Quote of the Day:
"In the beginning (if there was such a thing), God created
Newton's laws of motion together with the necessary masses
and forces. This is all; everything beyond this follows
from the development of appropriate mathematics methods by
means of deduction." -- Albert Einstein
Objectives:
The student will learn to solve problems involving velocity
and acceleration.
1. Bellringer
2. Problem:
There is a single path up a mountain in Shenandoah
National Park.
A mountaineer starts up at 7:00 AM and arrives at the
top at 7:00 PM. She stays there overnight.
The next morning, she starts back down at 7:00 AM and
arrives at the bottom at 7:00 PM.
On both days, she travels at varying speeds - enjoying
the scenery, stopping for lunch, smelling the roses...
What is the probability that there was a spot on the
trail that she passed at exactly the same time on both
days?
Hint: Draw a Displacement - Time graph!
Answer: It is 100%. Look at the Displacement-Time
Graph - it doesn't matter how long she takes to come
back down, there must be at least one place where the
descent curve must cross the ascent curve.
3. Given a displacement curve, s = f(t)
The velocity is defined to be the change in the
displacement divided by the change in time,
or in other words, the first derivative of
displacement with respect to time.
It measures how fast you are going.
The acceleration is defined to be the change in the
velocity divided by the change in time, or in
other words, the derivative of velocity with
respect to time.
When you press down on the accelerator in your
car, you are changing the velocity.
The jerk is a sudden change in acceleration. When
a ride in a car is jerky, it is not that the
accelerations involved are necessarily large but
that the changes in acceleration are abrupt.
Jerk is what spills your soft drink.
The derivative responsible for jerk is the third
derivative of position.
Example:
4. From physics, we have studied the formula for free fall motion:
5. Example
Nolan Ryan, one of the fastest pitchers of all-time,
could throw a baseball 150 ft/sec (over 102 mph).
Could Nolan Ryan hit the 208-ft ceiling of the Houston
Astrodome if he were capable of giving a baseball an
upward velocity of 100 ft/sec from a height of 7 feet?
Solution:
6. Examples -- p. 340
Difference between speed and velocity
Graphs of displacement -- ask whether velocity is positive or negative,
whether acceleration is positive or negative,
and whether the particle is slowing down or speeding up.
7. How to Live Your Life... from Alma Barkman
When Canada's Donovan Bailey ran the one-hundred-meter sprint in
9.84 seconds, he was acclaimed as the world's fastest man.
According to an article in the sports section, it almost didn't happen.
With so many wins under his belt, Bailey had become overly confident,
and before the big race he confessed he found it difficult to accept
coaching and apply himself to the rigorous training such a challenge required.
But he won the race, and dozens of experts analyzed the tapes in order to
ascertain his secret. The consensus? He was still accelerating when he
crossed the finish line.
This should be an example for us on how to live our lives. Don't slow
down as you grow older -- make sure that you are accelerating when you
cross that finish line!
8. Assignment
Finish "Class, Take Your Seats!" Worksheet.
Do p. 342 (1, 4, 7, 13)
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