"Once you've seen it, you can never again not see it."
A former semiconductor researcher, and later physics professor, ponders our human predicament, recalling the depth of understanding of his former students...

F = ma
Not really what Newton discovered: the REAL second law is F = dp/dt.

Born in about 1685
Gateway to revolution.
Foundation block of science


Affects billions.
Learned by millions.
Understood  "seen" by thousands.
Used by hundreds.


It's mathematical.
Simple math!
Subtle math!!
Very simple...very subtle.

Try this short  math check-up:
click on the door
to further explore
Some tax increases are on the way.  Your state of three million people is going to fund a new freeway costing six billion dollars.  Your city of one hundred thousand people is going to fund a new soccer stadium costing four million dollars.  Your country of 250 million is going to fund a new Coast Guard rescue vessel costing six hundred million dollars. 
If  the costs are proportioned equally among the populations, which one costs you the most?  Which one the least?

Those who understand ratio and proportion find this problem very easy, the route to the answer obvious.

Those who don't understand, often add numbers where they should be multiplied and subtract where they should be divided.

F = ma is a ratio.

Knowledge for Use: Hyperlinking Thinking

A radio reporter at the beginning of the noon news reported, "The stock market is down sharply today: The Dow is currently down one hundred and eighty-three points."  An hour later the same reporter finished the hour's news with, "The stock market is still falling, but not as fast.  The Dow is now down ninety-seven points."
How would you explain to the reporter what's wrong with the second report?
"Seeing" rates of change makes seeing the error here easy, the explanation obvious.

F = ma tells something about acceleration, which is a rate of change.

Knowledge for Use: Telling up from down

You need to know with some precision how fast the market is changing.  How many points per second is it falling or rising—at some specific time of day.

How would you go about calculaing the "instantanenous" rate of change?
At the end of the 17th century, Isaac Newton saw how to define and calculate instantaneous rates of change.  Those who understand Newton's insight see the answer to this problem as "obvious."

F = ma is about the instantaneous.

Knowledge for Use: Developing the derivative

The coach asks the athletes to line up by size.  Five of the players are on both the basketball team and the football team.  But  when they lined up ("by size") for the football coach they are in a different order than when they lined up for the basketball coach.  The wrestling coach didn't find either order very useful when those five reported for the wrestling team.
How might you help the many coaches rank their athletes?
"It depends on..." is an insight that dips into many dimensions. Those who have that kind of insight see ordering by most measures as being in a many-dimensional space.

Those who don't, pervasively and persistently see every measure as scalar, all orderings in lines.

F = ma relates two vectors  (one of the multi-component measures).

Knowledge for Use: Scalar-limited insight
What if the temp is 4 degrees F outside, and then the temperature drops 7 degrees.  What is the new temperature?

What is the direction, up or down, of the acceleration of a freely bouncing ball at the bottommost point of its bounce, that is, at the instant its velocity changes from down to up?

How many images can you associate with simple math concepts?

Those who don't in some way "see" the abstract concepts are doomed to rote memorize vast quantities of specific cases.

 F = ma is about the vector, acceleration  (an abstraction that is visualizable in many ways).

Knowledge for Use: Telling up from down
F = ma is not Newton's second law as Newton showed it to us. Newton's Law is hidden under the "F = ma" at the top of this page.
The simple math skills are things we need in day-to-day life...
but things that sometimes don't get honed into easily used tools.

Today, this page links to the Knowledge for Use Web site, a route to "Seeing Science" based on puzzle solving.
It will become an alternative to the Knowledge for Use approach.

This "Explore the Physicist's Domain" Web site is an "easier" approach to understanding science.
And this page will become, with your help, an "easier" approach to understanding Newton's laws of motion.
However, "easier" is possibly not more effective.  Probably not, we think.
Let's explore this matter!

October 31, 2001

If you are not sure of your answers to the above math problems, try working them out with friends.
The answers are unimportant; becoming able to see the routes to the answers is very important.
Teachers and learners alike need to explore ways to improve such ability.
How should answers be presented?