Teachers using this material in class may obtain a list of solutions by regular mail, by sending a personal request on school letterhead to
Dr. David P. Stern, 31 Lakeside Drive, Greenbelt, MD 20770, USA

1. [2]       We are taught to find north (at locations north of the equator) by looking for the direction of the pole star Polaris. Mars also has north and south poles, and its rotation axis is inclined to the ecliptic by about as much as the Earth's rotation axis, turning around it in a little more than 24 hours. Why would Polaris not be a good indicator of north, for someone at night on the surface of Mars?

2. [2]       The cut-out paper sundial provided in section 2 of "From Stargazers to Starships" is designed for latitude 38 north. If you took it to a location 38 degrees south of the equator, how could be adapted to work there--or could it?.

3. [3]       In middle latitudes, the longest day and shortest night of the year are at the summer solstice (21 June or near it). The shortest night and longest day are at the winter solstice, around 21 December.
         In between, days gradually get shorter and nights longer. How are day and night at the poles themselves, at solstices and (the subject of this problem) on in-between days?

4. [3]       You are taking a daytime flight from New York to Los Angeles and another daytime flight back. You like to sit by the window, but do not want the Sun to shine directly at you. Given a choice, which side of the airplane--right or left (facing forward)--would you prefer for each part of your trip?

5. [3]       Today 1s the 21st of June, and you are watching the sunset (1) from the US (2) from Argentina, south of the equator. In each case, do you see the Sun set (a) exactly to the west (b) to the north of west (c) to the south of west ?

   Hint: Use the drawing of the path of the Sun across the sky.

6. [3]       North of the equator, the longest day of the year is on or near 21 June. South of the equator, the longest day is on or near 21 December. When is the longest day on the equator?

   Hint: Consider the rotation of the celestial sphere. How do the stars (seem to) move on the equator? After that, consider where the Sun might be in the sky.

7. [3]       The first clocks were built in Europe, and it has been said that the "clockwise" direction in which clock hands rotate was chosen because that was the way the shadow on a sundial rotated there. All shadows--e.g. of flagpoles--also rotate there in that same direction.

   Suppose we were in Argentina, south of the equator, where the Sun at noon passes to the north, not to the south. Will the shadow of a flagpole (or the one on a sundial) rotate clockwise or counterclockwise? Explain your choice.

8. [4a]       Astronomers using very sensitive telescopes at the limit of their resolution reported in January 2002 that they had observed a planet around the star Sge 13, located 58 light years from Earth. It is much larger than the planets of the solar system--about 65 times the mass of Jupiter, orbiting at a distance somewhat greater than Jupiter's. Still, it is probably too small to produce heat like the Sun and shine with its own light; the only way to see the planet (you can assume) is by the light of its own sun, reflected from it, the way the Moon is seen. A picture taken through the telescope shows a big spot of light around the star (the telescope widens it--the star itself is much smaller), and a little second spot, in the position of 7 o'clock on the clock dial, marking the position of the planet (draw it!).

         Draw a straight line through the point where the planet was seen and the center of the star, and extend it on both sides of the star. Suppose we are viewing the planet's orbit edge-on: then at any time the planet will always appear somewhere on that line.

         Two possibilities exist then: the planet can be on the half of its orbit which is closer to Earth than the star, or the planet can be in the half that is more distant. Which of the two possibilities do you favor, and why?

9. [4a]       One evening, before the sky became dark enough to show many stars, I walked out of the house and saw a remarkably bright star next to the full Moon. What is it, most likely?

         Hint: The Earth, planets and Moon orbit very close to the same plane--as if they were all moving on the same flat table. For reference we assume that "table" is the plane of the Earth's orbit, the plane of the ecliptic.

10. [4a]       The space shuttle, orbiting the Earth at low altitude, makes one circuit in about an hour and a half. For higher circular orbits, the orbital period gets longer, until at the Moon's distance it is close to one month, the period of the Moon.

          At some distance in between the orbital period is exactly the same as the rotation period of the Earth, nearly 24 hours. A satellite in such a "synchronous" orbit above the equator--the distance is about 42,000 kilometers--will always be above the same spot on the equator, tracking it as the Earth rotates. Obviously, that is a good location for satellites monitoring the weather above some part of Earth, or communication satellites serving a specific locality, such as the US.

          Question: do such satellites spin around their own axis?

11. [4a]       Viewed from Earth, the Sun, Moon and stars rise and set with various periodicities, all of them close to 24 hours.
          Suppose you are an astronaut on the Moon, watching the Sun, Earth and stars. How frequently do they rise and set?

12. [5]       Local time in Paris is 6 hours ahead of that in Washington DC. The Concorde supersonic jetliner takes off from Dulles airport in Washington on a Monday at 8 pm local time, and reaches Paris in 3 hours. What is the local time and the day of the week when it lands? The airplane then takes off at noon for the return trip. When it lands in Washington, what is the day and the local time there?

13. [5b, M-12]       The two stars at the front of the constellation of the Big Dipper (Ursa Major--also "the Plough" or "the Big Bear") are known as the "pointers, " since they point towards the pole star (see in http://www.phy6.org/stargaze/Spolaris.htm, the picture of the Alaska state flag). The one closest to the pole star is the brightest star in the constellation, and since stars are assigned Greek letters in order of brightness, it is known as "Alpha Ursa Majoris. "

          An observer using a cross-staff has marks on the cross-piece spaced 8 inches (8") apart, i.e. each of them is 4" from the staff. The observer holds the cross-staff as shown in
                http://www.phy6.org/stargaze/Scrostaf.htm
    and slides the cross-piece until one mark covers the pole star and the other the front guide star. Taking down the coss-staff, the observer finds that the line between the marks is 15.5" ahead of the observing eye.

    What is the angular separation angle q between the two stars?

14. [5]       (You need a star chart to answer this one)   For a person standing on Mars north of the equator, at night, the following prescription for finding north may be given:
    "No star marks the north pole of the sky on Mars, but the position of that pole may be estimated as follows. Look for the 5 bright stars which mark the constellation of Cygnus, the swan. They form a cross--sometimes known as the "Northern Cross." Locate the center of the cross, and go from there to the top of the cross--the brightest star in Cygnus, known as Deneb. Follow that line a distance equal to the distance between the center of the cross and Deneb: the point you have reached is close to the pole of the heavens."
    
               Question: What are the declination and right ascension of that point of the heavens (give both in degrees), and what is its connection to Mars?

15. [6]       A satellite in synchronous Earth orbit (like most communication satellites) orbits the Earth's equator in a circular orbit which always keeps it above the same spot on Earth. Is its orbital period 24 hrs (solar day) or is it 23 h 56 min 4 sec (sidereal or "star" day)?

16. [8a]       An astronaut stands on the Moon, at altitude h. Assuming the Moon is spherical with radius R=3476 km, what is the distance D to the horizon? How far is the horizon for an astronaut on flat ground, with eyes 1.5 meters above the surface?

17. [8b]       You sit by the window of a jetliner on its landing approach and note that the shadow of the airplane. If it flies very high, of course, it appears from the ground too small to cover the Sun, and casts no shadow. To cover the Sun, the parallax of the airplane, as seen from the ground, must be at least as big as that of the Sun.

          The airplane measures 50 meters wingtip to wingtip--but if its wings just barely extend aross the Sun's disk, too little is covered to produce a dark spot. So as an obstacle to sunlight, assume the airplane has a 30 meter size.

          What is the largest distance from the ground at which the airplane can still produce a shadow?

    Assumptions: For this crude calculation, you may assume that the Sun's disk covers 0.5 degrees, and that the length of a circle is about 6 times the radius. (That is the approximation used in the Bible: see 1st Kings, ch. 7 v.23).
          You may also assume the Sun is directly overhead; then the distance airplane-shadow is also the altitude of the airplane.

18. [8c]       The average distance of Mars from the Sun is 1.52 astronomical units (AU), where 1 AU, about 150,000,000 kilometers, is the mean Earth-Sun distance. The average distance of the Moon is about 385,000 kilometers from the center of the Earth.

          Assume all orbits are circular (actually the orbit of Mars is somewhat elliptic and it gets within 1.4 AU of the Sun). Viewed from Mars, what is the Earth-Moon separation, in minutes of arc (1 degree = 60 minutes or 60')? If the average eye can separate objects 1' (1 minute of arc) apart and the difference in brightness is no obstacle, could a viewer from Mars tell the two apart when Earth and Mars are at their closest, and Earth and Moon appear furthest apart?

    Hint: The angle is so small, that a straight Earth-Moon line and a circular Earth-Moon arc centered on Mars may be assumed to have the same length (as in section 8-c, a somewhat similar calculation).

19. [13]       The space shuttle takes off vertically and gradually tilts over, accelerating all the time.
          Ignoring the initial vertical flight portion, we find its horizontal acceleration averages 2 2/3 times the acceleration due to gravity, approximated here as 10 meters/sec2. The horizontal acceleration lasts 5 minutes, then the rocket burns out and the shuttle is in orbit.

       (a) How much distance has the shuttle covered at burnout? (use formulas in section 13, using the acceleration a in place of g)

        (b) Assume the straight-line distance from the launch site at Cape Canaveral to the shuttle at burn-out is the same as the one in (a) (which, strictly speaking, is measured along a curved arc). If the shuttle's altitude at burn-out is 350 km above the ground, can it be seen at that time (through telescopes) from Cape Canaveral? Use the formula of section 8a.

20. [13]       In a dripping faucet, the water stream near the faucet seems solid, but further down it breaks up into droplets. Why?

    Hint: With g=10 m/s2, the distance S in meters, fallen by the water in t seconds, is S = 5 t2. If S is measured in millimeters, however, S = 5000 t2.
          Consider an amount of water fallen in a given period of 1/100 sec. What length does it occupy it comes out from the faucet, and again, 1/10 seconds after it began falling?

21. [15]       A ball rolls down a sloping board, and is assumed to lose a negligible amount of energy to friction. A smooth block of metal slides smoothly a slick ice-covered plate of the same slope and dimensions, also losing no appreciable amount of energy in the process. Both start together at the same height: which arrives first at the bottom? Give a reason for your choice!

22. [15]       As I was mowing my lawn, my push-mower stopped against an obstacle, a root of a tree growing along the surface, but its round side sticking out, creating a little hill, which stopped the mower from advancing.
          I rolled the mower back a short distance, then forward again. The extra distance allowed it to develop a fair speed as it rolled towards the root, and with that speed it easily went over the obstacle. Explain in terms of energy what has happened.

23. [15].       In a microwave oven, radio waves with very high frequency (microwaves) are piped from a powerful transmitter into the interior of the oven, whose walls reflect them almost perfectly. They then bounce around until they are absorbed by the food inside the oven.
          My new microwave oven has a label stating it produces 1200 watts of microwave energy . I place in it a cup with 220 gram water at room temperature, 250 deg. centigrade. How long will the oven have to heat the water before it boils, at 1000 centigrade? Neglect the heat needed to heat the cup and any heat lost by the water while it is being heated.
    
                Numbers you might need: one (small) calorie is equal to 4.18 joule, and is defined as the amount of heat required to raise the temperature of 1 gram of water by 1 degree centigrade. (Please note: the "calories" in which food energy is measured are "kilocalories" (Kcal) or "large" calories; one Kcal = 1000 cal.)

24. [15]       The solar constant is about 1.3 kilowatt/m2 and one (small) calorie equals 4.18 joule. A pan with water in it to the depth of 1 cm stands in the sunlight and receives 50% of the solar constant of power (the power is reduced, not just because the atmosphere absorbs or reflects some of the Sun's light, but also because the pan has to be horizontal while the Sun usually shines at an angle). How long does it take the water to heat up by 10 degrees centigrade--if we neglect the heat it loses to the air and the pan?

          Note: The result does not depend on the area of the pan, because if you double that area, both the water that needs heating and the rate at which sunlight supplies it with energy are doubled. Assume therefore an area of 1 meter2. A calorie is the amount of heat required to heat 1 gram of water (=1 cubic centimeter) by 1 degree centigrade.

25. [15]       A girl throws a stone from a bridge. Consider the following ways she might throw the stone. The speed of the stone as it leaves her hand is the same in each case.
    
          Case A: Thrown straight up (but missing the bridge on its way down)
          Case B: Thrown straight down.
          Case C: Thrown out at an angle of 45 degrees above horizontal
          Case D: Thrown straight out horizontally
    
    In which case will the speed of the stone be greatest when it hits the water below?

    Hint: Don't try to calculate. Think!

26. [15]       The food consumed by an adult human typically provides 3000 calories/day. Assuming all this is sooner or later converted to heat, what is the average heat production rate of such a person, in watts? How does it compare to the heat produced by a 100-watt lightbulb?

               Please note: the "calories" in which food energy is measured are "kilocalories" (Kcal) or "large" calories; one Kcal = 1000 cal. = 4180 joule.

27. [15]       Seward, the port at the end of the Alaska Railroad, has steep Mt. Marathon towering just behind it, to a height of about 900 meters. Every 4th of July a footrace is held, from the town to the top of Mt. Marathon and (with a lot of sliding!) back. The current record is 43 minutes and a fraction.

    Q.:       A runner weighing 60 kg reaches the top of Mt. Marathon in one hour. Approximating g = 10 m/s² and one horsepower=750 watt (accurate value is 746), how many horsepower must that person develop just to overcome gravity, on the average?

28. [16] A large sledge hammer is used to split limestone rocks. A rock will split if hit at a suitable spot, provided the force is large enough: the large force starts a vibration which splits the layers. Obviously, we need hit it with the greatest force possible.

          The hammer has two faces. One side is tapered like a blunt wedge, rounded at the end. The other face ends in a flat surface. A person splitting rocks finds that hitting with the flat end is more productive. Why?

29. [18]       According to an article in "Smithsonian", a jumping flea accelerates at 140 g. Does this mean the flea's body is unusually strong--compared to, say, the human body?

30. [18]       NASA's Mars Climate Orbiter was lost by dipping too deep into the atmosphere of Mars. According to the Washington Post (1 October 1999), this happened because the small control rocket gave the orbit a wrong final adjustment. The article claimed that the two goups controlling the orbit--of the Jet Propulsion Laboratory and of a NASA contractor--mixed up the units in which the rocket's thrust was specified. The thrust was specified in Newtons but the controllers assumed the number referred to pounds of weight.

                Which of the two units is bigger, and by how much? Assume that a pound is 0.4536 kg and that g=9.81 m/s.

31. [18]       Newton's theory of gravitation was allegedly inspired by the sight of an apple dropping from a tree (see sect. #20 here). When apples are ripe, their stem weakens until they finally drop.

          As is well known, if you shake an apple tree, ripe apples will drop out of it. Why do they drop? (Apply Newton' laws!)

32. [18]       Sometimes old factories are torn down to make way to new construction, and as part of the process, brick factory chimneys are toppled by a charge of dynamite. As the chimney falls down, to the side, it is generally observed to break in mid fall. Why? If the chimney (from where you view it) appears to topple clockwise, does its broken-off top break off clockwise or counterclockwise?

                Guidance: As the chimney topples, it rotates around its base, at first slowly, then faster and faster. Its parts still accelerate, but it is not so simple.

          First of all, rotation around a pivot is not free fall. When the chimney makes an angle A to the vertical, its parts move as if they were on an inclined slope of angle A to the horizontal (make a drawing!) and therefore their expected acceleration is only (g sinA), gradually increasing as the motion becomes more and more vertical.

          Secondly, the different parts of the chimney are not independent. If they were independent (each on its own long thin rod connecting it to the center of rotation, if you can imagine it), each would accelerate at this point by (g sinA). But they are connected! That means, pieces near the center accelerate by less than (g sinA) [zero at the center], and pieces near the top, by more than (g sinA). Somewhere in the middle, at one distance only, the acceleration is exactly (g sinA), the value gravity alone would prescribe. So.... ?

33. [19]       If the equatorial radius of Earth is 6378 km, its rotation period 23 hr. 56 min, by what amount Dg does the centrifugal force change the locally measured acceleration of gravity, usually taken as g = 9.81 meter/sec² ? Is the local value of g increased or reduced?

34. [21]       (a)       The outer moon of Mars, Deimos, orbits the planet at about 23,500 kilometers with a period of 1.26244 days. Our own Moon, at about 385,000 kilometers, takes 27.32 days to orbit Earth. Assuming both orbits are circular, what is the ratio of masses M_Earth/M_Mars ?

    (b)       The result you get may disagree somewhat with the one you find in textbooks. Can you guess the reason? [Hint: our Moon is BIG!]

          Guidance:       In section 21, Kepler's 3rd law for small objects (e.g. satellites) in circular orbit around Earth is derived as

    T² = (4p²/g R_E²) r³       With R_E the Earth radius and g the acceleration due to gravity at the surface of Earth. We need now generalize this for any planet.       If M is the mass of Earth and G the constant of gravitation in Newton's theory (section 20), an object of mass m on the surface is subject to a force
    mg = GMm/R_E² so
    g R_E² = GM and the formula becomes
    T² = (4p²/GM) r³ We bring M to the top by multiplying both sides by GM/4p²
    T² GM/4p²= r³ and we make it the only variable on the left (all other terms there are constant) by dividing both sides with T²
    GM/4p²= r³ / T²       The only quantity the connects this with Earth is the mass M. If that is replaced by the mass of another planet, it will hold for that planet instead.

          Let subscript 1 refer to Earth, subscript 2 to Mars. You now have to express (M₁/ M₂). This can be done in different ways, but the best is probably through the ratios (T₁/ ₂) and (r₁/ r₂). Doing so not only avoids dealing with large numbers, but you can measure T and r in any units you please, since the ratio does not depend on units.

35. [22d]             An airliner of mass m races down the runway at full power, accelerating at a constant rate a. When it reaches enough speed for take-off, the pilot rotates it so that it now climbs upwards at an angle of 10 degrees to the horizontal. The force generated by its engines ("thrust") remains the same as before, but because the airliner is now climbing, it moves at a constant velocity.

          In flight, its air resistance is 0.1 times its weight. (Both air resistance ["drag"] and "lift" are roughly proportional to air density and velocity squared, so they are proportional to each other; in ordinary flight, however, lift equals weight, so drag is approximately proportional to weight). Assuming that air resistance can be neglected during acceleration on the ground, how does the magnitude of its acceleration a on the ground compare with the acceleration of gravity?

36. [22d]       Suppose that the water resistance to a moving ship, like air resistance to an airplane, increases in proportion to v². After all, a moving ship also encounters resistance from the water it displaces and pushes aside, presumably with a velocity proportional to v.

          The navy has a troop transport with top speed v=12 miles per hour (MPH) and engines that deliver to the propeller shafts a power P=10,000 horsepower (HP). It would like to build a similar ship for fast emergency troop movements, capable of top speed of 30 MPH. If all other factors are the same, how much power should the engines of the new ship be capable of?

          Hint: Assuming a perfectly efficient propeller, what should P be proportional to?

37. [23]       The amusement park "Great America" in Santa Clara, California, has a ride named "the drop zone. " Riders are strapped into seats that ride on vertical rails, attached to the side of a tall tower (you may denote its height by 3H). The seats are hoisted to the top of the rail, and then are allowed to fall along the rail. For 2/3 of the height, friction is negligible and the seats are essentially in free fall (a frightening or bracing experience, depending on the rider). Over the last 1/3 of the distance, the seat is slowed down, and it stops just short of the ground.

    How many times their weight is the average force on riders during the deceleration ?

38. [23]       A jet engine revolves at 6000 revolutions per minute (RPM). Its burning fuel drives a turbine whose blades are set in a wheel with radius 50 centimeters. If a blade has mass m, how does the centrifugal force on it compare to the blade's weight, which is mg? (take g = 10 m/s²)

39. [23]       A pilot on an airplane high above the ground wants to "loop the loop." Two choices exist--looping upwards (a) or downwards(b). The choices:
    • (a) Gather speed and then start climbing sharply upwards at increasing steepness along a trajectory which ultimately bends over, so that the airplane passes the highest part upside-down, then completes the loop and ends up in level flight again.

    • (b) Dive downwards at an increasingly steep angle, ultimately bending over, so that the airplane passes the lowest part of the loop upside down, then with the speed gained (and the engine) climb again towards the vertical, ending up in level flight again.
    
        To complicate matters, the airplane's engine is of the old kind, relying on a float with a needle valve to regulate the mixture of gas and air. That engine will stop if turned upside down. Should the pilot try 1. (a), 2. (b), 3. Either is safe, or 4. Neither is safe. Give your reasons!

40. [23a]       (a)       An amusement park carrousel has seats hanging from chains. If they go in circle of radius 8 meters, and hang at that time at a 30-degree angle to the vertical, What is the number N of turns per minute that the carrousel makes?
         You may approximate sin30=0.5, cos30 = 0.866, g = 9.81 m/s².

    Hint: This is best solved by resolving both gravity and the centrifugal force into components parallel and perpendicular to the chain. Since the chain hangs at a constant angle, the force components perpendicular to it balance each other, and must therefore be equal. A drawing may help here.

    (b)       When the carousel starts up or slows to a stop, in each case, would the chains slant forwards or backwards? (That is, would the seat be ahead of the point from which it hangs or lagging behind it?)

41. [24]       In calculating the motion of a satellite around Earth, do we need to take into account the centrifugal acceleration as the Earth orbits the Sun (assume that orbit to be a circle)? Justify your answer. (This is a tricky question!).

42. [24]       Imagine yourself rocket scientist, designing a multi-stage liquid-fuel rocket for orbiting space payloads. The fuel tanks on your rocket have outflow pipes at their bottoms, and through them the fuel flows to pumps, which inject it into the engines.

          You expect no trouble from the first stage. On the launching pad, thanks to gravity, the fuel settles to the bottom of the tanks, from where it is supposed to flow out. Later in flight, the rocket's acceleration causes an inertial force which pushes the fuel even harder in that direction. The pumps will never lack fuel!

          Then the first stage burns out and is dropped, and the second stage is supposed to ignite. But will it? A soon as the first stage quits, the remaining rocket is coasting under gravity alone, like the airplane simulating weightlessness, and like the airplane, it is in a weightless state. The fuel will not settle to the bottom, but may instead form a big blob (or a number of blobs) floating inside the tank. Pressurizing the tank won't help--in the absence of gravity, the gas cannot tell the difference between up or down. How do you solve it?

43. [24]       In the rotating globe in section #24, three points with the same longitude are drawn, north of the equator: point A closest to the equator, point C closest to the pole, and point B between them.

    Suppose that meteorological processes raise the pressure at point B, causing air to flow away from it ("a high pressure system" in weather forecasts). Will that air swirl clockwise or counterclockwise? Give your reasons.

44. [S1-A]       If you followed the tracks of hurricanes, you may have noticed that they usually start in the Atlantic ocean north of the equator and drift westwards, gaining strength. Gradually their track curves northwards (often above the Caribbean Sea or the Gulf of Mexico), then as their motion continues northwards it curves back eastwards, often along the US eastern seaboard. (Check a map to clarify all this.)

    Could you give a general, qualitative explanation of this behavior using what you learned about global air flow?

45. [S4]       When a beam of light hits a pane of glass with parallel light, the light inside gets bent. As noted on the web page, different colors of the rainbow may be bent by different angles. However, since the beam is bent on its exit by exactly the same amount as it is bent on entering, only in the opposite direction, all its parts emerge again moving in the same direction as before. Therefore, it was reasoned, a flat pane of glass does not split light into colors the way a prism does.
          All the above is true. However, if you look through very thick glass, you will see fringes of rainbow color at the edges of bright objects. Why?

          (Hint: draw the path of two rays of light through a thick glass, belonging to different colors and therefore bent by different amounts inside the glass. What about the edges of the pane?)

46. [S-6a]       If the solar wind needs 5 days to cover one AU--one astronomical unit, the mean Sun-Earth distance (about 150 million km) and the Sun rotates at 13.5 degrees/day, calculate the equation of an equatorial magnetic field line of the Sun, in polar coordinates (r,f) in the Sun's equatorial plane. Measure angles in degrees and distances in AU, and as a simplification, assume that the motion starts from the center of the Sun (as if the Sun was very, very small).

47. [S7]       The fuel of the space shuttle is liquid hydrogen. As the hydrogen burns, it combines with liquid oxygen (and produces steam!). How can so many protons--nuclei of hydrogen--stay together in the fuel tank, without combining to helium?

48. [25].       By Newton's 3rd law, a spacecraft in free flight (no outside objects involved) cannot accelerate or decelerate (or change direction) unless it throws out some mass in the appropriate direction, e.g. the exhaust jet of a rocket.
          A version of the 3rd law also applies to rotations. If any part of a spacecraft starts rotating around an axis clockwise (as seen from some given observation point) the rest of it will rotate counterclockwise around the same axis, for as long as the first part keeps rotating, The rate at which the spacecraft rotates depends on the rotation of the rotating part, its mass and the average distance of that mass from the axis of rotation (the three multiplied together give the "angular momentum").
    [Note: There exists no unambiguous definition of "clockwise" and "counterclockwise." A rotation that appears "clockwise" when watched from one side, is "counterclockwise" when watched from the opposite one. On a transparent clock: viewed from the front, the hands move clockwise, but when viewed from the rear they move counterclockwise. You can demonstrate this by moving a finger around clockwise around a plate and having someone standing behind it tell you how it appears from there.]

    Now to the question: can a free-flying spacecraft invert its position, i.e. flip by 1800, without firing any jets? If yes--describe how, if no--explain why.

49. [26]      A suspended block of wood with mass M=1000 gram is used as a ballistic pendulum. It is hit by a bullet of mass m=5 gram and rises 45 cm before swinging back again. How fast did the bullet travel?

    Hint: From the 45 cm rise and the conservation of energy (section 15) , derive the velocity V with which the block of wood started moving (you may denote its mass by the letter M in this part of the calculation).

    Then, using conservation of momentum (section 18b) derive the velocity v of the bullet.

50. [27]       (a)       Sound moves in air (at ordinary temperatures) at about 1/3 kilometer per second. Aircraft speeds are often given in Mach units, 1 Mach=speed of sound, because at Mach=1 air resistance rises sharply. A spacecraft in low Earth orbit moves at 8 km/sec. How many Mach is that (calculated for the lower atmosphere) ?

    (b)       Early pictures of space rockets, from before the space age, show them tapering in the back, like an airship, and with wings to help support them. Actual rockets usually have a cylindrical shape and do not have any wings. Can you guess why?

51. [27]       The upper stage of an unmanned rocket is accelerated to 5 times the speed of sound by its first-stage rocket, which then separates and falls off.
                We have two choices of a first stage: one achieves its velocity in 1 minute, the other in 2 minutes. Which one needs to use less fuel? (1) The first (2) The second (3) Both are equal.

    Hint: What keeps a rocket from falling?

52. [29]       A spaceship of mass m moving at 8 km/s reenters atmosphere. How many calories are dissipated? If each kilogram needs 0.1 (large) calorie to heat up by 1 degree, and if no heat is lost otherwise, to what temperature does the object heat up? Can you guess what phenomena and processes reduce that temperature?

53. [30]      
    (a)       Why are there two railroad carriages in SHARP?
    (b)       Why does the car behind the secondary barrel weigh 10 times as much (100 tons) as the one behind the main barrel (10 tons)?

          Hint: Both questions involve conservation of momentum (sect. 18b). Question (b) is best solved as an actual example: suppose the piston in the main barrel has mass M=400 kilogram and moves at velocity v, while the projectile has mass m=1 kg kilogram and comes out at velocity V.

    Assume for simplicity that all the kinetic energy of the 400 kg mass is absorbed in compressing the gas, and all of it is then transferred to the projectile.
         --How many times is V larger than v?
         --How does the momentum of the 400-kg block compare to that of the projectile?

54. [34]       A satellite in an equatorial synchronous orbit (approximate orbital period 24 hours, distance 42000 km) matches the rotation of the Earth and always stays above the same spot on Earth.

          Does the same hold for a satellite in a 24 hour orbit inclined to the Earth's equator by 30 degrees? To what extent will such a satellite return to positions above the same point? Explain.

55. [M-11]       Show that cos x = 2 cos² (x/2) - 1 and sin x = 2 cos(x/2) sin(x/2). From these derive the sine and cosine of 7.5°.

    (Given the sine and cosine of 7.5°, you can now the table of sines and cosines in section M-9, which contains functions of 15, 30, 45 ... degrees. Using the formulas of M-11, you can add the functions of 7.5, 22.5, 37.5 ... degrees as well.)

56. [M-12]       Using the results of section M-12, derive a table of tangents from 0 to 75 degrees, in steps of 15 degrees. What can you say about the tangent of 90°?