Mathematics 131 -- Calculus for Physical and Life Sciences

A Balloon Trip with the Montgolfier Brothers

November 19, 2004

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.

Questions

1. Over what time intervals was the balloon rising? descending?
2. What apparently happened at t=40?
3. Next, let's try to answer a different and mysterious question, namely: What happened at the end of the trip? Did the balloon come safely back to ground level, get caught in a tree, or something else? To do this, we will try to estimate the total change in height over the trip using the information in the graph.
   • First, let's try to estimate the greatest altitude -- the total change in y over the interval t = 0 to t = 42.
   • What looks like a good estimate of the average velocity of the balloon between t = 0 and t = 10? Using your average velocity on t = 0 and t = 10, estimate the distance the balloon went up over that time interval.
   • 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 estimate the maximum altitude the balloon reaches. (How?)
   • Now, do the same for the downward portion of the trip -- estimate the total change in altitude over the downward part of the trip.
   • Using your work on the previous questions, What happened at the end of this flight? Did the balloon end up higher, lower, or at the same height it started? How can you tell?
   • Suppose now you want to really convince someone of your conclusion. Explain how you are using the information from the graph, justifying your reasoning and calculations, and showing that your conclusion is correct.