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!