "circular velocity") is given to a projectile, it will attain a circular orbit, while too great a velocity (the "escape velocity") will result in the object's escaping completely from the Earth.  There are, or course, a variety of orbits that occur "in-between".   The purpose of this exercise is to illustrate the relationship between a projectile's/rocket's launch speed, and the orbit it will, or will not, achieve around the Eart.

Data

Symbol Meaning Value
G Newton's Gravitational Constant

6.67 x 10-8 dynes cm2 gm-2
MEarth Mass of the Earth

5.97 x 1027 gm
REarth Radius of the Earth

6.38 x 108 cm

Procedure:

  • Do each  "Exploration" outlined in the first column below.  Before you actually do each "Exploration", answer the questions posed in the "Anticipate the Result" box for that "Exploration".  After you have done this, determine if the actual result was as you expected, or somehow different.  If the actual result was different, determine why before you proceed to the next "Exploration".
  • At the end of the complete exercise, you should have answered the "Anticipate the Results" questions as well as any questions at the end of this page.

Explorations:

Exploration

 Anticipate the Results Instructions/Sugestions to Help Answer These/Other Questions
1. Minimum velocity required for Earth orbit

 

 

 

 

 

 

 

 

 
  • At low velocities, will projectiles attain an orbit around the Earth?
  • What is the minimum velocity a projectile must be given to orbit the Earth?

 

 

 

 

 

 

 
  • Set a low launch velocity (say 8000 miles per hour, or mph) for the projectile by sliding the velocity bar in the applet, using the left mouse button.
  • Click on the "fire" tab with the left mouse button, and see where the cannon ball lands.
  • Gradually increase the launch velocity by sliding the velocity bar in the applet.   Repeat the firing procedure.
  • Repeat as needed until the cannon ball just barely clears the Earth, and completes an orbit.
  • Consider questions 1and 2 in the "Critical Thinking/Applications" section below.
2. Circular velocity about the Earth

 

 

 

 

 

 

 

 
  • Calculate the circular velocity for the Earth, using the values given in the "Data" section above.  Recall the circular velocity is given by:

Vcirc_boxed.gif (2661 bytes)

  • Convert your derived circular velocity from cm sec-1 to miles hour-1 - Note: this is only required because those are the units in this simulation!
  • Set the circular launch velocity for the projectile by sliding the velocity bar in the applet, using the left mouse button.
  • Click on the "fire" tab with the left mouse button.
  • Repeat the procedure, increasing or decreasing the launch velocity, until the cannon ball achieves as close to a perfectly circular orbit as possible.
  • If you find your calculated value does not exactly agree with that found in the simulation, proceed to question 3 in the "Critical Thinking / Applications" section below.
3. Elliptical Orbits

 

 

 

 

 

 

 

 
  • If an object is given just less than the circular velocity, will it still orbit the Earth and if so, what type of orbit will it be, and where, relative to the "launch point", are perigee and apogee?
  • If an object is given just greater than the circular velocity, will it still orbit the Earth and if so, what type of orbit will it be, and where, relative to the "launch point", are perigee and apogee?

 

 
  • Set a launch velocity for the projectile that is just less than the circular velocity found in Exploration 2, by sliding the velocity bar in the applet, using the left mouse button.
  • Observe the trajectory of the projectile, noting whether or not it completes an orbit, and if so, where apogee and perigee are located, relative to the launch point.
  • Repeat the procedure, but this time select a launch velocity for the projectile that is just greater than the circular velocity found in Exploration 2.
4. Escape velocity for the Earth

 

 

 

 

 

 

 

 

 

 
  • Calculate the escape velocity for the Earth, using the values given in the "Data" section above.  Recall the circular velocity is given by:

Vesc_boxed.gif (3105 bytes)

  • Convert your derived escape velocity from cm sec-1 to miles hour-1 - Note: this is only required because those are the units in this simulation!
  • Set the escape  launch velocity for the projectile by sliding the velocity bar in the applet, using the left mouse button.
  • Click on the "fire" tab with the left mouse button, or fine tune the velocity with the left and right arrows.
  • Wait (perhaps a long time!) to see if the projectile returns.  If so, gradually increase the lauch velocity until the projectile escapes, never to return to Earth again.  This is indicated by a "whistling" sound in the simulation!

 

 

 

Critical Thinking/Applications: 

  1. In the absence of an atmosphere, would a projectile return to the same location from which it was "launched"?

  2. Rockets seemingly are launched "straight up", yet they put satellites into orbit.  Using the ideas behind "Newton's Cannon", can you figure out how a satellite is actually inserted into orbit?

  3. In the exercise on circular velocities, the value of the circular velocity that you calculated may not have agreed exactly with that found by experimenting with the simulation.  This is because the simulaion assumes the projectile is launched from a "high" mountain, and not from the Earth's surface.  As the altitude of the launch point increases, the circular velocity decreases (does this make sense from the equation?).  Using the value for the circular velocity you found in the simulation, you can calculate how high the mountain is!  Do you see how, and if so, how high is the mountain?  This is an excellent example of taking basic principles, and using the available information to learn about the world around you.