Mathematics 133 -- Intensive Calculus for Science 1

Discussion 4 -- A Balloon Trip with the Montgolfier Brothers

October 17, 2001

Background

One important example of second derivatives comes from the study of straight line motion. If f(t) represents the position of a moving object as a function of time, then the first derivative f'(t) gives the rate of change of position with respect to time or velocity. The second derivative f ''(t) is the rate of change of velocity with respect to time, or acceleration. The units of acceleration are (distance/time)/time, or distance/time2 (for instance miles/hour2 or meters/second2).

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.

Discussion Questions

  1. Over what time intervals was the balloon rising? Over what intervals was it falling?
  2. What happened at t = 40? (Think about how a hot-air balloon works.)
  3. At what time was the greatest altitude achieved on this trip?
  4. Over what intervals was the acceleration positive?
  5. What apparently happened right before t = 60?

    Assignment

    Group write-ups due in class on Monday, October 22.