Those 16 rectangles hanging below the balloon are solar panels on this top side. The payload for "science" is between them. The balloon is nearly spherical, a little distorted by the load. In the foreground the pilot's helmet and shoulder are on the right, and the U2 wing extends outward. Of note, just below the tip of the wing, the shadow of the aircraft falls on the balloon. The curvature of the Earth is apparent where the edge of our planet is visible left of the helmet at top center. Analysts identified the location underneath by the pattern of rivers and forest. A few hours later, the balloon crossed over Kentucky on its path to a demise off the shore of South Carolina.

This homework is an exercise in simple physics to see what we can tell about the balloon from the image.

1. Read about U2 aircraft on Wikipedia at https://en.wikipedia.org/wiki/Lockheed_U-2. What is the length of the airplane? That's the distance from nose to the end of the tail.

2. Use the length of the shadow on the canopy to estimate the diameter of the balloon. Light from the Sun is shadowed by the body of the airplane and the rays are nearly parallel as they continue on toward the balloon. Make an argument why the line to the shadow is nearly perpendicular to the long dimension of the airplane where it falls on the balloon, and is therefore a useful measuring "stick".

3. Calculate the volume of the balloon in cubic meters.

Earth's atmosphere varies in density with altitude, as does its temperature. The surface temperature is a consequence of how it radiates energy into space to balance the energy it gets from the sun. Global warming results from an increase in CO₂ concentration in the atmosphere because this molecule and others absorb the radiated light and inhibits the cooling rate, requiring an increase in temperature to reach equilibrium. A reasonable assumption is that the temperature is close to 275 K, a few degrees above the freezing point of water. The scale height of a planet's atmosphere is set by the idea that the pressure must balance the weight of the air above it. It takes a little calculus, and the outcome for constant temperature in an ideal gas is

P=P0exp(z/H)

where z is the altitude above the surface, H is the scale height, and P0 is the surface pressure. The scale height is given by

mgH=kT or H=kT/mg

where m is the mean mass of a molecule in the atmosphere, g the acceleration of gravity, k Boltzmann's constant, and T the temperature in kelvin. Seehttps://en.wikipedia.org/wiki/Scale_height for more detail.

4. Inspect the units on the right hand side of the equation for H. Show that the dimension of H in this formula is meters, given the SI values for k, T in kelvin, m in kg, and g in m/s².

5. Use the nitrogen molecule as representative of Earth's atmospheric composition. It has two nitrogen atoms, which will tell you the mass of the molecule. Use 275 K as a temperature, and find the scale height of the atmosphere.

6. The U2's ceiling is 80,000 feet (24,000 m). News reports the altitude of the balloon at the time of this image was 18,000 m, consistent with down-looking photo we see here. What was the pressure of the atmosphere at this altitude?

7. From the pressure and the ideal gas law, P = n k T (n is the number of molecules/volume in this expression), find the number of molecules per volume at this pressure.

8. How much mass was displaced by the volume of the balloon? That's the mass of the air that would have been in the space now occupied by the balloon.

9. Archimedes principle says the the balloon is buoyed up by the weight of the displaced air. How much lift is that, in newtons?

10. Why does the capacity of a balloon to lift a given load diminish with altitude? Use this to explain why a party balloon does not rise to the fringes of outer space.