Experiment III-2

Motion (momenta) is Conserved

constructing another property useful for understanding nature


Aristotle (b384, d322 B.C.) attempted to summarize the best of Greek understanding of the world.  Some things moved on their own accord such as heavenly bodies and living creatures.  All materials belonged in a certain order of position depending on their composition.  Fire and air naturally belonged above water and earth in that sequence.  Anything out of place would have a tendency to return to where it belonged.  For example a rock would fall downward in either air or water because its natural position is under them.  Smoke tends to rise because its fire content belongs above air in the natural sequence of materials.  Air bubbles upwards through water.  These natural motions needed no further cause.  Other violent motion requires a cause or prime mover.  For example a sailing ship needs to be pushed by the wind, tides, or rowed.  A wheelbarrow needs to be pushed.  For the most part these explanations seemed reasonable and adequate so were used for nearly two thousand years!

The scribes who recorded the Bible appear to have merely assumed the validity of the science and astronomy that was common knowledge at the time of their writings.  The early Christians seem to have largely ignored any newer science and astronomy being proposed by the Greeks of their period.  Much of that newer Greek science and astronomy was only later introduced into Europe by the returning Crusaders.  The Church first tried to ignore it, briefly banned it, then decided that science using observation and reason was one of two methods provided by God to know the truth.  Only when it was acceptable to study the Greek works along side the Bible was there any realization that the Bible was silent towards or perhaps even inconsistent with some of the assumptions basic to the explanations of Greek science.  It took a considerable change in perspective to even notice any shortcomings in the Greek explanations.
ibn Sina

The Achilles Heel of the Greek understanding of motion was their explanation of projectiles.  It was difficult to explain why a thrown object doesn't fall directly downward immediately after being released.  Why doesn't natural motion instantly follow when violent motion ceases?  More careful observation and attempts to explain the curved parabolic trajectories of projectiles eventually led to today's understanding of all motion.  Earlier Abū ‘Alī al-Husayn ibn ‘Abd Allāh ibn Sīnā al-Balkhī (Persian ابوعلى سينا/پورسينا or arabicized: أبو علي الحسين بن عبد الله بن سينا; or simply ibn Sina, or latinized as Avicenna, b980, d1037 stamp portrait at right→) suggested that a projectile receives from its mover mayl, a quality related to an object's heaviness, which helps carry its motion.  I.e., every physical body which accepts the act of yielding (al-mayl) from the outside cause has to have a natural inclination, mayl tibaci, in itself.  This mayl would be reduced by resistance resulting in natural motion eventually occurring.

Thomas Bradwardine (b~1295, d1349) was the Archbishop of Canterbury for 40 days before dying of the plague.  Two decades earlier in 1328 he had written a treatise De proportionibus velocitatum in motibus (On the Proportion of Velocities in Moving Bodies, printed 1495 at Paris & 1505 in Venice) in which he proposed that motion should not be thought of as a process of each body but the mathematical proportion of distance : time.  His treatise was presented as part of many university courses and thus was widely read.  Treating motion as a subject for measurement was a major paradigm shift responsible for better understanding of motion as well as all variety of other changes.  One of the results was the invention at Merton College, England, of a primitive form of graphing to visually represent changes over time.

Jean Buridan (b<1328, d~1358) proposed that motion was maintained by some property of the body, imparted when it was set in motion, a property which he called impetus.  For example, the weight of a falling body would add impetus to the body causing uniform difform (accelerated) motion.  Buridan suggested that the impetus of a body also increased with the speed with which the body was set in motion, as well as with the quantity of matter in the body.  The impetus which caused the motion of the object did not dissipate spontaneously, but rather a body would eventually be stopped by the forces of air resistance and gravity which opposing its impetus.  Buridan wrote:

...after leaving the arm of the thrower, the projectile would be moved by an impetus given to it by the thrower and would continue to be moved as long as the impetus remained stronger than the resistance, and would be of infinite duration were it not diminished and corrupted by a contrary force resisting it or by something inclining it to a contrary motion.

As sea exploration, travel and commerce evolved, there was increased interest in mapping and navigation.  Understanding motions in the heavens as well as the related tides became more important to ocean travel beyond sight of land and the docking at ports of call.  Galileo Galilei (b1564, d1642) and Isaac Newton (b1642, d1727) largely completed the mathematical understanding of motions both earthly and that in the heavens.  The success of their methods revolutionized European understanding of the world and led many other people to try to use the same methods to improve even such diverse subjects as economics and government.

When one pays attention to the ratio of distance / time, what we call speed, that if you take an object (such as a ball) and minimize the friction, across a level smooth surface it will travel in a straight line hardly slowing much.  If one could totally eliminate friction, one might imagine it wouldn't slow or change direction at all.  Galileo suggested it likely would go completely around the world!  And Descartes suggested that if the Earth was not present to exert gravity, such an object would continue through space in a straight line.  Newton expressed this as his 1st law of motion: An object at rest will remain at rest unless acted upon by an external and unbalanced force.  An object in motion will remain in motion unless acted upon by an external and unbalanced force.  When compared to the laws of motion described by Aristotle, it might be noted that Aristotle's first law described natural motions which needed no further cause (or explanation).  Likewise, Newton's first law also describes which motions need no further explanation: stationary motion and motion with constant velocity.


But is speed the property of motion that is conserved?  Is speed the mayl of ibn Sina or the impetus of Buridan?  While the basic rules of algebra had been compiled by Persian mathematician Muhammad ibn Mūsā al-Khwārizmī (b~780, d~850), they were not accepted as a valid form of mathematical proof until the success of Newton.  Newton and his predecessors all used geometry when trying to convince readers.  So the formulae that we prefer today may have been expressed earlier in words, but had not been presented the the algebraic form that we today find easy to test by experimental measurements.

It was observed that when an object such as billiard ball or marble collides with a similar object that part and sometimes all of the motion is transferred.   When a light object collides with a more massive object, less of the motion is transferred.  Measurements of head on collisions suggested that the product of mass and velocity before the collision equals the sum of such products after collision.

Rene DescartesFor French philosopher, mathematician, and scientist René Descartes, (b1596, d1650 portrait at right→) mass multiplied by velocity was the fundamental conserved property of motion.  Galileo Galilei in his Two New Sciences used the Italian term impeto, while Newton's Principia used the Latin motus for motion.  Today we use the Latin term for movement, momentum:

p m v

where p is the momentum, m is the mass, and v the velocity.  The origin of the use of p for momentum is unclear.  Since m had already been used for mass, the p may be derived from the Latin petere (= to go) or from progress (a term used by Gottfried Wilhelm Leibniz).  Alternatively, what we call impulse:
F • Δt  =  Δ ( m v=  Δp
where Δt is the change in time and Δ ( m v ) equals the change in momentum.

Descartes' book of 1644 Principia philosophiae (Principles of philosophy) stated that bodies act on each other only by direct contact: a principle that required a universal medium, the æther, as a carrier of interactions which seem to occur over some distance such as light and gravity.  So Descartes had envisioned the planets being swept around in a cosmic vortex, a concept that Newton discounted in his Principia.  Descartes' flawed treatment of circular motion led others to identify his view of circular motion as conflicting with principle of inertia, but leading to a principle of angular momenta.


A. inertia

You're probably familiar with the trick of pulling a table cloth out from underneath a full table of dishes?  If you want to try this, start with a sheet of paper under a few moderately massive but unbreakable objects on a smooth table surface.  It demonstrates what Newton expressed as An object at rest tends to remain at rest.  But others such as Galileo earlier appreciated this property of nature.

B. linear collision

Linear collisions are those that occur along a straight line.  (The more complicated 2 and 3 dimensional collisions are studied in the next investigation.)  We can get a feel for the amount of momentum by how much impulse we have to exert to stop a moving object.  It is harder to stop if the object is moving faster.  It is also harder to stop if the object is more massive.
  1. Obtain several equal sized marbles or hard balls and make or obtain a track, perhaps by simply fastening two straight boards a small but equal distance apart.

  2. Place one ball roughly in the middle of the track and roll a second ball into it, comparing the motions before and after the collision

  3. Repeat the collision varying the speed of the colliding ball.  Is the total momentum after collision the same total as before collision?  (What is the momentum of a ball with zero velocity?  Pick convenient units for mass and velocity of the moving ball, ...perhaps a mass of 1 ball...)

  4. Place several balls tightly together in the middle of the track and roll the single ball into the stationary balls again comparing the motions before and after to collision.  After the collision, how many balls moved and roughly how fast did they move compared to to the colliding ball.  Is the total momenta after equal that before the collision?

  5. Most challenging, launch multiple balls TOGETHER into the collection of stationary balls and attempt to compare the totals of momenta before and after the collision.

  6. Does it make any difference if softer, rubber balls are used?
Of course if you have more elaborate equipment available, more elaborate measurements can be made.  For example, a camera which captures video sequences could be used to record the collisions, the video file imported into a computer, and locations of each ball could be measured (say from the side of the monitor screen) for each successive image to enable calculating each ball's speed before and after the collision.  But lacking such equipment, it should be possible to simply estimate if a ball is moving say 1/2 as fast (or the fraction anticipated if momentum is conserved) after a collision.

C. angular momentum

A spinning object has a variation of momentum which involves the distance the object is from its axis of rotation.  This may be most noticeable when a spinning person (perhaps on ice skates or someone on an easily rotated chair) spins more rapidly when they pull outstretched arms in closer to their center.  Their rotation speed increases as the radius decreases, conserving what we call angular momentum.  To develop an appropriate formula, consider swinging a mass, m, on a string so that it makes a circular path with radius r with an instantaneous velocity of v.  At an instant in time the linear momentum is

p = m v.  So we define
angular momentum as
l r x p

Since both r and p are vectors with directions perpendicular to each other, the convention is to assign a direction to the cross product vector, (l), as follows:  Align your flat, open right hand in the direction of the first (r) vector, then curl your fingers in the direction of the second vector, (p).  (If necessary twist your arm to make that possible).  The direct of any cross product, (in this case l) is the direction of your outstretched thumb, perpendicular to both of the vectors being multiplied.

topThe challenge is that a spinning object, such as a top, has mass at different radii and traveling at different speeds.  The trick is to use a summing process such as calculus to combine the angular momenta of all bits of mass.

While a moving object's linear momentum resists an attempt to slow it, a spinning object also resists changing its axis of rotation.  And a force exerted to try to change the direction of the axis or rotation causes the spinning object to precess.  This is perhaps more noticeable with a spinning top where gravity exerts a force to topple the top, but it instead caused the top to precess.

This precession effect is basically an effect of Newton's second law where the the force attempts to deflect the axis of rotation, changing the vector direction of the angular momentum.  The force changes momentum one direction on one side of the rotating object, and the other direction on the other side of the rotating object, creating a twist in what may seem to be an unanticipated direction.  This torque which causes the precession is also described by a perpendicular cross product of the rotating object's angular momentum and the deflecting force vector:

  1. Locate a flat, open area on which a bicyclist could learn to ride without fear of collision with obstacles or vehicles.

  2. Einstein on bikeWearing protective helmut and other available crash gear (unlike AE), comfortably ride a bicycle in a straight path at a moderate (not slow) speed.  Try shifting your mass slightly forward and backwards noticing any effects on the motion of the bicycle.  (Later on, ponder how this is different from when you pump yourself on a swing.  And if you haven't done that, add that to you list of simple skills to learn and understand.)

  3. Practice riding the bicycle in a straight line while only holding the steering handles loosely so you can sense any tendency for the bike to change direction.  Be prepared to tighten your grip and to assert full control as necessary.  Again try leaning very slightly forward and backward, paying any attention to a tendency for the bike to shift direction.

  4. Finally try shifting your mass extremely briefly and only very slightly to only one side or the other, again paying attention to any tendency for the bike to shift direction.  (Note it takes only a tiny lateral force to cause the bike to precess.  This is potentially hazardous; be cautious!  Don't shift mass for very much time or distance!)

Communicating technical information such as observations and findings is a skill used by scientists but useful for most others.  If you need course credit, use your observations in your journal to construct a formal report.



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created 4 March 2005
last addition 9 July 2007
by D Trapp
Mac made