# Problems

 satellite T days r in 106 km Io 1.77 0.4214 Europa 3.55 0.6705 Ganymede 7.15 1.0695 Callisto 16.67 1.8812
Check Kepler's 3rd law by deriving the ratios of period squared to distance cubed.

1.   (a) Kepler's 3rd law is T2 = K R3, where T is the orbital period of a planet, R its average distance from the Sun and K is some number, the same for all planets. Assume the orbits are all circles around the Sun.

The formula obviously says that if a planet is more distant from the Sun (R larger), it also takes a longer time to complete each orbit (T is larger too). Could it be that all planets move with the same speed V, but more distant ones take longer to complete each orbit because their orbits are longer--or do more distant planets also move more slowly?

Imagine two planetary systems with circular orbits, where at distance R a planet moves with velocity V and takes time T to complete one circuit. The systems obey different laws: in #1 Kepler's laws hold, in #2 all planets move with the same velocity V, no matter what the distance is. If we go to a planet with orbital radius 2R--are the orbital periods in both system also equal, or if not, in which system is the orbital period longer?

(b) (The solution of part a to be used here.)<  Suppose that in some different universe, with different laws, planetary system #2 existed, in which all circular orbits had the same velocity V. How likely would it be that we could find a pair of circular orbits, one in each system, which shared the same distance R, orbital velocity V and orbital period T? Explain.

2.  The mean distance of Neptune from the Sun is 30.07 AU (=astronomical unit, means Earth-Sun distance), that of Pluto 39.4 AU. Are these two numbers connected? (Hint: Derive the ratio of the orbital periods!)

3.  The period of Comet Halley is approximately 75 years. Assume its perigee is at 0.5 AU from the Sun (1 AU or "astronomical unit" is the mean Earth-Sun distance). How many AU is is it from the Sun to its apogee? Does it get further from the Sun than the mean distance of Pluto, about 39 AU?

4.  A satellite in a circular orbit just above the surface of the Earth (r = 1 Re) would need 8 km/sec to stay in orbit. If a missile is sent at that same speed straight up, how high will it get?

Hints: (1) The semimajor axis of an orbit depends only on the launch energy.(2) The trajectory of an object tossed straight up may be viewed as an ellipse of zero width.

5.  The scientific satellite ISEE 1 had its perigee is at 1.2 Re, apogee at 23 Re. About how much slower do you think its motion was at apogee, compared to its perigee pass?

6.  Meteorites tend to fall more frequently in the afternoon, suggesting they overtake the Earth in its orbital motion. What can this tell about their origin?

7. 
• (a) A golf ball is launched at a 45° angle to the horizontal and reaches a distance of 50 meters. If v is its initial velocity, express the time t during which it is in the air. Neglect any air resistance.

• (b) Express the horizontal distance covered in terms of v and t.

• (c) Using the fact that the ball covered 50 meters, derive v and t

• (d) Astronaut Alan Shepard drove a golf ball on the Moon, where the acceleration of gravity is only g/6. If the the ball is launched at 45° as before, with the same velocity, how far would it get?

8.  Baseball players have caught baseballs tossed from the top of the Washington Monument in Washington, DC (window height about 550 ft. 1 ft = 30.5 cm). How does their speed compare with that of a professionally pitched baseball, which may hit 90 mph? (1 mile = 1.6 km approx.) Assume g = 10 and neglect air resistance.

9.  If a force F is resolved into the sum of two forces Fx and Fy perpendicular to each, the values of Fx and Fy are not uniquely determined. Explain why, and show that in all such cases, the sum of squares Fx2 + Fy2 is always the same.

10.  When resolving a vector AB into components AC and CB, we draw a rectangle (or parallelogram) ACBD, of which AB is the diagonal. In vector addition, then, AB = AC + CB. How would you express the other diagonal? (Hint: you can use a minus sign.)

11.  A 3-dimensional vector V has components of magnitudes (Vx Vy Vz) along three mutually perpendicular axes. If V is the magnitude of the vector sum, show that

V2 = Vx2 + Vy2 + Vz2

(Hint: use Pythagoras!).

12. [15} A "Wispa" bar of chocolate milk is eaten by a high school student weighing 44 kg. Assuming the body converts 20% of the energy to muscle power, approximately how high is the mountain the student can climb, given the energy of the chocolate bar? Take g = 10 m/sec2 .

13.  If T1 is the orbital period around Earth at a radial distance 1.1 Re, and T2 the orbital period around the Moon at 1.1 Rm (Rm = the Moon's radius), which is bigger, and by how much? Assume that on the Moon the acceleration of gravity is 1/6 g, and that Rm = 0.273 Re.

14. The Earth moves around the Sun in an orbit that is nearly circular, at 30 km/sec. A spaceship has just enough velocity to escape the Earth's gravity. How much added velocity does it need to leave the solar system?

15.  Suppose a space probe escaped the Earth's gravity, but it still shares the Earth's orbital motion around the Sun, in a near-circle at 30 km/sec. We then fire an on-board rocket to give it an opposite velocity of 30 km/sec, so that its net velocity is zero and it falls down to the Sun.

How would you find the time T needed for reaching the Sun? (Ignore melting on the way!). Calculate T, if you can.

Hint: Can Kepler's 3rd law help?

16. [21a] The satellites of the Global Positioning System (see sect. #29d) are in 12-hour orbits. If the orbits are circular, what is their distance from the center of Earth?

17. [22a] In problem (9) it was pointed out that a rocket launched east from Cape Canaveral needs less thrust than one launched southward, because it already has the velocity given to it by the spin of the Earth, which equals a few 100s of meters/sec. A rocket launched westward similarly needs more thrust, by the same amount
Do airliners flying eastwards and westwards similarly experience a difference due to the Earth's rotation?

18.  A string of length L, with a weight m at its end, hangs from a rotating hook, which causes it to rotate with a period T (like some amusement-park carrousels, whose cars are suspended by long chains). As the string rotates, it describes a cone, and it forms an angle a with the vertical direction. Express a (or its sine or cosine).

Hint: In a rotating frame, the string makes a constant angle a with the vertical, under the action of the centrifugal force and gravity. Each of these forces can be resolved into components along the string and perpendicular to it.

The components along the string just keep the string stretched. However, if the string is in equilibrium in the rotating frame, the perpendicular components must cancel each other, i.e. be equal: if either were stronger, the string would move in its direction and change the angle a.

19.  Jules Verne in his book "From the Earth to the Moon" claimed that for passengers on a spaceship passing from Earth to the Moon, the "down" direction reversed when they passed from the region where the Earth's gravity was stronger than the Moon's to the one where the Moon's began to dominates, with "zero g" at the point where both were equal. What is wrong with this idea?

20.  Before the satellite age, someone suggested we were actually living inside a spinning hollow sphere, and what we thought was gravity was really the centrifugal force. How many arguments could you suggest against that theory?

21.  Rocket engines are cooled by fuel and oxidizer (e.g. liquid oxygen) circulating in pipes along their hot parts, before being burned. What do you think needs the cooling most--the combustion chamber or the wide "bell" through which the gases exit?

22.  The SHARP projectile weighs 10-20 kg. Why does the gun need recoil wagons, and why do you think the one behind the auxiliary barrel is 10 times heavier than the other one?

23.  Suppose you are in a space at the L2 point of the Sun-Earth system. You look in the direction of Earth: what do you see?
You may assume that the width of the Earth is 3.5 times that of the Moon (see problem 15), that the Moon is 60 Earth radii from the center of Earth and that, as seen from Earth, it is equal in size to the Sun.

24.  The Earth-Moon also has Lagrangian points L1 and L2. Its L2 point is on the opposite side of the Moon, about same distance as L1. Is this L2 point a good place to monitor the hidden side of the Moon--e.g. for nuclear test ban violations?

25.  Mars has surface gravity 0.39 g or about 3.9 m/s2, radius r = 3332 km and a rotation period of 24 hr. 37.38 min.

(a)What is the orbital velocity at distance r?
(b)What is the escape velocity from the surface?
(c) For communication, astronauts on Mars may use a synchronous satellite. At what distance R (in Mars radii) would it orbit? (Use a calculator with cube roots or 1/3 powers.)

#### ...and just for fun

Get hold of a map of the Moon and see if you can find craters named after personalities you met here. Some of the larger ones: Tycho (distinguished by bright streaks that radiate from it), Ptolemy ("Ptolemaeus"), Copernicus, Kepler, Aristarchus, Hipparchus, Erathosthenes.

Names were bestowed in the 17th century, and latecomers had to make do with left-overs: the craters Newton and Cavendish are at the southern edge of the visible disk, Goddard and Lagrange too are near the edge. Also, "Galilaei" is a small undistinguished crater (because of Galileo's banishment?), Meton and Pythagoras are on the edge, near the northern pole However, since the Russians were the first to observe the rear side of the Moon, a prominent crater there bears the name of Tsiolkovsky.

. Author and curator: David P. Stern
Last updated 22 February 1999