Website owner: James Miller
Time required for a marble to descend to the bottom of a chute. A marble starts from rest at point P1 at the top of a chute and rolls frictionlessly down the chute to point P2 at the bottom. How long does it take the marble to go from P1 to P2?
Construct the coordinate system shown in Fig. 1 with the origin O at point P1, the x axis horizontal, and the y axis pointed downward. The point P1 is then at coordinates (0, 0). Assume point P2 has coordinates (x2, y2). In addition, assume that the path of the chute is given by the function
1) y = f(x), 0 ≤ x ≤ x2
where f is a continuously differentiable function. Then the time required for the marble to descend to the bottom of the chute is given by
where g is the acceleration of gravity and yʹ = dy/dx.
Derivation. The function f must satisfy the condition
3) f(0) = 0, f(x2) = y2
at the ends of the segment [0, x2].
If s represents the distance along the curve from the origin to point P(x, y), t is time, and v is the speed of the marble at point P(x, y), the three variables s, v, and t are related by
4) ds = vdt
and the time required for the marble to travel from P1 to P2 is given by the formula
The sum of the kinetic energy (½mv2) and the potential energy of the marble at any point P on the curve is equal to the gravitational potential energy of the marble at point P1 [i.e. ½mv2 + (y2 - y)mg = mgy2]. Thus at every point in the descent of the marble down the chute the kinetic energy of the marble is equal to mgy:
Consequently,
At any point P(x, y) on the curve ds, dx, and dy are related by
8) ds2 = dx2 + dy2
or
From 7) and 9) we have
and substituting 10) into 5) gives
Jesus Christ and His Teachings
Way of enlightenment, wisdom, and understanding
America, a corrupt, depraved, shameless country
On integrity and the lack of it
The test of a person's Christianity is what he is
Ninety five percent of the problems that most people have come from personal foolishness
Liberalism, socialism and the modern welfare state
The desire to harm, a motivation for conduct
On Self-sufficient Country Living, Homesteading
Topically Arranged Proverbs, Precepts, Quotations. Common Sayings. Poor Richard's Almanac.
Theory on the Formation of Character
People are like radio tuners --- they pick out and listen to one wavelength and ignore the rest
Cause of Character Traits --- According to Aristotle
We are what we eat --- living under the discipline of a diet
Avoiding problems and trouble in life
Role of habit in formation of character
Personal attributes of the true Christian
What determines a person's character?
Love of God and love of virtue are closely united
Intellectual disparities among people and the power in good habits
Tools of Satan. Tactics and Tricks used by the Devil.
The Natural Way -- The Unnatural Way
Wisdom, Reason and Virtue are closely related
Knowledge is one thing, wisdom is another
My views on Christianity in America
The most important thing in life is understanding
We are all examples --- for good or for bad
Television --- spiritual poison
The Prime Mover that decides "What We Are"
Where do our outlooks, attitudes and values come from?
Sin is serious business. The punishment for it is real. Hell is real.
Self-imposed discipline and regimentation
Achieving happiness in life --- a matter of the right strategies
Self-control, self-restraint, self-discipline basic to so much in life