Mathematics 133 -- Intensive Calculus for Science 1
Whole Class Discussion -- A Balloon Trip with the Montgolfier Brothers, Continued
November 22, 2005
The Story
The Montgolfier brothers, Joseph and Etienne, were pioneers in
hot-air ballooning in the late 1700's in France. If they had possessed
appropriate instruments, they might have left a record of one
of their early experiments as shown in the graph below. The graph
shows the vertical velocity v of their balloon as
a function of time t.
Today, we want to use the graph of the velocity
to try to understand what happened in this balloon trip.
Questions
- Over what time intervals was the balloon rising? Over what intervals
was it falling?
- What happened at t = 40? (Think about how a hot-air balloon
works.)
- At what time was the greatest altitude achieved on this trip?
- Over what intervals was the acceleration positive?
- What apparently happened right before t = 60?
- What were the smallest and largest velocities
of the balloon for t between t = 0 and t = 10?
Use this information to give two estimates of the distance travelled
on this interval -- one definitely smaller than the actual distance
travelled, and one definitely larger.
- Now do the same on the intervals
t = 10 to t = 20,
t = 20 to t = 30,
t = 30 to t = 40, and
t = 40 to t = 42.
- Use the results of the previous questions to give two estimates of the
maximum altitude the balloon reaches -- one definitely an
overestimate and one definitely an underestimate. (How?)
- Now, do the same for the downward portion of the trip -- give a
definite overestimate and underestimate for
the total change in altitude over the downward part of the trip.
- Using the work on the previous questions,
did the balloon end up higher, lower, or at the same
height it started? How can you tell?