Mathematics 136 -- Calculus 2
Discussion 1 -- A Balloon Trip with the Montgolfier Brothers
January 24, 2014
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 same graph of the velocity
to try to estimate the total change in height
over the trip using the information in the graph.
Discussion Questions
- Over what time intervals was the balloon rising? Over what intervals
was it falling?
- What apparently 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 your work on the previous questions,
did the balloon end up higher, lower, or at the same
height it started? How can you tell?
Assignment
Group write-ups due in class on Tuesday, January 28.