Reading Assignment: Chapter 6 of Tippens.

We introduce two new kinematical equations today. The first provides us with a mathematical expression to find average velocity under a constant acceleration given the starting and ending velocities. The second combines this equation with our first kinematical equation for position in terms of velocity and time. Thiese two equations can be written:
average v = (v + v0)/2
x = x0 + v0 t + at^2/2.
Here, x0 and v0 mean initial position and initial velocity, respectively, and t^2 means time-squared. (I can't do subscripts and superscripts easily in html, yet.) The second of these equations is Galileo's Third Kinematical Equation and relates (x, a, t). Recall that the first two equations relate (x, v, t) and (v, a, t). These are:
x = x0 + vt
v = v0 + at,
where x0 means initial position and v0 means initial velocity. The equations that contain acceleration are true only if acceleration is constant.
One of the prime examples of motion under constant acceleration concerns bodies falling under the influence of gravity alone. Note that bodies thrown into the air are themselves falling during their entire flight as they lose speed on the way up and gain speed on the way down.
We view the second `Mechanical Universe' video today entitled `The Law of Falling Bodies'. This law is attributed to Galileo and can be stated as follows:
In a vacuum, all bodies fall with the same contant
acceleration.
There are some differences between the contents of the video and our discussion in class.These differences are principally terminological and notational:
(1) In the video, positive is taken downward and negative upward. So, the
acceleration of gravity is positive. That is, a = g. We use a coordinate
system in class consistently in which negative is downward and positive is
upward, so a = -g, where g is the magnitude of the acceleration of gravity.
Near the Earth's surface g is about 9.8 m/s^2.
(2) The video uses s to mean displacement and assumes that the bodies in
question are starting from rest -- that is, v0 = 0. For us, displacement
is x - x0. Thus, the second, third, and fourth equations above are written
in the video as:
s = at^2/2 = gt^2/2
s = (avg v) t,
v = at = gt.
(3) The video discusses the nature and importance of differential calculus in
the development of the kinematical equations above and the understanding of
the nature of gravity. This is beyond the scope of this class, but you should
know that the calculus was co-invented in the mid-to-late 1600s by Isaac Newton
and Gottried Wilhelm von Leibniz. More information about Newton can be obtained
by following the link-of-the-day for Lecture 3 and about Leibniz by following
today's link-of-the-day.
Link for the Day: Gottried Wilhelm von Leibniz

Questions or comments should be addressed to Mike Ritzwoller at ritzwoller@phys-geophys.colorado.edu 

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