1. how lift, drag, thrust, and gravitational forces affect airborne objects,
2. how thrust forces are produced,
3. how energy, momentum, and mass figure into propeller, jet, and rocket propulsion.

1. As one of these pilotless planes maintains level flight on a windless day, gravity pulls downward on the plane and continuously transfers downward momentum to it. (A) What does the plane do with this downward momentum to avoid descending? (B) How does the plane actually carry out this task (i.e. dealing with the downward momentum)?

Answer: (A) The plane transfers the downward momentum to the air. (B) It carries out this task by deflecting the passing air downward--pushing downward on the air and transferring downward momentum to that air.

Why: To maintain level flight, the plane must always have zero vertical momentum. Gravity really does transfer downward momentum to the plane, exerting a never-ending downward weight force on it and thus an ongoing downward impulse. The plane has to transfer this downward momentum elsewhere or down it will go. So the plane pushes downward on the passing air and transfers any downward momentum it receives from gravity to that passing air. The air ends up descending instead of the plane.

2. Although the pilotless plane manages to maintain level flight through the still air, it continuously loses energy to that air. Look at the dynamics of the air itself to show that this energy loss must occur, even if the plane does not experience any viscous or pressure drag.

Answer: Before the plane passes, the air is stationary. After the plane passes, the air is moving downward. The air's kinetic energy clearly increases and that energy must come from the passing plane.

Why: When the plane deflects the passing air downward, it transfers more than just downward momentum to that air. It also transfers energy to the air. Since the plane's goal is to get rid of undesirable momentum, not energy, it does best by pushing as much air downward as possible. Large amounts of air will carry away the necessary downward momentum while extracting relatively little energy from the plane.

3. To remain above a particular city, the pilotless plane flies in a slow, level circle centered high above the city hall. What exerts the centripetal force that allows the plane to undergo uniform circular motion? Explain.

Answer: The air exerts the centripetal force on the circling plane. The plane experiences a non-vertical lift force that both supports it against gravity and pushes it toward the center of the circle.

Why: The plane obtains the centripetal force it needs to circle by deflecting the passing air outward as well as downward. The reaction force it obtains from this air is thus inward and upward. The inward component of the lift force causes the circling motion while the upward component keeps the plane from falling.

4. In order to circle above city hall, the plane has to lower its inside wing (the wing that's closest to city hall). The plane thus flies with a slight tilt--its outside wing higher than its inside wing. How does this tilt lead to the plane's circling motion?

Answer: By tilting its wings, the plane obtains a lift force that isn't vertical. Instead, that force is tilted somewhat toward the center of the circle and has a component that points toward the center of the circle. It is that centripetal component of lift that leads to the circular motion.

Why: The only thing that can exert a horizontal force on the plane is the passing air and this air can only push the plane centripetally by way of a lift force. Drag forces always act in the downstream direction, so they only push the plane backward, never to the side. Despite its horizontal aspect, this centripetal force really is due to a lift force.

5. To make efficient use of its limited solar energy, the pilotless plane flies relatively slowly. Its wings are highly curved and nonsymmetrical--they bow significantly upward in the middle, between their leading edges and trailing edges. (A) How does this wing shape affect the air pressures above and below each wing and enable the plane to fly so slowly? (B) How would this wing shape affect the plane's ability to maintain level flight while inverted (upside-down)?

Answer: (A) The highly curved wing shape increases its lift at low speed by creating a dramatic inward bend and low pressure on its top and a corresponding outward bend and high pressure on its bottom. The large pressure imbalance that develops even at low speeds produces enough lift to support the plane's weight. (B) When inverted, this wing shape would tend to produce downward rather than upward lift and the plane would have trouble flying upside-down.

Why: Highly curved and non-symmetric wings can generate increased lifts at low airspeeds. They do this by bending the airflows severely and generating particularly low pressures above and high pressures below. The cost is elevated drag at high speeds (not a problem in this case) and difficulty flying while inverted. While a severe increase in angle of attack (tilt) might yield an upward lift when the wings are inverted, the wings would be likely to stall instead. In short, the plane would have real trouble flying upside-down and might simply fall out of the sky.

6. To make efficient use of its solar power, the slow-moving plane uses big propellers rather than jet engines. These propellers push large volumes of air backward to obtain their forward thrust. In the process, this air picks up a small backward speed. While the same amount of thrust could be obtained with a jet engine that pushes small volumes of air backward and gives that air a large backward speed, the jet engine would be less energy efficient. Why?

Answer: Energy increases very rapidly with speed (in proportion to the speed squared) so that moving less air but giving it a higher speed involves giving that air more energy.

Why: When the plane pushes the air backward to obtain forward thrust, the plane is actually exchanging horizontal momentum with the air. Each second, the plane gives the air a certain amount of backward momentum and the air gives the plane an equal amount of forward momentum. The air can carry away this backward momentum either as a little air moving backward quickly or a lot of air moving backward slowly. However, this air also carries away kinetic energy, an unfortunate fact for the plane since it has to supply this energy. To minimize that energy transfer, the plane adopts the latter arrangement: a lot of air moving backward slowly. Since the air's energy is proportional to the square of its speed, slow moving air has far less energy than fast moving air.

7. The higher the pilotless plane goes, the more trouble it has maintaining level flight. With its solar panels providing steady and renewable energy, something else must be limiting its altitude. (A) What actually limits the plane's maximum altitude? (B) Why would having small rocket engines onboard only temporarily increase the plane's maximum altitude? (Neglect the small increase in maximum altitude that comes with reducing the plane's total weight.)

Answer: (A) As the altitude increases, the air through which the plane pulls itself becomes less and less dense. The plane's engines obtain less thrust by pushing that air backward and the plane's wings obtain less lift by deflecting that air downward. Eventually, the plane cannot raise itself any higher and reaches its maximum altitude. (B) While rocket engines could temporarily make up for this shortage of thrust and lift, the mass of rocket fuel would soon be exhausted and the plane would return to its earlier maximum altitude.

Why: To support itself against gravity, the plane must transfer all of the downward momentum it gets from gravity to the air. As the air becomes less dense, this transfer becomes harder until finally, at some high altitude, the plane can rise no further. Rocket engines could temporarily boost the plane upward, but they consume precious mass as they fire. The plane may be able to renew its energy via solar power, but it can't replace the mass that its rocket engines expel. After that mass is gone, the plane returns to its original maximum altitude. (Note: The missing weight of rocket fuel does increase the maximum altitude slightly.)

8. To leave the earth's atmosphere completely, the pilotless vehicle would have to rely on rocket thrust instead of airplane engines. But with no atmosphere to push against, how could even a rocket-powered pilotless vehicle propel itself forward?

Answer: The rocket engine pushes against its own fuel and that is all that's needed to obtain forward thrust.

Why: A rocket engine doesn't need anything external to push against. The very act of pushing against its own fuel causes it to obtain a forward push. The fuel accelerates backward while the rocket engine and the vehicle to which it's attached accelerate forward.