1. Displaying motion using axes origins and transformed coordinates
Start with the '2-body' example. Tap the 'Examples' tab bar button, tap the
'2-body' example in the table, and tap the 'Load' button. You will be taken
back to the Motion view. Tap the 'go' button (triangle at lower right) to start
the simulation. The button image will change to a square, indicating 'stop'.
The 'reset' button is next to it. The motion should look like the image at
left.
Tap the 'Masses' tab bar button to go to the Masses view. This view shows a
table of all masses in the simulation. The mass in kilograms is shown on the
main line of each row, and the initial position ({x, y, z}) in meters is shown
on the subtitle line. Tap the '>' detail button for Mass 1. Tap the Mass text
field and enter a value of 1.25. Then tap 'return' on the keyboard. Tap the
'Save' button, and then tap the Motion view button in the tab bar. Tap the
'reset' button (at lower right, backward triangle with a vertical line).
Then tap the 'go' button. Now you should see both masses spiraling off to the
left. Tap and drag the display to rotate them into view, like the image at
left.
Tap the 'Settings' tab bar button to go to the Settings view. In the middle of
the set of items is the 'Axes origin' control. Tap the 'Centroid' segment of
the control, and then tap the 'Save' button. Back in the Motion view, tap
'reset' and then 'go', and you should now see the motion depicted at left. The
axes origin is now set to the center of mass (centroid), which follows the
masses as they move off along the -y axis. The trails of each mass therefore
seem to move off along the +y axis, although they are actually standing still.
Thus, the centroid coordinate system is a moving coordinate system which
follows the center of mass.
Tap the 'Settings' tab bar button again to go back to the Settings view. Now
tap the 'Transform trails' switch so that it is set to 'on'. Tap the 'Save'
button. Tap 'reset' and then 'go', and you should now see the motion depicted
at left, which is the familiar elliptical orbits of the two masses. However,
the masses are still moving along -y, and the coordinate system is following
them, but now the {x, y, z} coordinates of all trails are being transformed to
the moving coordinate system as well, so that only the relative motion between
the two masses is being shown.
Tap the 'Settings' tab bar button again to go back to the Settings view. Tap
the 'Mass #' segment of the 'Axes origin' control. The text field next to the
control should read '1' (tap and edit it if it doesn't). Tap the 'Save'
button. Tap 'reset' and then 'go', and you should now see the motion depicted
at left, which shows the motion from the viewpoint of mass 1 (you may need to
zoom out by tapping the '-' button at upper left). If this mass were the
Earth, this would be called 'geocentric' coordinates. You can set the 'Mass #'
text field to '2' (remembering to then tap 'Save') to see the motion from the
viewpoint of mass 2.
Finally, go to the Masses view, tap the '>' button for mass 2, tap the text
field for 'VY0' and enter '.625'. Tap 'return' and then 'Save'. Go to the
Settings view and tap the '{0, 0, 0}' segment of the 'Axes origin' control,
then tap 'Save'. Tap 'reset' and then 'go', and you should now see the motion
depicted at left. You are now seeing the actual {x, y, z} coordinates of each
mass in a closed elliptical orbit without moving along -y (verify this by
turning off the 'Transform trails' switch). This is because now the total
momentum of the system is equal to zero. Specifically, 1.25 * 0.5 = 1.00 *
0.625 = 0.625 Kg m/s. When you set the mass of mass 1 to 1.25 earlier, the
total momentum became negative along the y axis, so the center of mass moved
that way. By increasing the initial y speed of mass 2 to 0.625, the net
momentum was returned to zero, so the masses now stay in place again.
NOTE: After editing the initial position or velocity of a mass, you MUST
tap the 'reset' button for the changes to take effect.
SUMMARY:
1. The coordinate axes can be set to display motion based on fixed {x, y, z}
values, based on the moving coordinates of the center of all masses, or
relative to the moving coordinates of a specific mass.
2. Motion trails can be displayed as unmoving {x, y, z} coordinates, or they
can be transformed to the current (possibly moving) coordinate system to show
relative motion.
3. If the total momentum (sum of mass * signed velocity for all masses) is
zero, the center of mass will stay in one place, otherwise it will move.
