First lets write down the formulae for the motion which we analyzed on
the inclined air track. When I started off the clock the cart
had zero velocity and it was at the origin for our
x-axis. This implies that
This implies then that
Rather than use the acceleration that we determined graphically
I'll use one of our primary measurements, namely the time at
which we reached a distance of 4 m from x=0. To determine the only
unknown remaining constant, a I solve equation 8
for the acceleration:
The expression makes sense since a gets larger for smaller
t and larger x. Putting in numbers I obtain:
A result which is consistent with what we derived from
our graphical analysis.
Now I would like to emphasize that by suitable manipulation
of eq 2 you can handle any problem involving
motion with constant acceleration. However you will find yourself
often repeating certain symbolic manipulations of formulae. To avoid
this it can be useful to use the following expressions which
are easily derived once and for all as described in the book:
As compared to the book you will notice that I have suppressed
assuming that my clock always runs from zero time (
).
The last of these formulae is quite obvious when we look at the
parabolic x(t) curve we got for the inclined track. You will find
that it comes in very handy in problem 30 due this Monday.
The second of these formulae is also very important and comes
in handy. It is always the one to use when the time taken to perform
the motion is not of interest. An example: